LMFDB - Embedded newform 3864.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3864,2,Mod(1,3864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3864, 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("3864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3864.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:	$30.8541953410$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 6 $ x^3 - x^2 - 7*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.75153$ of defining polynomial
Character	$\chi$	$=$	3864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.75153 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.75153 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.32246 q^{11} +1.57093 q^{13} -2.75153 q^{15} +0.819397 q^{17} -2.81940 q^{19} -1.00000 q^{21} +1.00000 q^{23} +2.57093 q^{25} -1.00000 q^{27} -0.819397 q^{29} +8.32246 q^{31} -4.32246 q^{33} +2.75153 q^{35} +8.68367 q^{37} -1.57093 q^{39} +10.3225 q^{41} -0.429071 q^{43} +2.75153 q^{45} -10.1480 q^{47} +1.00000 q^{49} -0.819397 q^{51} -7.39645 q^{53} +11.8934 q^{55} +2.81940 q^{57} -6.39033 q^{59} +13.4352 q^{61} +1.00000 q^{63} +4.32246 q^{65} -1.60967 q^{67} -1.00000 q^{69} -2.06786 q^{71} -13.4643 q^{73} -2.57093 q^{75} +4.32246 q^{77} -6.81940 q^{79} +1.00000 q^{81} -4.32246 q^{83} +2.25460 q^{85} +0.819397 q^{87} +7.93214 q^{89} +1.57093 q^{91} -8.32246 q^{93} -7.75766 q^{95} +1.67754 q^{97} +4.32246 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} - 2 q^{11} - 3 q^{13} - q^{15} + 2 q^{17} - 8 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{27} - 2 q^{29} + 10 q^{31} + 2 q^{33} + q^{35} + 12 q^{37} + 3 q^{39} + 16 q^{41} - 9 q^{43} + q^{45} + 14 q^{47} + 3 q^{49} - 2 q^{51} + 15 q^{53} + 13 q^{55} + 8 q^{57} - 11 q^{59} + 19 q^{61} + 3 q^{63} - 2 q^{65} - 13 q^{67} - 3 q^{69} - 13 q^{71} - 10 q^{73} - 2 q^{77} - 20 q^{79} + 3 q^{81} + 2 q^{83} - 15 q^{85} + 2 q^{87} + 17 q^{89} - 3 q^{91} - 10 q^{93} + 13 q^{95} + 20 q^{97} - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9 - 2 * q^11 - 3 * q^13 - q^15 + 2 * q^17 - 8 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^27 - 2 * q^29 + 10 * q^31 + 2 * q^33 + q^35 + 12 * q^37 + 3 * q^39 + 16 * q^41 - 9 * q^43 + q^45 + 14 * q^47 + 3 * q^49 - 2 * q^51 + 15 * q^53 + 13 * q^55 + 8 * q^57 - 11 * q^59 + 19 * q^61 + 3 * q^63 - 2 * q^65 - 13 * q^67 - 3 * q^69 - 13 * q^71 - 10 * q^73 - 2 * q^77 - 20 * q^79 + 3 * q^81 + 2 * q^83 - 15 * q^85 + 2 * q^87 + 17 * q^89 - 3 * q^91 - 10 * q^93 + 13 * q^95 + 20 * 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
$5$	2.75153	1.23052	0.615261	−	0.788323i	$-0.289051\pi$
$5$	2.75153	1.23052	0.615261	+	0.788323i	$0.289051\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.32246	1.30327	0.651635	−	0.758532i	$-0.274083\pi$
$11$	4.32246	1.30327	0.651635	+	0.758532i	$0.274083\pi$
$12$	0	0
$13$	1.57093	0.435697	0.217849	−	0.975983i	$-0.430096\pi$
$13$	1.57093	0.435697	0.217849	+	0.975983i	$0.430096\pi$
$14$	0	0
$15$	−2.75153	−0.710443
$16$	0	0
$17$	0.819397	0.198733	0.0993664	−	0.995051i	$-0.468318\pi$
$17$	0.819397	0.198733	0.0993664	+	0.995051i	$0.468318\pi$
$18$	0	0
$19$	−2.81940	−0.646814	−0.323407	−	0.946260i	$-0.604828\pi$
$19$	−2.81940	−0.646814	−0.323407	+	0.946260i	$0.604828\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.57093	0.514186
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.819397	−0.152158	−0.0760791	−	0.997102i	$-0.524240\pi$
$29$	−0.819397	−0.152158	−0.0760791	+	0.997102i	$0.524240\pi$
$30$	0	0
$31$	8.32246	1.49476	0.747379	−	0.664398i	$-0.231312\pi$
$31$	8.32246	1.49476	0.747379	+	0.664398i	$0.231312\pi$
$32$	0	0
$33$	−4.32246	−0.752444
$34$	0	0
$35$	2.75153	0.465094
$36$	0	0
$37$	8.68367	1.42759	0.713793	−	0.700357i	$-0.246976\pi$
$37$	8.68367	1.42759	0.713793	+	0.700357i	$0.246976\pi$
$38$	0	0
$39$	−1.57093	−0.251550
$40$	0	0
$41$	10.3225	1.61210	0.806049	−	0.591849i	$-0.201602\pi$
$41$	10.3225	1.61210	0.806049	+	0.591849i	$0.201602\pi$
$42$	0	0
$43$	−0.429071	−0.0654328	−0.0327164	−	0.999465i	$-0.510416\pi$
$43$	−0.429071	−0.0654328	−0.0327164	+	0.999465i	$0.510416\pi$
$44$	0	0
$45$	2.75153	0.410174
$46$	0	0
$47$	−10.1480	−1.48024	−0.740118	−	0.672477i	$-0.765230\pi$
$47$	−10.1480	−1.48024	−0.740118	+	0.672477i	$0.765230\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.819397	−0.114738
$52$	0	0
$53$	−7.39645	−1.01598	−0.507991	−	0.861363i	$-0.669612\pi$
$53$	−7.39645	−1.01598	−0.507991	+	0.861363i	$0.669612\pi$
$54$	0	0
$55$	11.8934	1.60370
$56$	0	0
$57$	2.81940	0.373438
$58$	0	0
$59$	−6.39033	−0.831950	−0.415975	−	0.909376i	$-0.636560\pi$
$59$	−6.39033	−0.831950	−0.415975	+	0.909376i	$0.636560\pi$
$60$	0	0
$61$	13.4352	1.72020	0.860101	−	0.510125i	$-0.170401\pi$
