LMFDB - Embedded newform 4368.2.a.bs.1.4

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:	$4$
Coefficient field:	4.4.138892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 2x + 12 $ x^4 - x^3 - 10x^2 + 2x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.13296$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +4.16291 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.16291 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.71640 q^{11} +1.00000 q^{13} +4.16291 q^{15} -0.450485 q^{17} +0.103011 q^{19} +1.00000 q^{21} -4.42883 q^{23} +12.3298 q^{25} +1.00000 q^{27} +1.89699 q^{29} -2.55350 q^{31} +4.71640 q^{33} +4.16291 q^{35} -0.450485 q^{37} +1.00000 q^{39} +4.26592 q^{41} -4.42883 q^{43} +4.16291 q^{45} -8.61339 q^{47} +1.00000 q^{49} -0.450485 q^{51} +12.6733 q^{53} +19.6340 q^{55} +0.103011 q^{57} -7.16689 q^{59} -14.2091 q^{61} +1.00000 q^{63} +4.16291 q^{65} -12.8577 q^{67} -4.42883 q^{69} +7.16689 q^{71} +1.57117 q^{73} +12.3298 q^{75} +4.71640 q^{77} +2.34747 q^{79} +1.00000 q^{81} +8.93921 q^{83} -1.87533 q^{85} +1.89699 q^{87} +6.61339 q^{89} +1.00000 q^{91} -2.55350 q^{93} +0.428825 q^{95} -4.67329 q^{97} +4.71640 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 3 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 14 q^{25} + 4 q^{27} + 4 q^{29} - 9 q^{31} + 3 q^{33} + 2 q^{35} + 3 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 2 q^{45} - 15 q^{47} + 4 q^{49} + 3 q^{51} + 13 q^{53} - 7 q^{55} + 4 q^{57} - 8 q^{59} + 9 q^{61} + 4 q^{63} + 2 q^{65} + 8 q^{69} + 8 q^{71} + 32 q^{73} + 14 q^{75} + 3 q^{77} + q^{79} + 4 q^{81} - 13 q^{83} + 17 q^{85} + 4 q^{87} + 7 q^{89} + 4 q^{91} - 9 q^{93} - 24 q^{95} + 19 q^{97} + 3 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^15 + 3 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 + 14 * q^25 + 4 * q^27 + 4 * q^29 - 9 * q^31 + 3 * q^33 + 2 * q^35 + 3 * q^37 + 4 * q^39 + 6 * q^41 + 8 * q^43 + 2 * q^45 - 15 * q^47 + 4 * q^49 + 3 * q^51 + 13 * q^53 - 7 * q^55 + 4 * q^57 - 8 * q^59 + 9 * q^61 + 4 * q^63 + 2 * q^65 + 8 * q^69 + 8 * q^71 + 32 * q^73 + 14 * q^75 + 3 * q^77 + q^79 + 4 * q^81 - 13 * q^83 + 17 * q^85 + 4 * q^87 + 7 * q^89 + 4 * q^91 - 9 * q^93 - 24 * q^95 + 19 * q^97 + 3 * 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
$5$	4.16291	1.86171	0.930854	−	0.365390i	$-0.119065\pi$
$5$	4.16291	1.86171	0.930854	+	0.365390i	$0.119065\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.71640	1.42205	0.711025	−	0.703167i	$-0.248231\pi$
$11$	4.71640	1.42205	0.711025	+	0.703167i	$0.248231\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	4.16291	1.07486
$16$	0	0
$17$	−0.450485	−0.109259	−0.0546294	−	0.998507i	$-0.517398\pi$
$17$	−0.450485	−0.109259	−0.0546294	+	0.998507i	$0.517398\pi$
$18$	0	0
$19$	0.103011	0.0236324	0.0118162	−	0.999930i	$-0.496239\pi$
$19$	0.103011	0.0236324	0.0118162	+	0.999930i	$0.496239\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−4.42883	−0.923474	−0.461737	−	0.887017i	$-0.652774\pi$
$23$	−4.42883	−0.923474	−0.461737	+	0.887017i	$0.652774\pi$
$24$	0	0
$25$	12.3298	2.46596
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.89699	0.352262	0.176131	−	0.984367i	$-0.443642\pi$
$29$	1.89699	0.352262	0.176131	+	0.984367i	$0.443642\pi$
$30$	0	0
$31$	−2.55350	−0.458622	−0.229311	−	0.973353i	$-0.573647\pi$
$31$	−2.55350	−0.458622	−0.229311	+	0.973353i	$0.573647\pi$
$32$	0	0
$33$	4.71640	0.821020
$34$	0	0
$35$	4.16291	0.703660
$36$	0	0
$37$	−0.450485	−0.0740593	−0.0370297	−	0.999314i	$-0.511790\pi$
$37$	−0.450485	−0.0740593	−0.0370297	+	0.999314i	$0.511790\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	4.26592	0.666224	0.333112	−	0.942887i	$-0.391901\pi$
$41$	4.26592	0.666224	0.333112	+	0.942887i	$0.391901\pi$
$42$	0	0
$43$	−4.42883	−0.675390	−0.337695	−	0.941256i	$-0.609647\pi$
$43$	−4.42883	−0.675390	−0.337695	+	0.941256i	$0.609647\pi$
$44$	0	0
$45$	4.16291	0.620570
$46$	0	0
$47$	−8.61339	−1.25639	−0.628196	−	0.778055i	$-0.716206\pi$
$47$	−8.61339	−1.25639	−0.628196	+	0.778055i	$0.716206\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.450485	−0.0630806
$52$	0	0
$53$	12.6733	1.74081	0.870405	−	0.492336i	$-0.163857\pi$
$53$	12.6733	1.74081	0.870405	+	0.492336i	$0.163857\pi$
$54$	0	0
$55$	19.6340	2.64744
$56$	0	0
$57$	0.103011	0.0136441
$58$	0	0
$59$	−7.16689	−0.933049	−0.466525	−	0.884508i	$-0.654494\pi$
$59$	−7.16689	−0.933049	−0.466525	+	0.884508i	$0.654494\pi$
$60$	0	0
$61$	−14.2091	−1.81929	−0.909645	−	0.415387i	$-0.863646\pi$
$61$	−14.2091	−1.81929	−0.909645	+	0.415387i	$0.863646\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	4.16291	0.516345
$66$	0	0
