LMFDB - Embedded newform 2240.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.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:	$17.8864900528$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.56155 q^{9} +O(q^{10})$	$q-2.56155 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.56155 q^{9} -2.56155 q^{11} +0.561553 q^{13} +2.56155 q^{15} +4.56155 q^{17} +2.56155 q^{21} +1.00000 q^{25} -1.43845 q^{27} +5.68466 q^{29} +5.12311 q^{31} +6.56155 q^{33} +1.00000 q^{35} -7.12311 q^{37} -1.43845 q^{39} +2.00000 q^{41} -4.00000 q^{43} -3.56155 q^{45} -11.6847 q^{47} +1.00000 q^{49} -11.6847 q^{51} -4.87689 q^{53} +2.56155 q^{55} +10.2462 q^{59} +2.00000 q^{61} -3.56155 q^{63} -0.561553 q^{65} +6.24621 q^{67} -4.00000 q^{71} +16.2462 q^{73} -2.56155 q^{75} +2.56155 q^{77} -12.8078 q^{79} -7.00000 q^{81} +6.24621 q^{83} -4.56155 q^{85} -14.5616 q^{87} +12.2462 q^{89} -0.561553 q^{91} -13.1231 q^{93} -15.9309 q^{97} -9.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - 2 * q^5 - 2 * q^7 + 3 * q^9	$ 2 q - q^{3} - 2 q^{5} - 2 q^{7} + 3 q^{9} - q^{11} - 3 q^{13} + q^{15} + 5 q^{17} + q^{21} + 2 q^{25} - 7 q^{27} - q^{29} + 2 q^{31} + 9 q^{33} + 2 q^{35} - 6 q^{37} - 7 q^{39} + 4 q^{41} - 8 q^{43} - 3 q^{45} - 11 q^{47} + 2 q^{49} - 11 q^{51} - 18 q^{53} + q^{55} + 4 q^{59} + 4 q^{61} - 3 q^{63} + 3 q^{65} - 4 q^{67} - 8 q^{71} + 16 q^{73} - q^{75} + q^{77} - 5 q^{79} - 14 q^{81} - 4 q^{83} - 5 q^{85} - 25 q^{87} + 8 q^{89} + 3 q^{91} - 18 q^{93} - 3 q^{97} - 10 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 - 2 * q^7 + 3 * q^9 - q^11 - 3 * q^13 + q^15 + 5 * q^17 + q^21 + 2 * q^25 - 7 * q^27 - q^29 + 2 * q^31 + 9 * q^33 + 2 * q^35 - 6 * q^37 - 7 * q^39 + 4 * q^41 - 8 * q^43 - 3 * q^45 - 11 * q^47 + 2 * q^49 - 11 * q^51 - 18 * q^53 + q^55 + 4 * q^59 + 4 * q^61 - 3 * q^63 + 3 * q^65 - 4 * q^67 - 8 * q^71 + 16 * q^73 - q^75 + q^77 - 5 * q^79 - 14 * q^81 - 4 * q^83 - 5 * q^85 - 25 * q^87 + 8 * q^89 + 3 * q^91 - 18 * q^93 - 3 * q^97 - 10 * 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
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0	0
$13$	0.561553	0.155747	0.0778734	−	0.996963i	$-0.475187\pi$
$13$	0.561553	0.155747	0.0778734	+	0.996963i	$0.475187\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	4.56155	1.10634	0.553170	−	0.833069i	$-0.313418\pi$
$17$	4.56155	1.10634	0.553170	+	0.833069i	$0.313418\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	2.56155	0.558977
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	5.12311	0.920137	0.460068	−	0.887883i	$-0.347825\pi$
$31$	5.12311	0.920137	0.460068	+	0.887883i	$0.347825\pi$
$32$	0	0
$33$	6.56155	1.14222