$61$	13.4352	1.72020	0.860101	+	0.510125i	$0.170401\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	4.32246	0.536135
$66$	0	0
$67$	−1.60967	−0.196653	−0.0983265	−	0.995154i	$-0.531349\pi$
$67$	−1.60967	−0.196653	−0.0983265	+	0.995154i	$0.531349\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−2.06786	−0.245410	−0.122705	−	0.992443i	$-0.539157\pi$
$71$	−2.06786	−0.245410	−0.122705	+	0.992443i	$0.539157\pi$
$72$	0	0
$73$	−13.4643	−1.57588	−0.787940	−	0.615752i	$-0.788852\pi$
$73$	−13.4643	−1.57588	−0.787940	+	0.615752i	$0.788852\pi$
$74$	0	0
$75$	−2.57093	−0.296865
$76$	0	0
$77$	4.32246	0.492590
$78$	0	0
$79$	−6.81940	−0.767242	−0.383621	−	0.923491i	$-0.625323\pi$
$79$	−6.81940	−0.767242	−0.383621	+	0.923491i	$0.625323\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.32246	−0.474452	−0.237226	−	0.971455i	$-0.576238\pi$
$83$	−4.32246	−0.474452	−0.237226	+	0.971455i	$0.576238\pi$
$84$	0	0
$85$	2.25460	0.244545
$86$	0	0
$87$	0.819397	0.0878485
$88$	0	0
$89$	7.93214	0.840805	0.420402	−	0.907338i	$-0.361889\pi$
$89$	7.93214	0.840805	0.420402	+	0.907338i	$0.361889\pi$
$90$	0	0
$91$	1.57093	0.164678
$92$	0	0
$93$	−8.32246	−0.862999
$94$	0	0
$95$	−7.75766	−0.795919
$96$	0	0
$97$	1.67754	0.170328	0.0851641	−	0.996367i	$-0.472859\pi$
$97$	1.67754	0.170328	0.0851641	+	0.996367i	$0.472859\pi$
$98$	0	0
$99$	4.32246	0.434424
$100$	0	0
$101$	8.39033	0.834869	0.417434	−	0.908707i	$-0.362929\pi$
$101$	8.39033	0.834869	0.417434	+	0.908707i	$0.362929\pi$
$102$	0	0
$103$	−11.0061	−1.08447	−0.542233	−	0.840228i	$-0.682421\pi$
$103$	−11.0061	−1.08447	−0.542233	+	0.840228i	$0.682421\pi$
$104$	0	0
$105$	−2.75153	−0.268522
$106$	0	0
$107$	−4.29334	−0.415053	−0.207527	−	0.978229i	$-0.566541\pi$
$107$	−4.29334	−0.415053	−0.207527	+	0.978229i	$0.566541\pi$
$108$	0	0
$109$	−5.89339	−0.564484	−0.282242	−	0.959343i	$-0.591078\pi$
$109$	−5.89339	−0.564484	−0.282242	+	0.959343i	$0.591078\pi$
$110$	0	0
$111$	−8.68367	−0.824217
$112$	0	0
$113$	3.74540	0.352338	0.176169	−	0.984360i	$-0.443629\pi$
$113$	3.74540	0.352338	0.176169	+	0.984360i	$0.443629\pi$
$114$	0	0
$115$	2.75153	0.256582
$116$	0	0
$117$	1.57093	0.145232
$118$	0	0
$119$	0.819397	0.0751140
$120$	0	0
$121$	7.68367	0.698515
$122$	0	0
$123$	−10.3225	−0.930745
$124$	0	0
$125$	−6.68367	−0.597805
$126$	0	0
$127$	5.07399	0.450244	0.225122	−	0.974331i	$-0.427722\pi$
$127$	5.07399	0.450244	0.225122	+	0.974331i	$0.427722\pi$
$128$	0	0
$129$	0.429071	0.0377776
$130$	0	0
$131$	12.3225	1.07662	0.538309	−	0.842747i	$-0.319063\pi$
$131$	12.3225	1.07662	0.538309	+	0.842747i	$0.319063\pi$
$132$	0	0
$133$	−2.81940	−0.244473
$134$	0	0
$135$	−2.75153	−0.236814
$136$	0	0
$137$	1.67754	0.143322	0.0716609	−	0.997429i	$-0.477170\pi$
$137$	1.67754	0.143322	0.0716609	+	0.997429i	$0.477170\pi$
$138$	0	0
$139$	−11.7577	−0.997272	−0.498636	−	0.866812i	$-0.666166\pi$
$139$	−11.7577	−0.997272	−0.498636	+	0.866812i	$0.666166\pi$
$140$	0	0
$141$	10.1480	0.854615
$142$	0	0
$143$	6.79028	0.567832
$144$	0	0
$145$	−2.25460	−0.187234
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−0.496936	−0.0407106	−0.0203553	−	0.999793i	$-0.506480\pi$
$149$	−0.496936	−0.0407106	−0.0203553	+	0.999793i	$0.506480\pi$
$150$	0	0
$151$	−10.1480	−0.825831	−0.412916	−	0.910769i	$-0.635490\pi$
$151$	−10.1480	−0.825831	−0.412916	+	0.910769i	$0.635490\pi$
$152$	0	0
$153$	0.819397	0.0662443
$154$	0	0
$155$	22.8995	1.83933
$156$	0	0
$157$	19.4256	1.55033	0.775165	−	0.631759i	$-0.217667\pi$
$157$	19.4256	1.55033	0.775165	+	0.631759i	$0.217667\pi$
$158$	0	0
$159$	7.39645	0.586577
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	2.25460	0.176594	0.0882968	−	0.996094i	$-0.471858\pi$
$163$	2.25460	0.176594	0.0882968	+	0.996094i	$0.471858\pi$
$164$	0	0
$165$	−11.8934	−0.925899
$166$	0	0
$167$	−5.31633	−0.411390	−0.205695	−	0.978616i	$-0.565946\pi$
$167$	−5.31633	−0.411390	−0.205695	+	0.978616i	$0.565946\pi$
$168$	0	0
$169$	−10.5322	−0.810168
$170$	0	0
$171$	−2.81940	−0.215605
$172$	0	0
$173$	19.9613	1.51763	0.758813	−	0.651309i	$-0.225780\pi$
$173$	19.9613	1.51763	0.758813	+	0.651309i	$0.225780\pi$
$174$	0	0
$175$	2.57093	0.194344
$176$	0	0
$177$	6.39033	0.480326
$178$	0	0
$179$	4.02912	0.301150	0.150575	−	0.988599i	$-0.451887\pi$
$179$	4.02912	0.301150	0.150575	+	0.988599i	$0.451887\pi$
$180$	0	0
$181$	0.955126	0.0709939	0.0354970	−	0.999370i	$-0.488699\pi$
$181$	0.955126	0.0709939	0.0354970	+	0.999370i	$0.488699\pi$
$182$	0	0
$183$	−13.4352	−0.993159
$184$	0	0
$185$	23.8934	1.75668
$186$	0	0
$187$	3.54181	0.259003
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	8.85814	0.640953	0.320476	−	0.947257i	$-0.396157\pi$
$191$	8.85814	0.640953	0.320476	+	0.947257i	$0.396157\pi$
$192$	0	0