$67$	−12.8577	−1.57081	−0.785406	−	0.618981i	$-0.787546\pi$
$67$	−12.8577	−1.57081	−0.785406	+	0.618981i	$0.787546\pi$
$68$	0	0
$69$	−4.42883	−0.533168
$70$	0	0
$71$	7.16689	0.850553	0.425277	−	0.905063i	$-0.360177\pi$
$71$	7.16689	0.850553	0.425277	+	0.905063i	$0.360177\pi$
$72$	0	0
$73$	1.57117	0.183892	0.0919460	−	0.995764i	$-0.470691\pi$
$73$	1.57117	0.183892	0.0919460	+	0.995764i	$0.470691\pi$
$74$	0	0
$75$	12.3298	1.42372
$76$	0	0
$77$	4.71640	0.537484
$78$	0	0
$79$	2.34747	0.264112	0.132056	−	0.991242i	$-0.457842\pi$
$79$	2.34747	0.264112	0.132056	+	0.991242i	$0.457842\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.93921	0.981205	0.490603	−	0.871383i	$-0.336777\pi$
$83$	8.93921	0.981205	0.490603	+	0.871383i	$0.336777\pi$
$84$	0	0
$85$	−1.87533	−0.203408
$86$	0	0
$87$	1.89699	0.203379
$88$	0	0
$89$	6.61339	0.701018	0.350509	−	0.936559i	$-0.386009\pi$
$89$	6.61339	0.701018	0.350509	+	0.936559i	$0.386009\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−2.55350	−0.264785
$94$	0	0
$95$	0.428825	0.0439966
$96$	0	0
$97$	−4.67329	−0.474501	−0.237250	−	0.971449i	$-0.576246\pi$
$97$	−4.67329	−0.474501	−0.237250	+	0.971449i	$0.576246\pi$
$98$	0	0
$99$	4.71640	0.474016
$100$	0	0
$101$	−16.0844	−1.60046	−0.800231	−	0.599692i	$-0.795290\pi$
$101$	−16.0844	−1.60046	−0.800231	+	0.599692i	$0.795290\pi$
$102$	0	0
$103$	−5.87533	−0.578913	−0.289457	−	0.957191i	$-0.593475\pi$
$103$	−5.87533	−0.578913	−0.289457	+	0.957191i	$0.593475\pi$
$104$	0	0
$105$	4.16291	0.406258
$106$	0	0
$107$	3.09903	0.299594	0.149797	−	0.988717i	$-0.452138\pi$
$107$	3.09903	0.299594	0.149797	+	0.988717i	$0.452138\pi$
$108$	0	0
$109$	3.22370	0.308774	0.154387	−	0.988010i	$-0.450660\pi$
$109$	3.22370	0.308774	0.154387	+	0.988010i	$0.450660\pi$
$110$	0	0
$111$	−0.450485	−0.0427582
$112$	0	0
$113$	−17.2051	−1.61852	−0.809261	−	0.587449i	$-0.800132\pi$
$113$	−17.2051	−1.61852	−0.809261	+	0.587449i	$0.800132\pi$
$114$	0	0
$115$	−18.4368	−1.71924
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−0.450485	−0.0412959
$120$	0	0
$121$	11.2445	1.02222
$122$	0	0
$123$	4.26592	0.384645
$124$	0	0
$125$	30.5133	2.72919
$126$	0	0
$127$	−0.369134	−0.0327554	−0.0163777	−	0.999866i	$-0.505213\pi$
$127$	−0.369134	−0.0327554	−0.0163777	+	0.999866i	$0.505213\pi$
$128$	0	0
$129$	−4.42883	−0.389936
$130$	0	0
$131$	−9.87533	−0.862811	−0.431406	−	0.902158i	$-0.641982\pi$
$131$	−9.87533	−0.862811	−0.431406	+	0.902158i	$0.641982\pi$
$132$	0	0
$133$	0.103011	0.00893219
$134$	0	0
$135$	4.16291	0.358286
$136$	0	0
$137$	6.34727	0.542284	0.271142	−	0.962539i	$-0.412599\pi$
$137$	6.34727	0.542284	0.271142	+	0.962539i	$0.412599\pi$
$138$	0	0
$139$	−21.8784	−1.85570	−0.927851	−	0.372950i	$-0.878346\pi$
$139$	−21.8784	−1.85570	−0.927851	+	0.372950i	$0.878346\pi$
$140$	0	0
$141$	−8.61339	−0.725379
$142$	0	0
$143$	4.71640	0.394405
$144$	0	0
$145$	7.89699	0.655809
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	10.5997	0.868361	0.434180	−	0.900826i	$-0.357038\pi$
$149$	10.5997	0.868361	0.434180	+	0.900826i	$0.357038\pi$
$150$	0	0
$151$	−21.8753	−1.78019	−0.890095	−	0.455776i	$-0.849362\pi$
$151$	−21.8753	−1.78019	−0.890095	+	0.455776i	$0.849362\pi$
$152$	0	0
$153$	−0.450485	−0.0364196
$154$	0	0
$155$	−10.6300	−0.853820
$156$	0	0
$157$	7.54951	0.602517	0.301258	−	0.953543i	$-0.402593\pi$
$157$	7.54951	0.602517	0.301258	+	0.953543i	$0.402593\pi$
$158$	0	0
$159$	12.6733	1.00506
$160$	0	0
$161$	−4.42883	−0.349040
$162$	0	0
$163$	13.4328	1.05214	0.526069	−	0.850442i	$-0.323665\pi$
$163$	13.4328	1.05214	0.526069	+	0.850442i	$0.323665\pi$
$164$	0	0
$165$	19.6340	1.52850
$166$	0	0
$167$	11.3897	0.881361	0.440680	−	0.897664i	$-0.354737\pi$
$167$	11.3897	0.881361	0.440680	+	0.897664i	$0.354737\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.103011	0.00787745
$172$	0	0
$173$	1.42484	0.108329	0.0541644	−	0.998532i	$-0.482750\pi$
$173$	1.42484	0.108329	0.0541644	+	0.998532i	$0.482750\pi$
$174$	0	0
$175$	12.3298	0.932045
$176$	0	0
$177$	−7.16689	−0.538696
$178$	0	0
$179$	−19.2051	−1.43546	−0.717729	−	0.696322i	$-0.754818\pi$
$179$	−19.2051	−1.43546	−0.717729	+	0.696322i	$0.754818\pi$
$180$	0	0
$181$	−10.6516	−0.791729	−0.395865	−	0.918309i	$-0.629555\pi$
$181$	−10.6516	−0.791729	−0.395865	+	0.918309i	$0.629555\pi$
$182$	0	0
$183$	−14.2091	−1.05037
$184$	0	0
$185$	−1.87533	−0.137877
$186$	0	0
$187$	−2.12467	−0.155371
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	4.97436	0.359932	0.179966	−	0.983673i	$-0.442401\pi$
$191$	4.97436	0.359932	0.179966	+	0.983673i	$0.442401\pi$
$192$	0	0
$193$	14.9010	1.07260	0.536298	−	0.844029i	$-0.319822\pi$
$193$	14.9010	1.07260	0.536298	+	0.844029i	$0.319822\pi$