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−7.12311	−1.17103	−0.585516	−	0.810661i	$-0.699108\pi$
$37$	−7.12311	−1.17103	−0.585516	+	0.810661i	$0.699108\pi$
$38$	0	0
$39$	−1.43845	−0.230336
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	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
$45$	−3.56155	−0.530925
$46$	0	0
$47$	−11.6847	−1.70438	−0.852191	−	0.523230i	$-0.824727\pi$
$47$	−11.6847	−1.70438	−0.852191	+	0.523230i	$0.824727\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−11.6847	−1.63618
$52$	0	0
$53$	−4.87689	−0.669893	−0.334946	−	0.942237i	$-0.608718\pi$
$53$	−4.87689	−0.669893	−0.334946	+	0.942237i	$0.608718\pi$
$54$	0	0
$55$	2.56155	0.345400
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.2462	1.33394	0.666972	−	0.745083i	$-0.267590\pi$
$59$	10.2462	1.33394	0.666972	+	0.745083i	$0.267590\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−3.56155	−0.448713
$64$	0	0
$65$	−0.561553	−0.0696521
$66$	0	0
$67$	6.24621	0.763096	0.381548	−	0.924349i	$-0.375391\pi$
$67$	6.24621	0.763096	0.381548	+	0.924349i	$0.375391\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	16.2462	1.90148	0.950738	−	0.309997i	$-0.100328\pi$
$73$	16.2462	1.90148	0.950738	+	0.309997i	$0.100328\pi$
$74$	0	0
$75$	−2.56155	−0.295783
$76$	0	0
$77$	2.56155	0.291916
$78$	0	0
$79$	−12.8078	−1.44099	−0.720493	−	0.693462i	$-0.756084\pi$
$79$	−12.8078	−1.44099	−0.720493	+	0.693462i	$0.756084\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	6.24621	0.685611	0.342805	−	0.939406i	$-0.388623\pi$
$83$	6.24621	0.685611	0.342805	+	0.939406i	$0.388623\pi$
$84$	0	0
$85$	−4.56155	−0.494770
$86$	0	0
$87$	−14.5616	−1.56116
$88$	0	0
$89$	12.2462	1.29810	0.649048	−	0.760748i	$-0.275167\pi$
$89$	12.2462	1.29810	0.649048	+	0.760748i	$0.275167\pi$
$90$	0	0
$91$	−0.561553	−0.0588667
$92$	0	0
$93$	−13.1231	−1.36080
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.9309	−1.61753	−0.808767	−	0.588129i	$-0.799865\pi$
$97$	−15.9309	−1.61753	−0.808767	+	0.588129i	$0.799865\pi$
$98$	0	0
$99$	−9.12311	−0.916907
$100$	0	0
$101$	−13.3693	−1.33030	−0.665148	−	0.746711i	$-0.731632\pi$
$101$	−13.3693	−1.33030	−0.665148	+	0.746711i	$0.731632\pi$
$102$	0	0
$103$	−19.6847	−1.93959	−0.969794	−	0.243927i	$-0.921564\pi$
$103$	−19.6847	−1.93959	−0.969794	+	0.243927i	$0.921564\pi$
$104$	0	0
$105$	−2.56155	−0.249982
$106$	0	0
$107$	−6.87689	−0.664814	−0.332407	−	0.943136i	$-0.607861\pi$
$107$	−6.87689	−0.664814	−0.332407	+	0.943136i	$0.607861\pi$
$108$	0	0
$109$	10.8078	1.03520	0.517598	−	0.855624i	$-0.326826\pi$
$109$	10.8078	1.03520	0.517598	+	0.855624i	$0.326826\pi$
$110$	0	0
$111$	18.2462	1.73185
$112$	0	0
$113$	−5.36932	−0.505103	−0.252551	−	0.967583i	$-0.581270\pi$