$193$	9.14186	0.658045	0.329023	−	0.944322i	$-0.393281\pi$
$193$	9.14186	0.658045	0.329023	+	0.944322i	$0.393281\pi$
$194$	0	0
$195$	−4.32246	−0.309538
$196$	0	0
$197$	−6.21585	−0.442861	−0.221431	−	0.975176i	$-0.571073\pi$
$197$	−6.21585	−0.442861	−0.221431	+	0.975176i	$0.571073\pi$
$198$	0	0
$199$	21.3965	1.51675	0.758377	−	0.651816i	$-0.225993\pi$
$199$	21.3965	1.51675	0.758377	+	0.651816i	$0.225993\pi$
$200$	0	0
$201$	1.60967	0.113538
$202$	0	0
$203$	−0.819397	−0.0575104
$204$	0	0
$205$	28.4026	1.98372
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−12.1867	−0.842974
$210$	0	0
$211$	5.18060	0.356647	0.178324	−	0.983972i	$-0.442933\pi$
$211$	5.18060	0.356647	0.178324	+	0.983972i	$0.442933\pi$
$212$	0	0
$213$	2.06786	0.141688
$214$	0	0
$215$	−1.18060	−0.0805165
$216$	0	0
$217$	8.32246	0.564965
$218$	0	0
$219$	13.4643	0.909834
$220$	0	0
$221$	1.28721	0.0865874
$222$	0	0
$223$	−16.0291	−1.07339	−0.536695	−	0.843777i	$-0.680327\pi$
$223$	−16.0291	−1.07339	−0.536695	+	0.843777i	$0.680327\pi$
$224$	0	0
$225$	2.57093	0.171395
$226$	0	0
$227$	−13.2607	−0.880145	−0.440073	−	0.897962i	$-0.645047\pi$
$227$	−13.2607	−0.880145	−0.440073	+	0.897962i	$0.645047\pi$
$228$	0	0
$229$	−0.254596	−0.0168242	−0.00841210	−	0.999965i	$-0.502678\pi$
$229$	−0.254596	−0.0168242	−0.00841210	+	0.999965i	$0.502678\pi$
$230$	0	0
$231$	−4.32246	−0.284397
$232$	0	0
$233$	3.74540	0.245370	0.122685	−	0.992446i	$-0.460850\pi$
$233$	3.74540	0.245370	0.122685	+	0.992446i	$0.460850\pi$
$234$	0	0
$235$	−27.9225	−1.82146
$236$	0	0
$237$	6.81940	0.442967
$238$	0	0
$239$	8.61580	0.557310	0.278655	−	0.960391i	$-0.410111\pi$
$239$	8.61580	0.557310	0.278655	+	0.960391i	$0.410111\pi$
$240$	0	0
$241$	−0.683667	−0.0440389	−0.0220194	−	0.999758i	$-0.507010\pi$
$241$	−0.683667	−0.0440389	−0.0220194	+	0.999758i	$0.507010\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.75153	0.175789
$246$	0	0
$247$	−4.42907	−0.281815
$248$	0	0
$249$	4.32246	0.273925
$250$	0	0
$251$	−24.5092	−1.54701	−0.773503	−	0.633792i	$-0.781497\pi$
$251$	−24.5092	−1.54701	−0.773503	+	0.633792i	$0.781497\pi$
$252$	0	0
$253$	4.32246	0.271751
$254$	0	0
$255$	−2.25460	−0.141188
$256$	0	0
$257$	28.7929	1.79605	0.898026	−	0.439942i	$-0.145001\pi$
$257$	28.7929	1.79605	0.898026	+	0.439942i	$0.145001\pi$
$258$	0	0
$259$	8.68367	0.539577
$260$	0	0
$261$	−0.819397	−0.0507194
$262$	0	0
$263$	23.1929	1.43013	0.715067	−	0.699056i	$-0.246396\pi$
$263$	23.1929	1.43013	0.715067	+	0.699056i	$0.246396\pi$
$264$	0	0
$265$	−20.3516	−1.25019
$266$	0	0
$267$	−7.93214	−0.485439
$268$	0	0
$269$	9.57093	0.583550	0.291775	−	0.956487i	$-0.405754\pi$
$269$	9.57093	0.583550	0.291775	+	0.956487i	$0.405754\pi$
$270$	0	0
$271$	−19.1154	−1.16118	−0.580588	−	0.814198i	$-0.697177\pi$
$271$	−19.1154	−1.16118	−0.580588	+	0.814198i	$0.697177\pi$
$272$	0	0
$273$	−1.57093	−0.0950769
$274$	0	0
$275$	11.1127	0.670123
$276$	0	0
$277$	4.84852	0.291319	0.145660	−	0.989335i	$-0.453470\pi$
$277$	4.84852	0.291319	0.145660	+	0.989335i	$0.453470\pi$
$278$	0	0
$279$	8.32246	0.498253
$280$	0	0
$281$	−8.14799	−0.486068	−0.243034	−	0.970018i	$-0.578143\pi$
$281$	−8.14799	−0.486068	−0.243034	+	0.970018i	$0.578143\pi$
$282$	0	0
$283$	20.0291	1.19061	0.595304	−	0.803501i	$-0.297032\pi$
$283$	20.0291	1.19061	0.595304	+	0.803501i	$0.297032\pi$
$284$	0	0
$285$	7.75766	0.459524
$286$	0	0
$287$	10.3225	0.609316
$288$	0	0
$289$	−16.3286	−0.960505
$290$	0	0
$291$	−1.67754	−0.0983391
$292$	0	0
$293$	−5.92251	−0.345997	−0.172998	−	0.984922i	$-0.555346\pi$
$293$	−5.92251	−0.345997	−0.172998	+	0.984922i	$0.555346\pi$
$294$	0	0
$295$	−17.5832	−1.02373
$296$	0	0
$297$	−4.32246	−0.250815
$298$	0	0
$299$	1.57093	0.0908492
$300$	0	0
$301$	−0.429071	−0.0247313
$302$	0	0
$303$	−8.39033	−0.482012
$304$	0	0
$305$	36.9674	2.11675
$306$	0	0
$307$	3.40608	0.194395	0.0971976	−	0.995265i	$-0.469012\pi$
$307$	3.40608	0.194395	0.0971976	+	0.995265i	$0.469012\pi$
$308$	0	0
$309$	11.0061	0.626117
$310$	0	0
$311$	26.4413	1.49935	0.749675	−	0.661806i	$-0.230210\pi$
$311$	26.4413	1.49935	0.749675	+	0.661806i	$0.230210\pi$
$312$	0	0
$313$	7.63879	0.431770	0.215885	−	0.976419i	$-0.430736\pi$
$313$	7.63879	0.431770	0.215885	+	0.976419i	$0.430736\pi$
$314$	0	0
$315$	2.75153	0.155031
$316$	0	0
$317$	−9.75766	−0.548045	−0.274022	−	0.961723i	$-0.588354\pi$
$317$	−9.75766	−0.548045	−0.274022	+	0.961723i	$0.588354\pi$
$318$	0	0
$319$	−3.54181	−0.198303
$320$	0	0
$321$	4.29334	0.239631
$322$	0	0
$323$	−2.31020	−0.128543
$324$	0	0
$325$	4.03875	0.224029
$326$	0	0
$327$	5.89339	0.325905
$328$	0	0
$329$	−10.1480	−0.559477
$330$	0	0
$331$	4.64492	0.255308	0.127654	−	0.991819i	$-0.459255\pi$
$331$	4.64492	0.255308	0.127654	+	0.991819i	$0.459255\pi$
$332$	0	0
$333$	8.68367	0.475862