$194$	0	0
$195$	4.16291	0.298112
$196$	0	0
$197$	−20.3857	−1.45242	−0.726211	−	0.687472i	$-0.758720\pi$
$197$	−20.3857	−1.45242	−0.726211	+	0.687472i	$0.758720\pi$
$198$	0	0
$199$	3.55748	0.252183	0.126091	−	0.992019i	$-0.459757\pi$
$199$	3.55748	0.252183	0.126091	+	0.992019i	$0.459757\pi$
$200$	0	0
$201$	−12.8577	−0.906909
$202$	0	0
$203$	1.89699	0.133143
$204$	0	0
$205$	17.7586	1.24032
$206$	0	0
$207$	−4.42883	−0.307825
$208$	0	0
$209$	0.485842	0.0336064
$210$	0	0
$211$	5.00398	0.344488	0.172244	−	0.985054i	$-0.444898\pi$
$211$	5.00398	0.344488	0.172244	+	0.985054i	$0.444898\pi$
$212$	0	0
$213$	7.16689	0.491067
$214$	0	0
$215$	−18.4368	−1.25738
$216$	0	0
$217$	−2.55350	−0.173343
$218$	0	0
$219$	1.57117	0.106170
$220$	0	0
$221$	−0.450485	−0.0303029
$222$	0	0
$223$	−2.34747	−0.157199	−0.0785993	−	0.996906i	$-0.525045\pi$
$223$	−2.34747	−0.157199	−0.0785993	+	0.996906i	$0.525045\pi$
$224$	0	0
$225$	12.3298	0.821986
$226$	0	0
$227$	16.5997	1.10176	0.550880	−	0.834584i	$-0.314292\pi$
$227$	16.5997	1.10176	0.550880	+	0.834584i	$0.314292\pi$
$228$	0	0
$229$	14.6516	0.968207	0.484103	−	0.875011i	$-0.339146\pi$
$229$	14.6516	0.968207	0.484103	+	0.875011i	$0.339146\pi$
$230$	0	0
$231$	4.71640	0.310317
$232$	0	0
$233$	−16.3475	−1.07096	−0.535479	−	0.844548i	$-0.679869\pi$
$233$	−16.3475	−1.07096	−0.535479	+	0.844548i	$0.679869\pi$
$234$	0	0
$235$	−35.8568	−2.33904
$236$	0	0
$237$	2.34747	0.152485
$238$	0	0
$239$	15.8185	1.02321	0.511607	−	0.859219i	$-0.329050\pi$
$239$	15.8185	1.02321	0.511607	+	0.859219i	$0.329050\pi$
$240$	0	0
$241$	28.6300	1.84422	0.922109	−	0.386930i	$-0.126464\pi$
$241$	28.6300	1.84422	0.922109	+	0.386930i	$0.126464\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	4.16291	0.265958
$246$	0	0
$247$	0.103011	0.00655444
$248$	0	0
$249$	8.93921	0.566499
$250$	0	0
$251$	−23.6340	−1.49176	−0.745881	−	0.666079i	$-0.767971\pi$
$251$	−23.6340	−1.49176	−0.745881	+	0.666079i	$0.767971\pi$
$252$	0	0
$253$	−20.8881	−1.31323
$254$	0	0
$255$	−1.87533	−0.117438
$256$	0	0
$257$	12.9854	0.810007	0.405004	−	0.914315i	$-0.367270\pi$
$257$	12.9854	0.810007	0.405004	+	0.914315i	$0.367270\pi$
$258$	0	0
$259$	−0.450485	−0.0279918
$260$	0	0
$261$	1.89699	0.117421
$262$	0	0
$263$	10.6733	0.658143	0.329072	−	0.944305i	$-0.393264\pi$
$263$	10.6733	0.658143	0.329072	+	0.944305i	$0.393264\pi$
$264$	0	0
$265$	52.7577	3.24088
$266$	0	0
$267$	6.61339	0.404733
$268$	0	0
$269$	−23.8784	−1.45589	−0.727946	−	0.685634i	$-0.759525\pi$
$269$	−23.8784	−1.45589	−0.727946	+	0.685634i	$0.759525\pi$
$270$	0	0
$271$	−5.31302	−0.322743	−0.161371	−	0.986894i	$-0.551592\pi$
$271$	−5.31302	−0.322743	−0.161371	+	0.986894i	$0.551592\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	58.1523	3.50672
$276$	0	0
$277$	−1.08533	−0.0652113	−0.0326057	−	0.999468i	$-0.510381\pi$
$277$	−1.08533	−0.0652113	−0.0326057	+	0.999468i	$0.510381\pi$
$278$	0	0
$279$	−2.55350	−0.152874
$280$	0	0
$281$	9.28668	0.553997	0.276998	−	0.960870i	$-0.410660\pi$
$281$	9.28668	0.553997	0.276998	+	0.960870i	$0.410660\pi$
$282$	0	0
$283$	13.8019	0.820440	0.410220	−	0.911987i	$-0.365452\pi$
$283$	13.8019	0.820440	0.410220	+	0.911987i	$0.365452\pi$
$284$	0	0
$285$	0.428825	0.0254014
$286$	0	0
$287$	4.26592	0.251809
$288$	0	0
$289$	−16.7971	−0.988063
$290$	0	0
$291$	−4.67329	−0.273953
$292$	0	0
$293$	−32.3287	−1.88866	−0.944331	−	0.328996i	$-0.893290\pi$
$293$	−32.3287	−1.88866	−0.944331	+	0.328996i	$0.893290\pi$
$294$	0	0
$295$	−29.8351	−1.73707
$296$	0	0
$297$	4.71640	0.273673
$298$	0	0
$299$	−4.42883	−0.256126
$300$	0	0
$301$	−4.42883	−0.255273
$302$	0	0
$303$	−16.0844	−0.924027
$304$	0	0
$305$	−59.1512	−3.38699
$306$	0	0
$307$	21.5389	1.22929	0.614645	−	0.788804i	$-0.289299\pi$
$307$	21.5389	1.22929	0.614645	+	0.788804i	$0.289299\pi$
$308$	0	0
$309$	−5.87533	−0.334236
$310$	0	0
$311$	−3.91377	−0.221930	−0.110965	−	0.993824i	$-0.535394\pi$
$311$	−3.91377	−0.221930	−0.110965	+	0.993824i	$0.535394\pi$
$312$	0	0
$313$	23.9646	1.35456	0.677281	−	0.735725i	$-0.263158\pi$
$313$	23.9646	1.35456	0.677281	+	0.735725i	$0.263158\pi$
$314$	0	0
$315$	4.16291	0.234553
$316$	0	0
$317$	−26.5564	−1.49155	−0.745777	−	0.666195i	$-0.767922\pi$
$317$	−26.5564	−1.49155	−0.745777	+	0.666195i	$0.767922\pi$
$318$	0	0
$319$	8.94697	0.500934
$320$	0	0
$321$	3.09903	0.172971
$322$	0	0
$323$	−0.0464050	−0.00258204
$324$	0	0
$325$	12.3298	0.683934
$326$	0	0
$327$	3.22370	0.178271
$328$	0	0
$329$	−8.61339	−0.474872
$330$	0	0
$331$	−6.65959	−0.366044	−0.183022	−	0.983109i	$-0.558588\pi$
$331$	−6.65959	−0.366044	−0.183022	+	0.983109i	$0.558588\pi$
$332$	0	0
$333$	−0.450485	−0.0246864
$334$	0	0
$335$	−53.5252	−2.92439