$113$	−5.36932	−0.505103	−0.252551	+	0.967583i	$0.581270\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−4.56155	−0.418157
$120$	0	0
$121$	−4.43845	−0.403495
$122$	0	0
$123$	−5.12311	−0.461935
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.2462	−0.909204	−0.454602	−	0.890695i	$-0.650219\pi$
$127$	−10.2462	−0.909204	−0.454602	+	0.890695i	$0.650219\pi$
$128$	0	0
$129$	10.2462	0.902129
$130$	0	0
$131$	−2.24621	−0.196252	−0.0981262	−	0.995174i	$-0.531285\pi$
$131$	−2.24621	−0.196252	−0.0981262	+	0.995174i	$0.531285\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.43845	0.123802
$136$	0	0
$137$	−16.2462	−1.38801	−0.694004	−	0.719971i	$-0.744155\pi$
$137$	−16.2462	−1.38801	−0.694004	+	0.719971i	$0.744155\pi$
$138$	0	0
$139$	−10.8769	−0.922566	−0.461283	−	0.887253i	$-0.652611\pi$
$139$	−10.8769	−0.922566	−0.461283	+	0.887253i	$0.652611\pi$
$140$	0	0
$141$	29.9309	2.52063
$142$	0	0
$143$	−1.43845	−0.120289
$144$	0	0
$145$	−5.68466	−0.472085
$146$	0	0
$147$	−2.56155	−0.211273
$148$	0	0
$149$	−8.24621	−0.675556	−0.337778	−	0.941226i	$-0.609675\pi$
$149$	−8.24621	−0.675556	−0.337778	+	0.941226i	$0.609675\pi$
$150$	0	0
$151$	−0.315342	−0.0256621	−0.0128311	−	0.999918i	$-0.504084\pi$
$151$	−0.315342	−0.0256621	−0.0128311	+	0.999918i	$0.504084\pi$
$152$	0	0
$153$	16.2462	1.31343
$154$	0	0
$155$	−5.12311	−0.411498
$156$	0	0
$157$	−10.4924	−0.837386	−0.418693	−	0.908128i	$-0.637512\pi$
$157$	−10.4924	−0.837386	−0.418693	+	0.908128i	$0.637512\pi$
$158$	0	0
$159$	12.4924	0.990714
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−17.1231	−1.34119	−0.670593	−	0.741826i	$-0.733960\pi$
$163$	−17.1231	−1.34119	−0.670593	+	0.741826i	$0.733960\pi$
$164$	0	0
$165$	−6.56155	−0.510816
$166$	0	0
$167$	8.80776	0.681565	0.340783	−	0.940142i	$-0.389308\pi$
$167$	8.80776	0.681565	0.340783	+	0.940142i	$0.389308\pi$
$168$	0	0
$169$	−12.6847	−0.975743
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.5616	−0.955037	−0.477519	−	0.878622i	$-0.658464\pi$
$173$	−12.5616	−0.955037	−0.477519	+	0.878622i	$0.658464\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−26.2462	−1.97279
$178$	0	0
$179$	14.2462	1.06481	0.532406	−	0.846489i	$-0.321288\pi$
$179$	14.2462	1.06481	0.532406	+	0.846489i	$0.321288\pi$
$180$	0	0
$181$	12.8769	0.957132	0.478566	−	0.878052i	$-0.341157\pi$
$181$	12.8769	0.957132	0.478566	+	0.878052i	$0.341157\pi$
$182$	0	0
$183$	−5.12311	−0.378711
$184$	0	0
$185$	7.12311	0.523701
$186$	0	0
$187$	−11.6847	−0.854467
$188$	0	0
$189$	1.43845	0.104632
$190$	0	0
$191$	26.5616	1.92193	0.960963	−	0.276676i	$-0.0892329\pi$
$191$	26.5616	1.92193	0.960963	+	0.276676i	$0.0892329\pi$
$192$	0	0