$334$	0	0
$335$	−4.42907	−0.241986
$336$	0	0
$337$	3.15148	0.171672	0.0858362	−	0.996309i	$-0.472644\pi$
$337$	3.15148	0.171672	0.0858362	+	0.996309i	$0.472644\pi$
$338$	0	0
$339$	−3.74540	−0.203422
$340$	0	0
$341$	35.9735	1.94807
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.75153	−0.148138
$346$	0	0
$347$	21.6123	1.16021	0.580105	−	0.814542i	$-0.303012\pi$
$347$	21.6123	1.16021	0.580105	+	0.814542i	$0.303012\pi$
$348$	0	0
$349$	3.60967	0.193221	0.0966107	−	0.995322i	$-0.469200\pi$
$349$	3.60967	0.193221	0.0966107	+	0.995322i	$0.469200\pi$
$350$	0	0
$351$	−1.57093	−0.0838500
$352$	0	0
$353$	−32.2572	−1.71688	−0.858439	−	0.512915i	$-0.828566\pi$
$353$	−32.2572	−1.71688	−0.858439	+	0.512915i	$0.828566\pi$
$354$	0	0
$355$	−5.68980	−0.301983
$356$	0	0
$357$	−0.819397	−0.0433671
$358$	0	0
$359$	9.47395	0.500016	0.250008	−	0.968244i	$-0.419567\pi$
$359$	9.47395	0.500016	0.250008	+	0.968244i	$0.419567\pi$
$360$	0	0
$361$	−11.0510	−0.581632
$362$	0	0
$363$	−7.68367	−0.403288
$364$	0	0
$365$	−37.0475	−1.93915
$366$	0	0
$367$	13.0740	0.682457	0.341228	−	0.939980i	$-0.389157\pi$
$367$	13.0740	0.682457	0.341228	+	0.939980i	$0.389157\pi$
$368$	0	0
$369$	10.3225	0.537366
$370$	0	0
$371$	−7.39645	−0.384005
$372$	0	0
$373$	−32.3347	−1.67423	−0.837114	−	0.547028i	$-0.815759\pi$
$373$	−32.3347	−1.67423	−0.837114	+	0.547028i	$0.815759\pi$
$374$	0	0
$375$	6.68367	0.345143
$376$	0	0
$377$	−1.28721	−0.0662949
$378$	0	0
$379$	−11.7285	−0.602455	−0.301227	−	0.953552i	$-0.597396\pi$
$379$	−11.7285	−0.602455	−0.301227	+	0.953552i	$0.597396\pi$
$380$	0	0
$381$	−5.07399	−0.259949
$382$	0	0
$383$	19.3286	0.987645	0.493822	−	0.869563i	$-0.335599\pi$
$383$	19.3286	0.987645	0.493822	+	0.869563i	$0.335599\pi$
$384$	0	0
$385$	11.8934	0.606143
$386$	0	0
$387$	−0.429071	−0.0218109
$388$	0	0
$389$	−31.7480	−1.60969	−0.804845	−	0.593486i	$-0.797751\pi$
$389$	−31.7480	−1.60969	−0.804845	+	0.593486i	$0.797751\pi$
$390$	0	0
$391$	0.819397	0.0414387
$392$	0	0
$393$	−12.3225	−0.621586
$394$	0	0
$395$	−18.7638	−0.944109
$396$	0	0
$397$	−25.7868	−1.29420	−0.647101	−	0.762405i	$-0.724019\pi$
$397$	−25.7868	−1.29420	−0.647101	+	0.762405i	$0.724019\pi$
$398$	0	0
$399$	2.81940	0.141146
$400$	0	0
$401$	15.7480	0.786419	0.393210	−	0.919449i	$-0.371365\pi$
$401$	15.7480	0.786419	0.393210	+	0.919449i	$0.371365\pi$
$402$	0	0
$403$	13.0740	0.651262
$404$	0	0
$405$	2.75153	0.136725
$406$	0	0
$407$	37.5348	1.86053
$408$	0	0
$409$	−3.96125	−0.195871	−0.0979357	−	0.995193i	$-0.531224\pi$
$409$	−3.96125	−0.195871	−0.0979357	+	0.995193i	$0.531224\pi$
$410$	0	0
$411$	−1.67754	−0.0827469
$412$	0	0
$413$	−6.39033	−0.314447
$414$	0	0
$415$	−11.8934	−0.583824
$416$	0	0
$417$	11.7577	0.575775
$418$	0	0
$419$	−16.7515	−0.818366	−0.409183	−	0.912452i	$-0.634186\pi$
$419$	−16.7515	−0.818366	−0.409183	+	0.912452i	$0.634186\pi$
$420$	0	0
$421$	−2.77802	−0.135392	−0.0676962	−	0.997706i	$-0.521565\pi$
$421$	−2.77802	−0.135392	−0.0676962	+	0.997706i	$0.521565\pi$
$422$	0	0
$423$	−10.1480	−0.493412
$424$	0	0
$425$	2.10661	0.102186
$426$	0	0
$427$	13.4352	0.650175
$428$	0	0
$429$	−6.79028	−0.327838
$430$	0	0
$431$	9.58319	0.461606	0.230803	−	0.973001i	$-0.425865\pi$
$431$	9.58319	0.461606	0.230803	+	0.973001i	$0.425865\pi$
$432$	0	0
$433$	26.4317	1.27023	0.635113	−	0.772419i	$-0.280953\pi$
$433$	26.4317	1.27023	0.635113	+	0.772419i	$0.280953\pi$
$434$	0	0
$435$	2.25460	0.108100
$436$	0	0
$437$	−2.81940	−0.134870
$438$	0	0
$439$	−21.6898	−1.03520	−0.517599	−	0.855623i	$-0.673174\pi$
$439$	−21.6898	−1.03520	−0.517599	+	0.855623i	$0.673174\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	7.00613	0.332871	0.166436	−	0.986052i	$-0.446774\pi$
$443$	7.00613	0.332871	0.166436	+	0.986052i	$0.446774\pi$
$444$	0	0
$445$	21.8255	1.03463
$446$	0	0
$447$	0.496936	0.0235043
$448$	0	0
$449$	19.5322	0.921781	0.460890	−	0.887457i	$-0.347530\pi$
$449$	19.5322	0.921781	0.460890	+	0.887457i	$0.347530\pi$
$450$	0	0
$451$	44.6184	2.10100
$452$	0	0
$453$	10.1480	0.476794
$454$	0	0
$455$	4.32246	0.202640
$456$	0	0
$457$	−38.4026	−1.79640	−0.898199	−	0.439590i	$-0.855124\pi$
$457$	−38.4026	−1.79640	−0.898199	+	0.439590i	$0.855124\pi$
$458$	0	0
$459$	−0.819397	−0.0382462
$460$	0	0
$461$	30.6740	1.42863	0.714316	−	0.699823i	$-0.246738\pi$
$461$	30.6740	1.42863	0.714316	+	0.699823i	$0.246738\pi$
$462$	0	0
$463$	39.8960	1.85413	0.927063	−	0.374907i	$-0.122325\pi$
$463$	39.8960	1.85413	0.927063	+	0.374907i	$0.122325\pi$
$464$	0	0
$465$	−22.8995	−1.06194
$466$	0	0
$467$	−0.671411	−0.0310692	−0.0155346	−	0.999879i	$-0.504945\pi$
$467$	−0.671411	−0.0310692	−0.0155346	+	0.999879i	$0.504945\pi$
$468$	0	0
$469$	−1.60967	−0.0743279
$470$	0	0
$471$	−19.4256	−0.895083
$472$	0	0