$336$	0	0
$337$	−5.36515	−0.292258	−0.146129	−	0.989266i	$-0.546682\pi$
$337$	−5.36515	−0.292258	−0.146129	+	0.989266i	$0.546682\pi$
$338$	0	0
$339$	−17.2051	−0.934454
$340$	0	0
$341$	−12.0433	−0.652182
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−18.4368	−0.992603
$346$	0	0
$347$	20.5318	1.10221	0.551103	−	0.834437i	$-0.314207\pi$
$347$	20.5318	1.10221	0.551103	+	0.834437i	$0.314207\pi$
$348$	0	0
$349$	−4.55350	−0.243743	−0.121872	−	0.992546i	$-0.538890\pi$
$349$	−4.55350	−0.243743	−0.121872	+	0.992546i	$0.538890\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	1.16689	0.0621072	0.0310536	−	0.999518i	$-0.490114\pi$
$353$	1.16689	0.0621072	0.0310536	+	0.999518i	$0.490114\pi$
$354$	0	0
$355$	29.8351	1.58348
$356$	0	0
$357$	−0.450485	−0.0238422
$358$	0	0
$359$	15.8185	0.834869	0.417435	−	0.908707i	$-0.362929\pi$
$359$	15.8185	0.834869	0.417435	+	0.908707i	$0.362929\pi$
$360$	0	0
$361$	−18.9894	−0.999442
$362$	0	0
$363$	11.2445	0.590181
$364$	0	0
$365$	6.54065	0.342353
$366$	0	0
$367$	24.0765	1.25678	0.628391	−	0.777898i	$-0.283714\pi$
$367$	24.0765	1.25678	0.628391	+	0.777898i	$0.283714\pi$
$368$	0	0
$369$	4.26592	0.222075
$370$	0	0
$371$	12.6733	0.657964
$372$	0	0
$373$	22.4182	1.16077	0.580386	−	0.814342i	$-0.302902\pi$
$373$	22.4182	1.16077	0.580386	+	0.814342i	$0.302902\pi$
$374$	0	0
$375$	30.5133	1.57570
$376$	0	0
$377$	1.89699	0.0976999
$378$	0	0
$379$	36.0765	1.85312	0.926562	−	0.376142i	$-0.122750\pi$
$379$	36.0765	1.85312	0.926562	+	0.376142i	$0.122750\pi$
$380$	0	0
$381$	−0.369134	−0.0189113
$382$	0	0
$383$	−21.5741	−1.10238	−0.551191	−	0.834379i	$-0.685827\pi$
$383$	−21.5741	−1.10238	−0.551191	+	0.834379i	$0.685827\pi$
$384$	0	0
$385$	19.6340	1.00064
$386$	0	0
$387$	−4.42883	−0.225130
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	1.99512	0.100898
$392$	0	0
$393$	−9.87533	−0.498144
$394$	0	0
$395$	9.77232	0.491699
$396$	0	0
$397$	5.24845	0.263412	0.131706	−	0.991289i	$-0.457955\pi$
$397$	5.24845	0.263412	0.131706	+	0.991289i	$0.457955\pi$
$398$	0	0
$399$	0.103011	0.00515700
$400$	0	0
$401$	−34.3937	−1.71754	−0.858769	−	0.512363i	$-0.828770\pi$
$401$	−34.3937	−1.71754	−0.858769	+	0.512363i	$0.828770\pi$
$402$	0	0
$403$	−2.55350	−0.127199
$404$	0	0
$405$	4.16291	0.206857
$406$	0	0
$407$	−2.12467	−0.105316
$408$	0	0
$409$	−11.7418	−0.580597	−0.290298	−	0.956936i	$-0.593755\pi$
$409$	−11.7418	−0.580597	−0.290298	+	0.956936i	$0.593755\pi$
$410$	0	0
$411$	6.34727	0.313088
$412$	0	0
$413$	−7.16689	−0.352660
$414$	0	0
$415$	37.2131	1.82672
$416$	0	0
$417$	−21.8784	−1.07139
$418$	0	0
$419$	14.5703	0.711805	0.355902	−	0.934523i	$-0.384174\pi$
$419$	14.5703	0.711805	0.355902	+	0.934523i	$0.384174\pi$
$420$	0	0
$421$	−19.8784	−0.968815	−0.484407	−	0.874843i	$-0.660965\pi$
$421$	−19.8784	−0.968815	−0.484407	+	0.874843i	$0.660965\pi$
$422$	0	0
$423$	−8.61339	−0.418797
$424$	0	0
$425$	−5.55439	−0.269428
$426$	0	0
$427$	−14.2091	−0.687627
$428$	0	0
$429$	4.71640	0.227710
$430$	0	0
$431$	25.2787	1.21763	0.608816	−	0.793311i	$-0.291645\pi$
$431$	25.2787	1.21763	0.608816	+	0.793311i	$0.291645\pi$
$432$	0	0
$433$	29.7233	1.42841	0.714204	−	0.699937i	$-0.246789\pi$
$433$	29.7233	1.42841	0.714204	+	0.699937i	$0.246789\pi$
$434$	0	0
$435$	7.89699	0.378632
$436$	0	0
$437$	−0.456218	−0.0218239
$438$	0	0
$439$	−12.7843	−0.610160	−0.305080	−	0.952327i	$-0.598683\pi$
$439$	−12.7843	−0.610160	−0.305080	+	0.952327i	$0.598683\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	12.6653	0.601748	0.300874	−	0.953664i	$-0.402722\pi$
$443$	12.6653	0.601748	0.300874	+	0.953664i	$0.402722\pi$
$444$	0	0
$445$	27.5309	1.30509
$446$	0	0
$447$	10.5997	0.501348
$448$	0	0
$449$	−6.18457	−0.291868	−0.145934	−	0.989294i	$-0.546619\pi$
$449$	−6.18457	−0.291868	−0.145934	+	0.989294i	$0.546619\pi$
$450$	0	0
$451$	20.1198	0.947404
$452$	0	0
$453$	−21.8753	−1.02779
$454$	0	0
$455$	4.16291	0.195160
$456$	0	0
$457$	22.5654	1.05556	0.527782	−	0.849380i	$-0.323024\pi$
$457$	22.5654	1.05556	0.527782	+	0.849380i	$0.323024\pi$
$458$	0	0
$459$	−0.450485	−0.0210269
$460$	0	0
$461$	37.9078	1.76554	0.882772	−	0.469802i	$-0.155675\pi$
$461$	37.9078	1.76554	0.882772	+	0.469802i	$0.155675\pi$
$462$	0	0
$463$	−32.1658	−1.49487	−0.747435	−	0.664334i	$-0.768715\pi$
$463$	−32.1658	−1.49487	−0.747435	+	0.664334i	$0.768715\pi$
$464$	0	0
$465$	−10.6300	−0.492953
$466$	0	0
$467$	20.9470	0.969310	0.484655	−	0.874705i	$-0.338945\pi$
$467$	20.9470	0.969310	0.484655	+	0.874705i	$0.338945\pi$
$468$	0	0
$469$	−12.8577	−0.593711
$470$	0	0
$471$	7.54951	0.347863
$472$	0	0
$473$	−20.8881	−0.960437
$474$	0	0
$475$	1.27011	0.0582764
$476$	0	0
$477$	12.6733	0.580270
$478$	0	0
$479$	−23.7584	−1.08555	−0.542775	−	0.839878i	$-0.682626\pi$
$479$	−23.7584	−1.08555	−0.542775	+	0.839878i	$0.682626\pi$
$480$	0	0
$481$	−0.450485	−0.0205404
$482$	0	0
$483$	−4.42883	−0.201519
$484$	0	0
$485$	−19.4545	−0.883382
$486$	0	0
$487$	−18.2905	−0.828820	−0.414410	−	0.910090i	$-0.636012\pi$
$487$	−18.2905	−0.828820	−0.414410	+	0.910090i	$0.636012\pi$
$488$	0	0
$489$	13.4328	0.607453
$490$	0	0
$491$	4.90097	0.221178	0.110589	−	0.993866i	$-0.464726\pi$
$491$	4.90097	0.221178	0.110589	+	0.993866i	$0.464726\pi$
$492$	0	0
$493$	−0.854566	−0.0384877
$494$	0	0
$495$	19.6340	0.882480
$496$	0	0
$497$	7.16689	0.321479
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	11.3897	0.508854
$502$	0	0
$503$	15.1914	0.677352	0.338676	−	0.940903i	$-0.390021\pi$
$503$	15.1914	0.677352	0.338676	+	0.940903i	$0.390021\pi$
$504$	0	0
$505$	−66.9580	−2.97959
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	20.5811	0.912242	0.456121	−	0.889918i	$-0.349238\pi$
$509$	20.5811	0.912242	0.456121	+	0.889918i	$0.349238\pi$
$510$	0	0
$511$	1.57117	0.0695047
$512$	0	0
$513$	0.103011	0.00454805
$514$	0	0
$515$	−24.4584	−1.07777
$516$	0	0
$517$	−40.6242	−1.78665
$518$	0	0
$519$	1.42484	0.0625437
$520$	0	0
$521$	17.3944	0.762061	0.381031	−	0.924562i	$-0.375569\pi$
$521$	17.3944	0.762061	0.381031	+	0.924562i	$0.375569\pi$
$522$	0	0
$523$	36.0765	1.57751	0.788757	−	0.614705i	$-0.210725\pi$
$523$	36.0765	1.57751	0.788757	+	0.614705i	$0.210725\pi$
$524$	0	0
$525$	12.3298	0.538116
$526$	0	0
$527$	1.15031	0.0501084
$528$	0	0
$529$	−3.38551	−0.147196
$530$	0	0
$531$	−7.16689	−0.311016
$532$	0	0
$533$	4.26592	0.184777
$534$	0	0
$535$	12.9010	0.557758
$536$	0	0
$537$	−19.2051	−0.828762
$538$	0	0
$539$	4.71640	0.203150
$540$	0	0
$541$	16.5703	0.712412	0.356206	−	0.934407i	$-0.384070\pi$
$541$	16.5703	0.712412	0.356206	+	0.934407i	$0.384070\pi$
$542$	0	0
$543$	−10.6516	−0.457105
$544$	0	0
$545$	13.4200	0.574848
$546$	0	0
$547$	29.9509	1.28061	0.640305	−	0.768121i	$-0.278808\pi$
$547$	29.9509	1.28061	0.640305	+	0.768121i	$0.278808\pi$
$548$	0	0
$549$	−14.2091	−0.606430
$550$	0	0
$551$	0.195411	0.00832478
$552$	0	0
$553$	2.34747	0.0998248
$554$	0	0
$555$	−1.87533	−0.0796033
$556$	0	0
$557$	31.9463	1.35361	0.676804	−	0.736164i	$-0.263365\pi$
$557$	31.9463	1.35361	0.676804	+	0.736164i	$0.263365\pi$
$558$	0	0
$559$	−4.42883	−0.187319
$560$	0	0
$561$	−2.12467	−0.0897037
$562$	0	0
$563$	−32.4487	−1.36755	−0.683775	−	0.729693i	$-0.739663\pi$
$563$	−32.4487	−1.36755	−0.683775	+	0.729693i	$0.739663\pi$
$564$	0	0
$565$	−71.6233	−3.01322
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	6.76748	0.283708	0.141854	−	0.989888i	$-0.454694\pi$
$569$	6.76748	0.283708	0.141854	+	0.989888i	$0.454694\pi$
$570$	0	0
$571$	−16.3554	−0.684454	−0.342227	−	0.939617i	$-0.611181\pi$
$571$	−16.3554	−0.684454	−0.342227	+	0.939617i	$0.611181\pi$
$572$	0	0
$573$	4.97436	0.207807
$574$	0	0
$575$	−54.6065	−2.27725
$576$	0	0
$577$	38.1198	1.58695	0.793474	−	0.608604i	$-0.208270\pi$
$577$	38.1198	1.58695	0.793474	+	0.608604i	$0.208270\pi$
$578$	0	0
$579$	14.9010	0.619263
$580$	0	0
$581$	8.93921	0.370861
$582$	0	0
$583$	59.7723	2.47552
$584$	0	0
$585$	4.16291	0.172115
$586$	0	0
$587$	43.0666	1.77755	0.888773	−	0.458347i	$-0.151558\pi$
$587$	43.0666	1.77755	0.888773	+	0.458347i	$0.151558\pi$
$588$	0	0
$589$	−0.263038	−0.0108383
$590$	0	0
$591$	−20.3857	−0.838556
$592$	0	0
$593$	−47.7969	−1.96278	−0.981391	−	0.192021i	$-0.938496\pi$
$593$	−47.7969	−1.96278	−0.981391	+	0.192021i	$0.938496\pi$
$594$	0	0
$595$	−1.87533	−0.0768810
$596$	0	0
$597$	3.55748	0.145598
$598$	0	0
$599$	27.9894	1.14362	0.571808	−	0.820388i	$-0.306242\pi$
$599$	27.9894	1.14362	0.571808	+	0.820388i	$0.306242\pi$
$600$	0	0
$601$	39.5526	1.61338	0.806692	−	0.590972i	$-0.201256\pi$
$601$	39.5526	1.61338	0.806692	+	0.590972i	$0.201256\pi$
$602$	0	0
$603$	−12.8577	−0.523604
$604$	0	0
$605$	46.8097	1.90308
$606$	0	0
$607$	26.3369	1.06898	0.534490	−	0.845175i	$-0.320504\pi$
$607$	26.3369	1.06898	0.534490	+	0.845175i	$0.320504\pi$
$608$	0	0
$609$	1.89699	0.0768699
$610$	0	0
$611$	−8.61339	−0.348461
$612$	0	0
$613$	−12.9394	−0.522618	−0.261309	−	0.965255i	$-0.584154\pi$
$613$	−12.9394	−0.522618	−0.261309	+	0.965255i	$0.584154\pi$
$614$	0	0
$615$	17.7586	0.716097
$616$	0	0
$617$	35.4604	1.42758	0.713792	−	0.700358i	$-0.246976\pi$
$617$	35.4604	1.42758	0.713792	+	0.700358i	$0.246976\pi$
$618$	0	0
$619$	27.9598	1.12380	0.561899	−	0.827206i	$-0.310071\pi$
$619$	27.9598	1.12380	0.561899	+	0.827206i	$0.310071\pi$
$620$	0	0
$621$	−4.42883	−0.177723
$622$	0	0
$623$	6.61339	0.264960
$624$	0	0
$625$	65.3749	2.61500
$626$	0	0