$473$	−1.85464	−0.0852766
$474$	0	0
$475$	−7.24847	−0.332583
$476$	0	0
$477$	−7.39645	−0.338660
$478$	0	0
$479$	16.9091	0.772599	0.386299	−	0.922373i	$-0.373753\pi$
$479$	16.9091	0.772599	0.386299	+	0.922373i	$0.373753\pi$
$480$	0	0
$481$	13.6414	0.621995
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	4.61580	0.209593
$486$	0	0
$487$	−2.03875	−0.0923844	−0.0461922	−	0.998933i	$-0.514709\pi$
$487$	−2.03875	−0.0923844	−0.0461922	+	0.998933i	$0.514709\pi$
$488$	0	0
$489$	−2.25460	−0.101956
$490$	0	0
$491$	−12.7250	−0.574273	−0.287137	−	0.957890i	$-0.592703\pi$
$491$	−12.7250	−0.574273	−0.287137	+	0.957890i	$0.592703\pi$
$492$	0	0
$493$	−0.671411	−0.0302388
$494$	0	0
$495$	11.8934	0.534568
$496$	0	0
$497$	−2.06786	−0.0927564
$498$	0	0
$499$	21.9056	0.980631	0.490316	−	0.871545i	$-0.336881\pi$
$499$	21.9056	0.980631	0.490316	+	0.871545i	$0.336881\pi$
$500$	0	0
$501$	5.31633	0.237516
$502$	0	0
$503$	−18.9770	−0.846143	−0.423072	−	0.906096i	$-0.639048\pi$
$503$	−18.9770	−0.846143	−0.423072	+	0.906096i	$0.639048\pi$
$504$	0	0
$505$	23.0862	1.02732
$506$	0	0
$507$	10.5322	0.467751
$508$	0	0
$509$	40.5214	1.79608	0.898041	−	0.439912i	$-0.144990\pi$
$509$	40.5214	1.79608	0.898041	+	0.439912i	$0.144990\pi$
$510$	0	0
$511$	−13.4643	−0.595626
$512$	0	0
$513$	2.81940	0.124479
$514$	0	0
$515$	−30.2837	−1.33446
$516$	0	0
$517$	−43.8643	−1.92915
$518$	0	0
$519$	−19.9613	−0.876202
$520$	0	0
$521$	3.71629	0.162813	0.0814067	−	0.996681i	$-0.474059\pi$
$521$	3.71629	0.162813	0.0814067	+	0.996681i	$0.474059\pi$
$522$	0	0
$523$	11.3551	0.496523	0.248261	−	0.968693i	$-0.420141\pi$
$523$	11.3551	0.496523	0.248261	+	0.968693i	$0.420141\pi$
$524$	0	0
$525$	−2.57093	−0.112205
$526$	0	0
$527$	6.81940	0.297058
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.39033	−0.277317
$532$	0	0
$533$	16.2159	0.702386
$534$	0	0
$535$	−11.8133	−0.510732
$536$	0	0
$537$	−4.02912	−0.173869
$538$	0	0
$539$	4.32246	0.186182
$540$	0	0
$541$	−30.5092	−1.31169	−0.655846	−	0.754894i	$-0.727688\pi$
$541$	−30.5092	−1.31169	−0.655846	+	0.754894i	$0.727688\pi$
$542$	0	0
$543$	−0.955126	−0.0409884
$544$	0	0
$545$	−16.2159	−0.694611
$546$	0	0
$547$	−31.6802	−1.35455	−0.677273	−	0.735732i	$-0.736838\pi$
$547$	−31.6802	−1.35455	−0.677273	+	0.735732i	$0.736838\pi$
$548$	0	0
$549$	13.4352	0.573400
$550$	0	0
$551$	2.31020	0.0984180
$552$	0	0
$553$	−6.81940	−0.289990
$554$	0	0
$555$	−23.8934	−1.01422
$556$	0	0
$557$	16.4194	0.695714	0.347857	−	0.937548i	$-0.386909\pi$
$557$	16.4194	0.695714	0.347857	+	0.937548i	$0.386909\pi$
$558$	0	0
$559$	−0.674040	−0.0285089
$560$	0	0
$561$	−3.54181	−0.149535
$562$	0	0
$563$	12.5648	0.529543	0.264772	−	0.964311i	$-0.414703\pi$
$563$	12.5648	0.529543	0.264772	+	0.964311i	$0.414703\pi$
$564$	0	0
$565$	10.3056	0.433560
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	32.9286	1.38044	0.690220	−	0.723599i	$-0.257514\pi$
$569$	32.9286	1.38044	0.690220	+	0.723599i	$0.257514\pi$
$570$	0	0
$571$	−44.7542	−1.87290	−0.936452	−	0.350797i	$-0.885911\pi$
$571$	−44.7542	−1.87290	−0.936452	+	0.350797i	$0.885911\pi$
$572$	0	0
$573$	−8.85814	−0.370054
$574$	0	0
$575$	2.57093	0.107215
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	−9.14186	−0.379923
$580$	0	0
$581$	−4.32246	−0.179326
$582$	0	0
$583$	−31.9709	−1.32410
$584$	0	0
$585$	4.32246	0.178712
$586$	0	0
$587$	−41.2342	−1.70192	−0.850960	−	0.525231i	$-0.823979\pi$
$587$	−41.2342	−1.70192	−0.850960	+	0.525231i	$0.823979\pi$
$588$	0	0
$589$	−23.4643	−0.966830
$590$	0	0
$591$	6.21585	0.255686
$592$	0	0
$593$	39.4256	1.61901	0.809507	−	0.587110i	$-0.199734\pi$
$593$	39.4256	1.61901	0.809507	+	0.587110i	$0.199734\pi$
$594$	0	0
$595$	2.25460	0.0924294
$596$	0	0
$597$	−21.3965	−0.875699
$598$	0	0
$599$	0.293342	0.0119856	0.00599282	−	0.999982i	$-0.498092\pi$
$599$	0.293342	0.0119856	0.00599282	+	0.999982i	$0.498092\pi$
$600$	0	0
$601$	11.4739	0.468032	0.234016	−	0.972233i	$-0.424813\pi$
$601$	11.4739	0.468032	0.234016	+	0.972233i	$0.424813\pi$
$602$	0	0
$603$	−1.60967	−0.0655510
$604$	0	0
$605$	21.1419	0.859539
$606$	0	0
$607$	8.72504	0.354139	0.177069	−	0.984198i	$-0.443338\pi$
$607$	8.72504	0.354139	0.177069	+	0.984198i	$0.443338\pi$
$608$	0	0
$609$	0.819397	0.0332036
$610$	0	0
$611$	−15.9418	−0.644935
$612$	0	0
$613$	−37.1154	−1.49908	−0.749538	−	0.661962i	$-0.769724\pi$
$613$	−37.1154	−1.49908	−0.749538	+	0.661962i	$0.769724\pi$
$614$	0	0
$615$	−28.4026	−1.14530
$616$	0	0
$617$	−22.3516	−0.899841	−0.449920	−	0.893069i	$-0.648548\pi$
$617$	−22.3516	−0.899841	−0.449920	+	0.893069i	$0.648548\pi$
$618$	0	0
$619$	−33.2342	−1.33580	−0.667898	−	0.744253i	$-0.732806\pi$
$619$	−33.2342	−1.33580	−0.667898	+	0.744253i	$0.732806\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	7.93214	0.317794