$627$	0.485842	0.0194026
$628$	0	0
$629$	0.202937	0.00809163
$630$	0	0
$631$	−43.6340	−1.73704	−0.868520	−	0.495654i	$-0.834928\pi$
$631$	−43.6340	−1.73704	−0.868520	+	0.495654i	$0.834928\pi$
$632$	0	0
$633$	5.00398	0.198890
$634$	0	0
$635$	−1.53667	−0.0609810
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	7.16689	0.283518
$640$	0	0
$641$	−27.5230	−1.08709	−0.543546	−	0.839379i	$-0.682919\pi$
$641$	−27.5230	−1.08709	−0.543546	+	0.839379i	$0.682919\pi$
$642$	0	0
$643$	−21.1101	−0.832500	−0.416250	−	0.909250i	$-0.636656\pi$
$643$	−21.1101	−0.832500	−0.416250	+	0.909250i	$0.636656\pi$
$644$	0	0
$645$	−18.4368	−0.725948
$646$	0	0
$647$	−35.5446	−1.39740	−0.698702	−	0.715413i	$-0.746239\pi$
$647$	−35.5446	−1.39740	−0.698702	+	0.715413i	$0.746239\pi$
$648$	0	0
$649$	−33.8019	−1.32684
$650$	0	0
$651$	−2.55350	−0.100079
$652$	0	0
$653$	−27.1530	−1.06258	−0.531289	−	0.847190i	$-0.678292\pi$
$653$	−27.1530	−1.06258	−0.531289	+	0.847190i	$0.678292\pi$
$654$	0	0
$655$	−41.1101	−1.60630
$656$	0	0
$657$	1.57117	0.0612974
$658$	0	0
$659$	−28.4748	−1.10922	−0.554611	−	0.832110i	$-0.687133\pi$
$659$	−28.4748	−1.10922	−0.554611	+	0.832110i	$0.687133\pi$
$660$	0	0
$661$	−31.0071	−1.20604	−0.603018	−	0.797728i	$-0.706035\pi$
$661$	−31.0071	−1.20604	−0.603018	+	0.797728i	$0.706035\pi$
$662$	0	0
$663$	−0.450485	−0.0174954
$664$	0	0
$665$	0.428825	0.0166291
$666$	0	0
$667$	−8.40143	−0.325305
$668$	0	0
$669$	−2.34747	−0.0907586
$670$	0	0
$671$	−67.0159	−2.58712
$672$	0	0
$673$	−12.6269	−0.486731	−0.243365	−	0.969935i	$-0.578251\pi$
$673$	−12.6269	−0.486731	−0.243365	+	0.969935i	$0.578251\pi$
$674$	0	0
$675$	12.3298	0.474574
$676$	0	0
$677$	−16.4536	−0.632362	−0.316181	−	0.948699i	$-0.602401\pi$
$677$	−16.4536	−0.632362	−0.316181	+	0.948699i	$0.602401\pi$
$678$	0	0
$679$	−4.67329	−0.179344
$680$	0	0
$681$	16.5997	0.636102
$682$	0	0
$683$	29.1620	1.11585	0.557927	−	0.829890i	$-0.311597\pi$
$683$	29.1620	1.11585	0.557927	+	0.829890i	$0.311597\pi$
$684$	0	0
$685$	26.4231	1.00957
$686$	0	0
$687$	14.6516	0.558995
$688$	0	0
$689$	12.6733	0.482814
$690$	0	0
$691$	−41.9076	−1.59424	−0.797121	−	0.603820i	$-0.793645\pi$
$691$	−41.9076	−1.59424	−0.797121	+	0.603820i	$0.793645\pi$
$692$	0	0
$693$	4.71640	0.179161
$694$	0	0
$695$	−91.0778	−3.45478
$696$	0	0
$697$	−1.92173	−0.0727909
$698$	0	0
$699$	−16.3475	−0.618318
$700$	0	0
$701$	−28.6026	−1.08030	−0.540152	−	0.841567i	$-0.681633\pi$
$701$	−28.6026	−1.08030	−0.540152	+	0.841567i	$0.681633\pi$
$702$	0	0
$703$	−0.0464050	−0.00175020
$704$	0	0
$705$	−35.8568	−1.35044
$706$	0	0
$707$	−16.0844	−0.604917
$708$	0	0
$709$	−5.70775	−0.214359	−0.107179	−	0.994240i	$-0.534182\pi$
$709$	−5.70775	−0.214359	−0.107179	+	0.994240i	$0.534182\pi$
$710$	0	0
$711$	2.34747	0.0880372
$712$	0	0
$713$	11.3090	0.423525
$714$	0	0
$715$	19.6340	0.734268
$716$	0	0
$717$	15.8185	0.590753
$718$	0	0
$719$	−38.0411	−1.41869	−0.709347	−	0.704859i	$-0.751010\pi$
$719$	−38.0411	−1.41869	−0.709347	+	0.704859i	$0.751010\pi$
$720$	0	0
$721$	−5.87533	−0.218809
$722$	0	0
$723$	28.6300	1.06476
$724$	0	0
$725$	23.3895	0.868664
$726$	0	0
$727$	−32.8174	−1.21713	−0.608565	−	0.793504i	$-0.708255\pi$
$727$	−32.8174	−1.21713	−0.608565	+	0.793504i	$0.708255\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.99512	0.0737922
$732$	0	0
$733$	27.7290	1.02419	0.512097	−	0.858928i	$-0.328869\pi$
$733$	27.7290	1.02419	0.512097	+	0.858928i	$0.328869\pi$
$734$	0	0
$735$	4.16291	0.153551
$736$	0	0
$737$	−60.6419	−2.23377
$738$	0	0
$739$	23.3112	0.857517	0.428759	−	0.903419i	$-0.358951\pi$
$739$	23.3112	0.857517	0.428759	+	0.903419i	$0.358951\pi$
$740$	0	0
$741$	0.103011	0.00378421
$742$	0	0
$743$	34.9175	1.28100	0.640500	−	0.767958i	$-0.278727\pi$
$743$	34.9175	1.28100	0.640500	+	0.767958i	$0.278727\pi$
$744$	0	0
$745$	44.1256	1.61664
$746$	0	0
$747$	8.93921	0.327068
$748$	0	0
$749$	3.09903	0.113236
$750$	0	0
$751$	−7.08533	−0.258547	−0.129274	−	0.991609i	$-0.541265\pi$
$751$	−7.08533	−0.258547	−0.129274	+	0.991609i	$0.541265\pi$
$752$	0	0
$753$	−23.6340	−0.861269
$754$	0	0
$755$	−91.0650	−3.31419
$756$	0	0
$757$	19.5822	0.711728	0.355864	−	0.934538i	$-0.384187\pi$
$757$	19.5822	0.711728	0.355864	+	0.934538i	$0.384187\pi$
$758$	0	0
$759$	−20.8881	−0.758191
$760$	0	0
$761$	−47.4277	−1.71925	−0.859627	−	0.510922i	$-0.829304\pi$
$761$	−47.4277	−1.71925	−0.859627	+	0.510922i	$0.829304\pi$
$762$	0	0
$763$	3.22370	0.116706
$764$	0	0
$765$	−1.87533	−0.0678027
$766$	0	0
$767$	−7.16689	−0.258781
$768$	0	0
$769$	−15.4222	−0.556139	−0.278069	−	0.960561i	$-0.589694\pi$
$769$	−15.4222	−0.556139	−0.278069	+	0.960561i	$0.589694\pi$
$770$	0	0
$771$	12.9854	0.467658
$772$	0	0
$773$	−19.9432	−0.717307	−0.358653	−	0.933471i	$-0.616764\pi$
$773$	−19.9432	−0.717307	−0.358653	+	0.933471i	$0.616764\pi$
$774$	0	0
$775$	−31.4841	−1.13094
$776$	0	0
$777$	−0.450485	−0.0161611
$778$	0	0
$779$	0.439437	0.0157445
$780$	0	0
$781$	33.8019	1.20953
$782$	0	0
$783$	1.89699	0.0677929
$784$	0	0
$785$	31.4279	1.12171
$786$	0	0
$787$	−38.4704	−1.37132	−0.685660	−	0.727922i	$-0.740486\pi$
$787$	−38.4704	−1.37132	−0.685660	+	0.727922i	$0.740486\pi$
$788$	0	0
$789$	10.6733	0.379979
$790$	0	0
$791$	−17.2051	−0.611744
$792$	0	0
$793$	−14.2091	−0.504580
$794$	0	0
$795$	52.7577	1.87112
$796$	0	0
$797$	−42.4023	−1.50197	−0.750983	−	0.660322i	$-0.770420\pi$
$797$	−42.4023	−1.50197	−0.750983	+	0.660322i	$0.770420\pi$
$798$	0	0
$799$	3.88021	0.137272
$800$	0	0
$801$	6.61339	0.233673
$802$	0	0
$803$	7.41029	0.261504
$804$	0	0
$805$	−18.4368	−0.649811
$806$	0	0
$807$	−23.8784	−0.840560
$808$	0	0
$809$	26.5517	0.933508	0.466754	−	0.884387i	$-0.345423\pi$
$809$	26.5517	0.933508	0.466754	+	0.884387i	$0.345423\pi$
$810$	0	0
$811$	24.4916	0.860016	0.430008	−	0.902825i	$-0.358511\pi$
$811$	24.4916	0.860016	0.430008	+	0.902825i	$0.358511\pi$
$812$	0	0
$813$	−5.31302	−0.186336
$814$	0	0
$815$	55.9195	1.95878
$816$	0	0
$817$	−0.456218	−0.0159610
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−18.0679	−0.630573	−0.315286	−	0.948997i	$-0.602101\pi$
$821$	−18.0679	−0.630573	−0.315286	+	0.948997i	$0.602101\pi$
$822$	0	0
$823$	27.1052	0.944828	0.472414	−	0.881377i	$-0.343383\pi$
$823$	27.1052	0.944828	0.472414	+	0.881377i	$0.343383\pi$
$824$	0	0
$825$	58.1523	2.02460
$826$	0	0
$827$	−9.45426	−0.328757	−0.164378	−	0.986397i	$-0.552562\pi$
$827$	−9.45426	−0.328757	−0.164378	+	0.986397i	$0.552562\pi$
$828$	0	0
$829$	21.3081	0.740062	0.370031	−	0.929019i	$-0.379347\pi$
$829$	21.3081	0.740062	0.370031	+	0.929019i	$0.379347\pi$
$830$	0	0
$831$	−1.08533	−0.0376498
$832$	0	0
$833$	−0.450485	−0.0156084
$834$	0	0
$835$	47.4142	1.64084
$836$	0	0
$837$	−2.55350	−0.0882617
$838$	0	0
$839$	−18.5153	−0.639218	−0.319609	−	0.947550i	$-0.603552\pi$
$839$	−18.5153	−0.639218	−0.319609	+	0.947550i	$0.603552\pi$
$840$	0	0
$841$	−25.4014	−0.875911
$842$	0	0
$843$	9.28668	0.319850
$844$	0	0
$845$	4.16291	0.143208
$846$	0	0
$847$	11.2445	0.386364
$848$	0	0
$849$	13.8019	0.473681
$850$	0	0
$851$	1.99512	0.0683919
$852$	0	0
$853$	−48.2768	−1.65296	−0.826482	−	0.562963i	$-0.809662\pi$
$853$	−48.2768	−1.65296	−0.826482	+	0.562963i	$0.809662\pi$
$854$	0	0
$855$	0.428825	0.0146655
$856$	0	0
$857$	−33.6370	−1.14902	−0.574510	−	0.818498i	$-0.694807\pi$
$857$	−33.6370	−1.14902	−0.574510	+	0.818498i	$0.694807\pi$
$858$	0	0
$859$	20.6242	0.703690	0.351845	−	0.936058i	$-0.385554\pi$
$859$	20.6242	0.703690	0.351845	+	0.936058i	$0.385554\pi$
$860$	0	0
$861$	4.26592	0.145382
$862$	0	0
$863$	−51.5692	−1.75544	−0.877718	−	0.479178i	$-0.840935\pi$
$863$	−51.5692	−1.75544	−0.877718	+	0.479178i	$0.840935\pi$
$864$	0	0
$865$	5.93149	0.201677
$866$	0	0
$867$	−16.7971	−0.570458
$868$	0	0
$869$	11.0716	0.375580
$870$	0	0
$871$	−12.8577	−0.435665
$872$	0	0
$873$	−4.67329	−0.158167
$874$	0	0
$875$	30.5133	1.03154
$876$	0	0
$877$	11.7586	0.397060	0.198530	−	0.980095i	$-0.436383\pi$
$877$	11.7586	0.397060	0.198530	+	0.980095i	$0.436383\pi$
$878$	0	0
$879$	−32.3287	−1.09042
$880$	0	0
$881$	45.4439	1.53104	0.765521	−	0.643411i	$-0.222481\pi$
$881$	45.4439	1.53104	0.765521	+	0.643411i	$0.222481\pi$
$882$	0	0
$883$	55.4279	1.86530	0.932649	−	0.360785i	$-0.117491\pi$
$883$	55.4279	1.86530	0.932649	+	0.360785i	$0.117491\pi$
$884$	0	0
$885$	−29.8351	−1.00290
$886$	0	0
$887$	−28.2555	−0.948727	−0.474364	−	0.880329i	$-0.657322\pi$
$887$	−28.2555	−0.948727	−0.474364	+	0.880329i	$0.657322\pi$
$888$	0	0
$889$	−0.369134	−0.0123804
$890$	0	0
$891$	4.71640	0.158005
$892$	0	0
$893$	−0.887275	−0.0296915
$894$	0	0
$895$	−79.9492	−2.67241
$896$	0	0
$897$	−4.42883	−0.147874
$898$	0	0
$899$	−4.84395	−0.161555
$900$	0	0
$901$	−5.70913	−0.190199
$902$	0	0
$903$	−4.42883	−0.147382
$904$	0	0
$905$	−44.3417	−1.47397
$906$	0	0
$907$	47.4173	1.57447	0.787233	−	0.616656i	$-0.211513\pi$
$907$	47.4173	1.57447	0.787233	+	0.616656i	$0.211513\pi$
$908$	0	0
$909$	−16.0844	−0.533487
$910$	0	0
$911$	8.54862	0.283228	0.141614	−	0.989922i	$-0.454771\pi$
$911$	8.54862	0.283228	0.141614	+	0.989922i	$0.454771\pi$
$912$	0	0
$913$	42.1609	1.39532
$914$	0	0
$915$	−59.1512	−1.95548
$916$	0	0
$917$	−9.87533	−0.326112
$918$	0	0
$919$	6.15515	0.203040	0.101520	−	0.994834i	$-0.467629\pi$
$919$	6.15515	0.203040	0.101520	+	0.994834i	$0.467629\pi$
$920$	0	0
$921$	21.5389	0.709731
$922$	0	0
$923$	7.16689	0.235901
$924$	0	0