$624$	0	0
$625$	−31.2450	−1.24980
$626$	0	0
$627$	12.1867	0.486691
$628$	0	0
$629$	7.11537	0.283708
$630$	0	0
$631$	−3.60005	−0.143316	−0.0716578	−	0.997429i	$-0.522829\pi$
$631$	−3.60005	−0.143316	−0.0716578	+	0.997429i	$0.522829\pi$
$632$	0	0
$633$	−5.18060	−0.205910
$634$	0	0
$635$	13.9613	0.554035
$636$	0	0
$637$	1.57093	0.0622425
$638$	0	0
$639$	−2.06786	−0.0818035
$640$	0	0
$641$	−22.7250	−0.897585	−0.448793	−	0.893636i	$-0.648146\pi$
$641$	−22.7250	−0.897585	−0.448793	+	0.893636i	$0.648146\pi$
$642$	0	0
$643$	−42.9505	−1.69380	−0.846902	−	0.531750i	$-0.821535\pi$
$643$	−42.9505	−1.69380	−0.846902	+	0.531750i	$0.821535\pi$
$644$	0	0
$645$	1.18060	0.0464862
$646$	0	0
$647$	17.2607	0.678589	0.339295	−	0.940680i	$-0.389812\pi$
$647$	17.2607	0.678589	0.339295	+	0.940680i	$0.389812\pi$
$648$	0	0
$649$	−27.6219	−1.08426
$650$	0	0
$651$	−8.32246	−0.326183
$652$	0	0
$653$	24.7638	0.969082	0.484541	−	0.874769i	$-0.338987\pi$
$653$	24.7638	0.969082	0.484541	+	0.874769i	$0.338987\pi$
$654$	0	0
$655$	33.9056	1.32480
$656$	0	0
$657$	−13.4643	−0.525293
$658$	0	0
$659$	−13.2388	−0.515712	−0.257856	−	0.966183i	$-0.583016\pi$
$659$	−13.2388	−0.515712	−0.257856	+	0.966183i	$0.583016\pi$
$660$	0	0
$661$	24.6837	0.960083	0.480042	−	0.877246i	$-0.340622\pi$
$661$	24.6837	0.960083	0.480042	+	0.877246i	$0.340622\pi$
$662$	0	0
$663$	−1.28721	−0.0499912
$664$	0	0
$665$	−7.75766	−0.300829
$666$	0	0
$667$	−0.819397	−0.0317272
$668$	0	0
$669$	16.0291	0.619722
$670$	0	0
$671$	58.0731	2.24189
$672$	0	0
$673$	−18.4000	−0.709266	−0.354633	−	0.935006i	$-0.615394\pi$
$673$	−18.4000	−0.709266	−0.354633	+	0.935006i	$0.615394\pi$
$674$	0	0
$675$	−2.57093	−0.0989551
$676$	0	0
$677$	−48.1506	−1.85058	−0.925289	−	0.379262i	$-0.876178\pi$
$677$	−48.1506	−1.85058	−0.925289	+	0.379262i	$0.876178\pi$
$678$	0	0
$679$	1.67754	0.0643780
$680$	0	0
$681$	13.2607	0.508152
$682$	0	0
$683$	−36.0510	−1.37945	−0.689727	−	0.724070i	$-0.742269\pi$
$683$	−36.0510	−1.37945	−0.689727	+	0.724070i	$0.742269\pi$
$684$	0	0
$685$	4.61580	0.176361
$686$	0	0
$687$	0.254596	0.00971345
$688$	0	0
$689$	−11.6193	−0.442660
$690$	0	0
$691$	−19.0088	−0.723127	−0.361564	−	0.932347i	$-0.617757\pi$
$691$	−19.0088	−0.723127	−0.361564	+	0.932347i	$0.617757\pi$
$692$	0	0
$693$	4.32246	0.164197
$694$	0	0
$695$	−32.3516	−1.22717
$696$	0	0
$697$	8.45819	0.320377
$698$	0	0
$699$	−3.74540	−0.141664
$700$	0	0
$701$	−7.60967	−0.287413	−0.143707	−	0.989620i	$-0.545902\pi$
$701$	−7.60967	−0.287413	−0.143707	+	0.989620i	$0.545902\pi$
$702$	0	0
$703$	−24.4827	−0.923383
$704$	0	0
$705$	27.9225	1.05162
$706$	0	0
$707$	8.39033	0.315551
$708$	0	0
$709$	−34.2933	−1.28791	−0.643957	−	0.765062i	$-0.722708\pi$
$709$	−34.2933	−1.28791	−0.643957	+	0.765062i	$0.722708\pi$
$710$	0	0
$711$	−6.81940	−0.255747
$712$	0	0
$713$	8.32246	0.311679
$714$	0	0
$715$	18.6837	0.698730
$716$	0	0
$717$	−8.61580	−0.321763
$718$	0	0
$719$	0.807140	0.0301012	0.0150506	−	0.999887i	$-0.495209\pi$
$719$	0.807140	0.0301012	0.0150506	+	0.999887i	$0.495209\pi$
$720$	0	0
$721$	−11.0061	−0.409890
$722$	0	0
$723$	0.683667	0.0254259
$724$	0	0
$725$	−2.10661	−0.0782375
$726$	0	0
$727$	5.37457	0.199332	0.0996659	−	0.995021i	$-0.468223\pi$
$727$	5.37457	0.199332	0.0996659	+	0.995021i	$0.468223\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.351580	−0.0130036
$732$	0	0
$733$	−37.7093	−1.39282	−0.696412	−	0.717642i	$-0.745221\pi$
$733$	−37.7093	−1.39282	−0.696412	+	0.717642i	$0.745221\pi$
$734$	0	0
$735$	−2.75153	−0.101492
$736$	0	0
$737$	−6.95776	−0.256292
$738$	0	0
$739$	5.18060	0.190572	0.0952858	−	0.995450i	$-0.469624\pi$
$739$	5.18060	0.190572	0.0952858	+	0.995450i	$0.469624\pi$
$740$	0	0
$741$	4.42907	0.162706
$742$	0	0
$743$	−32.8343	−1.20457	−0.602287	−	0.798280i	$-0.705744\pi$
$743$	−32.8343	−1.20457	−0.602287	+	0.798280i	$0.705744\pi$
$744$	0	0
$745$	−1.36733	−0.0500953
$746$	0	0
$747$	−4.32246	−0.158151
$748$	0	0
$749$	−4.29334	−0.156875
$750$	0	0
$751$	21.8282	0.796521	0.398260	−	0.917272i	$-0.369614\pi$
$751$	21.8282	0.796521	0.398260	+	0.917272i	$0.369614\pi$
$752$	0	0
$753$	24.5092	0.893165
$754$	0	0
$755$	−27.9225	−1.01620
$756$	0	0
$757$	−39.9613	−1.45242	−0.726208	−	0.687475i	$-0.758719\pi$
$757$	−39.9613	−1.45242	−0.726208	+	0.687475i	$0.758719\pi$
$758$	0	0
$759$	−4.32246	−0.156895
$760$	0	0
$761$	18.3735	0.666038	0.333019	−	0.942920i	$-0.391933\pi$
$761$	18.3735	0.666038	0.333019	+	0.942920i	$0.391933\pi$
$762$	0	0
$763$	−5.89339	−0.213355
$764$	0	0
$765$	2.25460	0.0815151
$766$	0	0
$767$	−10.0387	−0.362478
$768$	0	0
$769$	−52.7154	−1.90097	−0.950483	−	0.310776i	$-0.899411\pi$
$769$	−52.7154	−1.90097	−0.950483	+	0.310776i	$0.899411\pi$
$770$	0	0
$771$	−28.7929	−1.03695
$772$	0	0
$773$	−31.9083	−1.14766	−0.573830	−	0.818974i	$-0.694543\pi$
$773$	−31.9083	−1.14766	−0.573830	+	0.818974i	$0.694543\pi$
$774$	0	0
$775$	21.3965	0.768583
$776$	0	0
$777$	−8.68367	−0.311525
$778$	0	0
$779$	−29.1031	−1.04273
$780$	0	0
$781$	−8.93826	−0.319836
$782$	0	0
$783$	0.819397	0.0292828
$784$	0	0
$785$	53.4501	1.90772
$786$	0	0
$787$	6.11887	0.218114	0.109057	−	0.994035i	$-0.465217\pi$
$787$	6.11887	0.218114	0.109057	+	0.994035i	$0.465217\pi$
$788$	0	0
$789$	−23.1929	−0.825688
$790$	0	0
$791$	3.74540	0.133171
$792$	0	0
$793$	21.1057	0.749487
$794$	0	0
$795$	20.3516	0.721796
$796$	0	0
$797$	−41.7868	−1.48016	−0.740082	−	0.672517i	$-0.765213\pi$
$797$	−41.7868	−1.48016	−0.740082	+	0.672517i	$0.765213\pi$
$798$	0	0
$799$	−8.31523	−0.294172
$800$	0	0
$801$	7.93214	0.280268
$802$	0	0
$803$	−58.1990	−2.05380
$804$	0	0
$805$	2.75153	0.0969788
$806$	0	0
$807$	−9.57093	−0.336913
$808$	0	0
$809$	23.1322	0.813286	0.406643	−	0.913587i	$-0.366699\pi$
$809$	23.1322	0.813286	0.406643	+	0.913587i	$0.366699\pi$
$810$	0	0
$811$	29.8255	1.04732	0.523658	−	0.851929i	$-0.324567\pi$
$811$	29.8255	1.04732	0.523658	+	0.851929i	$0.324567\pi$
$812$	0	0
$813$	19.1154	0.670405
$814$	0	0
$815$	6.20359	0.217302
$816$	0	0
$817$	1.20972	0.0423228
$818$	0	0
$819$	1.57093	0.0548927
$820$	0	0
$821$	−28.7929	−1.00488	−0.502440	−	0.864612i	$-0.667564\pi$
$821$	−28.7929	−1.00488	−0.502440	+	0.864612i	$0.667564\pi$
$822$	0	0
$823$	8.67404	0.302358	0.151179	−	0.988506i	$-0.451693\pi$
$823$	8.67404	0.302358	0.151179	+	0.988506i	$0.451693\pi$
$824$	0	0
$825$	−11.1127	−0.386896
$826$	0	0
$827$	−2.89952	−0.100826	−0.0504131	−	0.998728i	$-0.516054\pi$
$827$	−2.89952	−0.100826	−0.0504131	+	0.998728i	$0.516054\pi$
$828$	0	0
$829$	41.7021	1.44837	0.724186	−	0.689605i	$-0.242216\pi$
$829$	41.7021	1.44837	0.724186	+	0.689605i	$0.242216\pi$
$830$	0	0
$831$	−4.84852	−0.168193
$832$	0	0
$833$	0.819397	0.0283904
$834$	0	0
$835$	−14.6281	−0.506225
$836$	0	0
$837$	−8.32246	−0.287666
$838$	0	0
$839$	−19.3577	−0.668302	−0.334151	−	0.942520i	$-0.608450\pi$
$839$	−19.3577	−0.668302	−0.334151	+	0.942520i	$0.608450\pi$
$840$	0	0
$841$	−28.3286	−0.976848
$842$	0	0
$843$	8.14799	0.280632
$844$	0	0
$845$	−28.9796	−0.996930
$846$	0	0
$847$	7.68367	0.264014
$848$	0	0
$849$	−20.0291	−0.687398
$850$	0	0
$851$	8.68367	0.297672
$852$	0	0
$853$	−1.56830	−0.0536975	−0.0268488	−	0.999640i	$-0.508547\pi$
$853$	−1.56830	−0.0536975	−0.0268488	+	0.999640i	$0.508547\pi$
$854$	0	0
$855$	−7.75766	−0.265306
$856$	0	0
$857$	42.1602	1.44017	0.720083	−	0.693888i	$-0.244104\pi$
$857$	42.1602	1.44017	0.720083	+	0.693888i	$0.244104\pi$
$858$	0	0
$859$	8.94089	0.305059	0.152530	−	0.988299i	$-0.451258\pi$
$859$	8.94089	0.305059	0.152530	+	0.988299i	$0.451258\pi$
$860$	0	0
$861$	−10.3225	−0.351789
$862$	0	0
$863$	−47.5153	−1.61744	−0.808720	−	0.588194i	$-0.799839\pi$
$863$	−47.5153	−1.61744	−0.808720	+	0.588194i	$0.799839\pi$
$864$	0	0
$865$	54.9240	1.86747
$866$	0	0
$867$	16.3286	0.554548
$868$	0	0
$869$	−29.4766	−0.999924
$870$	0	0
$871$	−2.52868	−0.0856812
$872$	0	0
$873$	1.67754	0.0567761
$874$	0	0
$875$	−6.68367	−0.225949
$876$	0	0
$877$	−40.2307	−1.35850	−0.679248	−	0.733909i	$-0.737694\pi$
$877$	−40.2307	−1.35850	−0.679248	+	0.733909i	$0.737694\pi$
$878$	0	0
$879$	5.92251	0.199761
$880$	0	0
$881$	−6.70316	−0.225835	−0.112918	−	0.993604i	$-0.536020\pi$
$881$	−6.70316	−0.225835	−0.112918	+	0.993604i	$0.536020\pi$
$882$	0	0
$883$	−27.2802	−0.918052	−0.459026	−	0.888423i	$-0.651802\pi$
$883$	−27.2802	−0.918052	−0.459026	+	0.888423i	$0.651802\pi$
$884$	0	0
$885$	17.5832	0.591052
$886$	0	0
$887$	−16.2933	−0.547077	−0.273538	−	0.961861i	$-0.588194\pi$
$887$	−16.2933	−0.547077	−0.273538	+	0.961861i	$0.588194\pi$
$888$	0	0
$889$	5.07399	0.170176
$890$	0	0
$891$	4.32246	0.144808
$892$	0	0
$893$	28.6112	0.957437
$894$	0	0
$895$	11.0862	0.370572
$896$	0	0
$897$	−1.57093	−0.0524518
$898$	0	0
$899$	−6.81940	−0.227440
$900$	0	0
$901$	−6.06063	−0.201909
$902$	0	0
$903$	0.429071	0.0142786
$904$	0	0
$905$	2.62806	0.0873597
$906$	0	0
$907$	8.80977	0.292524	0.146262	−	0.989246i	$-0.453276\pi$
$907$	8.80977	0.292524	0.146262	+	0.989246i	$0.453276\pi$
$908$	0	0
$909$	8.39033	0.278290
$910$	0	0
$911$	−30.4440	−1.00865	−0.504327	−	0.863513i	$-0.668259\pi$
$911$	−30.4440	−1.00865	−0.504327	+	0.863513i	$0.668259\pi$
$912$	0	0
$913$	−18.6837	−0.618339
$914$	0	0
$915$	−36.9674	−1.22210
$916$	0	0
$917$	12.3225	0.406924
$918$	0	0
$919$	−28.6184	−0.944035	−0.472017	−	0.881589i	$-0.656474\pi$
$919$	−28.6184	−0.944035	−0.472017	+	0.881589i	$0.656474\pi$
$920$	0	0
$921$	−3.40608	−0.112234
$922$	0	0
$923$	−3.24847	−0.106925
$924$	0	0
$925$	22.3251	0.734044
$926$	0	0
$927$	−11.0061	−0.361489