$925$	−5.55439	−0.182627
$926$	0	0
$927$	−5.87533	−0.192971
$928$	0	0
$929$	−50.3800	−1.65291	−0.826457	−	0.563000i	$-0.809647\pi$
$929$	−50.3800	−1.65291	−0.826457	+	0.563000i	$0.809647\pi$
$930$	0	0
$931$	0.103011	0.00337605
$932$	0	0
$933$	−3.91377	−0.128131
$934$	0	0
$935$	−8.84481	−0.289256
$936$	0	0
$937$	14.0433	0.458775	0.229388	−	0.973335i	$-0.426328\pi$
$937$	14.0433	0.458775	0.229388	+	0.973335i	$0.426328\pi$
$938$	0	0
$939$	23.9646	0.782057
$940$	0	0
$941$	13.4299	0.437803	0.218902	−	0.975747i	$-0.429753\pi$
$941$	13.4299	0.437803	0.218902	+	0.975747i	$0.429753\pi$
$942$	0	0
$943$	−18.8930	−0.615241
$944$	0	0
$945$	4.16291	0.135419
$946$	0	0
$947$	−24.8791	−0.808462	−0.404231	−	0.914657i	$-0.632461\pi$
$947$	−24.8791	−0.808462	−0.404231	+	0.914657i	$0.632461\pi$
$948$	0	0
$949$	1.57117	0.0510025
$950$	0	0
$951$	−26.5564	−0.861150
$952$	0	0
$953$	−60.3042	−1.95344	−0.976722	−	0.214511i	$-0.931184\pi$
$953$	−60.3042	−1.95344	−0.976722	+	0.214511i	$0.931184\pi$
$954$	0	0
$955$	20.7078	0.670088
$956$	0	0
$957$	8.94697	0.289214
$958$	0	0
$959$	6.34727	0.204964
$960$	0	0
$961$	−24.4797	−0.789666
$962$	0	0
$963$	3.09903	0.0998648
$964$	0	0
$965$	62.0314	1.99686
$966$	0	0
$967$	−56.9470	−1.83129	−0.915645	−	0.401987i	$-0.868320\pi$
$967$	−56.9470	−1.83129	−0.915645	+	0.401987i	$0.868320\pi$
$968$	0	0
$969$	−0.0464050	−0.00149074
$970$	0	0
$971$	−25.5190	−0.818945	−0.409472	−	0.912322i	$-0.634287\pi$
$971$	−25.5190	−0.818945	−0.409472	+	0.912322i	$0.634287\pi$
$972$	0	0
$973$	−21.8784	−0.701390
$974$	0	0
$975$	12.3298	0.394870
$976$	0	0
$977$	−31.0422	−0.993129	−0.496564	−	0.868000i	$-0.665405\pi$
$977$	−31.0422	−0.993129	−0.496564	+	0.868000i	$0.665405\pi$
$978$	0	0
$979$	31.1914	0.996882
$980$	0	0
$981$	3.22370	0.102925
$982$	0	0
$983$	37.0621	1.18210	0.591048	−	0.806636i	$-0.298714\pi$
$983$	37.0621	1.18210	0.591048	+	0.806636i	$0.298714\pi$
$984$	0	0
$985$	−84.8638	−2.70399
$986$	0	0
$987$	−8.61339	−0.274167
$988$	0	0
$989$	19.6145	0.623705
$990$	0	0
$991$	−47.7666	−1.51736	−0.758678	−	0.651466i	$-0.774154\pi$
$991$	−47.7666	−1.51736	−0.758678	+	0.651466i	$0.774154\pi$
$992$	0	0
$993$	−6.65959	−0.211336
$994$	0	0
$995$	14.8095	0.469491
$996$	0	0
$997$	−47.9982	−1.52012	−0.760059	−	0.649854i	$-0.774830\pi$
$997$	−47.9982	−1.52012	−0.760059	+	0.649854i	$0.774830\pi$
$998$	0	0
$999$	−0.450485	−0.0142527

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bs.1.4		4
4.3	odd	2		2184.2.a.v.1.4	✓	4
12.11	even	2		6552.2.a.bs.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.a.v.1.4	✓	4	4.3	odd	2
4368.2.a.bs.1.4		4	1.1	even	1	trivial
6552.2.a.bs.1.1		4	12.11	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.16291 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.16291 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.71640 q^{11} +1.00000 q^{13} +4.16291 q^{15} -0.450485 q^{17} +0.103011 q^{19} +1.00000 q^{21} -4.42883 q^{23} +12.3298 q^{25} +1.00000 q^{27} +1.89699 q^{29} -2.55350 q^{31} +4.71640 q^{33} +4.16291 q^{35} -0.450485 q^{37} +1.00000 q^{39} +4.26592 q^{41} -4.42883 q^{43} +4.16291 q^{45} -8.61339 q^{47} +1.00000 q^{49} -0.450485 q^{51} +12.6733 q^{53} +19.6340 q^{55} +0.103011 q^{57} -7.16689 q^{59} -14.2091 q^{61} +1.00000 q^{63} +4.16291 q^{65} -12.8577 q^{67} -4.42883 q^{69} +7.16689 q^{71} +1.57117 q^{73} +12.3298 q^{75} +4.71640 q^{77} +2.34747 q^{79} +1.00000 q^{81} +8.93921 q^{83} -1.87533 q^{85} +1.89699 q^{87} +6.61339 q^{89} +1.00000 q^{91} -2.55350 q^{93} +0.428825 q^{95} -4.67329 q^{97} +4.71640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 3 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 14 q^{25} + 4 q^{27} + 4 q^{29} - 9 q^{31} + 3 q^{33} + 2 q^{35} + 3 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 2 q^{45} - 15 q^{47} + 4 q^{49} + 3 q^{51} + 13 q^{53} - 7 q^{55} + 4 q^{57} - 8 q^{59} + 9 q^{61} + 4 q^{63} + 2 q^{65} + 8 q^{69} + 8 q^{71} + 32 q^{73} + 14 q^{75} + 3 q^{77} + q^{79} + 4 q^{81} - 13 q^{83} + 17 q^{85} + 4 q^{87} + 7 q^{89} + 4 q^{91} - 9 q^{93} - 24 q^{95} + 19 q^{97} + 3 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^15 + 3 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 + 14 * q^25 + 4 * q^27 + 4 * q^29 - 9 * q^31 + 3 * q^33 + 2 * q^35 + 3 * q^37 + 4 * q^39 + 6 * q^41 + 8 * q^43 + 2 * q^45 - 15 * q^47 + 4 * q^49 + 3 * q^51 + 13 * q^53 - 7 * q^55 + 4 * q^57 - 8 * q^59 + 9 * q^61 + 4 * q^63 + 2 * q^65 + 8 * q^69 + 8 * q^71 + 32 * q^73 + 14 * q^75 + 3 * q^77 + q^79 + 4 * q^81 - 13 * q^83 + 17 * q^85 + 4 * q^87 + 7 * q^89 + 4 * q^91 - 9 * q^93 - 24 * q^95 + 19 * q^97 + 3 * q^99

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