$928$	0	0
$929$	39.7117	1.30290	0.651449	−	0.758692i	$-0.274161\pi$
$929$	39.7117	1.30290	0.651449	+	0.758692i	$0.274161\pi$
$930$	0	0
$931$	−2.81940	−0.0924020
$932$	0	0
$933$	−26.4413	−0.865650
$934$	0	0
$935$	9.74540	0.318709
$936$	0	0
$937$	38.1867	1.24751	0.623753	−	0.781621i	$-0.285607\pi$
$937$	38.1867	1.24751	0.623753	+	0.781621i	$0.285607\pi$
$938$	0	0
$939$	−7.63879	−0.249283
$940$	0	0
$941$	33.2246	1.08309	0.541546	−	0.840671i	$-0.317839\pi$
$941$	33.2246	1.08309	0.541546	+	0.840671i	$0.317839\pi$
$942$	0	0
$943$	10.3225	0.336146
$944$	0	0
$945$	−2.75153	−0.0895073
$946$	0	0
$947$	−8.19397	−0.266268	−0.133134	−	0.991098i	$-0.542504\pi$
$947$	−8.19397	−0.266268	−0.133134	+	0.991098i	$0.542504\pi$
$948$	0	0
$949$	−21.1515	−0.686606
$950$	0	0
$951$	9.75766	0.316414
$952$	0	0
$953$	12.7710	0.413694	0.206847	−	0.978373i	$-0.433680\pi$
$953$	12.7710	0.413694	0.206847	+	0.978373i	$0.433680\pi$
$954$	0	0
$955$	24.3735	0.788707
$956$	0	0
$957$	3.54181	0.114490
$958$	0	0
$959$	1.67754	0.0541706
$960$	0	0
$961$	38.2634	1.23430
$962$	0	0
$963$	−4.29334	−0.138351
$964$	0	0
$965$	25.1541	0.809740
$966$	0	0
$967$	−37.3673	−1.20165	−0.600826	−	0.799380i	$-0.705162\pi$
$967$	−37.3673	−1.20165	−0.600826	+	0.799380i	$0.705162\pi$
$968$	0	0
$969$	2.31020	0.0742145
$970$	0	0
$971$	35.0088	1.12348	0.561742	−	0.827312i	$-0.310131\pi$
$971$	35.0088	1.12348	0.561742	+	0.827312i	$0.310131\pi$
$972$	0	0
$973$	−11.7577	−0.376933
$974$	0	0
$975$	−4.03875	−0.129343
$976$	0	0
$977$	21.2995	0.681430	0.340715	−	0.940167i	$-0.389331\pi$
$977$	21.2995	0.681430	0.340715	+	0.940167i	$0.389331\pi$
$978$	0	0
$979$	34.2863	1.09580
$980$	0	0
$981$	−5.89339	−0.188161
$982$	0	0
$983$	−18.4122	−0.587258	−0.293629	−	0.955919i	$-0.594863\pi$
$983$	−18.4122	−0.587258	−0.293629	+	0.955919i	$0.594863\pi$
$984$	0	0
$985$	−17.1031	−0.544950
$986$	0	0
$987$	10.1480	0.323014
$988$	0	0
$989$	−0.429071	−0.0136437
$990$	0	0
$991$	−57.3700	−1.82242	−0.911208	−	0.411945i	$-0.864849\pi$
$991$	−57.3700	−1.82242	−0.911208	+	0.411945i	$0.864849\pi$
$992$	0	0
$993$	−4.64492	−0.147402
$994$	0	0
$995$	58.8730	1.86640
$996$	0	0
$997$	−18.5939	−0.588875	−0.294438	−	0.955671i	$-0.595132\pi$
$997$	−18.5939	−0.588875	−0.294438	+	0.955671i	$0.595132\pi$
$998$	0	0
$999$	−8.68367	−0.274739

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3864.2.a.n.1.3	✓	3
4.3	odd	2		7728.2.a.by.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.n.1.3	✓	3	1.1	even	1	trivial
7728.2.a.by.1.3		3	4.3	odd	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:	\( N \)	\(=\)	\( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3864.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.75153 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.75153 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.32246 q^{11} +1.57093 q^{13} -2.75153 q^{15} +0.819397 q^{17} -2.81940 q^{19} -1.00000 q^{21} +1.00000 q^{23} +2.57093 q^{25} -1.00000 q^{27} -0.819397 q^{29} +8.32246 q^{31} -4.32246 q^{33} +2.75153 q^{35} +8.68367 q^{37} -1.57093 q^{39} +10.3225 q^{41} -0.429071 q^{43} +2.75153 q^{45} -10.1480 q^{47} +1.00000 q^{49} -0.819397 q^{51} -7.39645 q^{53} +11.8934 q^{55} +2.81940 q^{57} -6.39033 q^{59} +13.4352 q^{61} +1.00000 q^{63} +4.32246 q^{65} -1.60967 q^{67} -1.00000 q^{69} -2.06786 q^{71} -13.4643 q^{73} -2.57093 q^{75} +4.32246 q^{77} -6.81940 q^{79} +1.00000 q^{81} -4.32246 q^{83} +2.25460 q^{85} +0.819397 q^{87} +7.93214 q^{89} +1.57093 q^{91} -8.32246 q^{93} -7.75766 q^{95} +1.67754 q^{97} +4.32246 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} - 2 q^{11} - 3 q^{13} - q^{15} + 2 q^{17} - 8 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{27} - 2 q^{29} + 10 q^{31} + 2 q^{33} + q^{35} + 12 q^{37} + 3 q^{39} + 16 q^{41} - 9 q^{43} + q^{45} + 14 q^{47} + 3 q^{49} - 2 q^{51} + 15 q^{53} + 13 q^{55} + 8 q^{57} - 11 q^{59} + 19 q^{61} + 3 q^{63} - 2 q^{65} - 13 q^{67} - 3 q^{69} - 13 q^{71} - 10 q^{73} - 2 q^{77} - 20 q^{79} + 3 q^{81} + 2 q^{83} - 15 q^{85} + 2 q^{87} + 17 q^{89} - 3 q^{91} - 10 q^{93} + 13 q^{95} + 20 q^{97} - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9 - 2 * q^11 - 3 * q^13 - q^15 + 2 * q^17 - 8 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^27 - 2 * q^29 + 10 * q^31 + 2 * q^33 + q^35 + 12 * q^37 + 3 * q^39 + 16 * q^41 - 9 * q^43 + q^45 + 14 * q^47 + 3 * q^49 - 2 * q^51 + 15 * q^53 + 13 * q^55 + 8 * q^57 - 11 * q^59 + 19 * q^61 + 3 * q^63 - 2 * q^65 - 13 * q^67 - 3 * q^69 - 13 * q^71 - 10 * q^73 - 2 * q^77 - 20 * q^79 + 3 * q^81 + 2 * q^83 - 15 * q^85 + 2 * q^87 + 17 * q^89 - 3 * q^91 - 10 * q^93 + 13 * q^95 + 20 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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