$193$	−16.2462	−1.16943	−0.584714	−	0.811240i	$-0.698793\pi$
$193$	−16.2462	−1.16943	−0.584714	+	0.811240i	$0.698793\pi$
$194$	0	0
$195$	1.43845	0.103009
$196$	0	0
$197$	−24.7386	−1.76255	−0.881277	−	0.472599i	$-0.843316\pi$
$197$	−24.7386	−1.76255	−0.881277	+	0.472599i	$0.843316\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	0	0
$203$	−5.68466	−0.398985
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	10.5616	0.727087	0.363544	−	0.931577i	$-0.381567\pi$
$211$	10.5616	0.727087	0.363544	+	0.931577i	$0.381567\pi$
$212$	0	0
$213$	10.2462	0.702059
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−5.12311	−0.347779
$218$	0	0
$219$	−41.6155	−2.81212
$220$	0	0
$221$	2.56155	0.172309
$222$	0	0
$223$	16.8078	1.12553	0.562766	−	0.826617i	$-0.309737\pi$
$223$	16.8078	1.12553	0.562766	+	0.826617i	$0.309737\pi$
$224$	0	0
$225$	3.56155	0.237437
$226$	0	0
$227$	13.4384	0.891941	0.445971	−	0.895048i	$-0.352859\pi$
$227$	13.4384	0.891941	0.445971	+	0.895048i	$0.352859\pi$
$228$	0	0
$229$	−0.246211	−0.0162701	−0.00813505	−	0.999967i	$-0.502589\pi$
$229$	−0.246211	−0.0162701	−0.00813505	+	0.999967i	$0.502589\pi$
$230$	0	0
$231$	−6.56155	−0.431718
$232$	0	0
$233$	25.3693	1.66200	0.831000	−	0.556273i	$-0.187769\pi$
$233$	25.3693	1.66200	0.831000	+	0.556273i	$0.187769\pi$
$234$	0	0
$235$	11.6847	0.762223
$236$	0	0
$237$	32.8078	2.13109
$238$	0	0
$239$	−4.80776	−0.310988	−0.155494	−	0.987837i	$-0.549697\pi$
$239$	−4.80776	−0.310988	−0.155494	+	0.987837i	$0.549697\pi$
$240$	0	0
$241$	−19.1231	−1.23183	−0.615914	−	0.787814i	$-0.711213\pi$
$241$	−19.1231	−1.23183	−0.615914	+	0.787814i	$0.711213\pi$
$242$	0	0
$243$	22.2462	1.42710
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−16.0000	−1.01396
$250$	0	0
$251$	23.3693	1.47506	0.737529	−	0.675315i	$-0.235992\pi$
$251$	23.3693	1.47506	0.737529	+	0.675315i	$0.235992\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	11.6847	0.731722
$256$	0	0
$257$	−30.4924	−1.90207	−0.951033	−	0.309091i	$-0.899975\pi$
$257$	−30.4924	−1.90207	−0.951033	+	0.309091i	$0.899975\pi$
$258$	0	0
$259$	7.12311	0.442608
$260$	0	0
$261$	20.2462	1.25321
$262$	0	0
$263$	7.36932	0.454412	0.227206	−	0.973847i	$-0.427041\pi$
$263$	7.36932	0.454412	0.227206	+	0.973847i	$0.427041\pi$
$264$	0	0
$265$	4.87689	0.299585
$266$	0	0
$267$	−31.3693	−1.91977
$268$	0	0
$269$	22.4924	1.37139	0.685694	−	0.727890i	$-0.259499\pi$
$269$	22.4924	1.37139	0.685694	+	0.727890i	$0.259499\pi$
$270$	0	0
$271$	−7.36932	−0.447654	−0.223827	−	0.974629i	$-0.571855\pi$
$271$	−7.36932	−0.447654	−0.223827	+	0.974629i	$0.571855\pi$
$272$	0	0
$273$	1.43845	0.0870588
$274$	0	0
$275$	−2.56155	−0.154467
$276$	0	0
$277$	−12.2462	−0.735804	−0.367902	−	0.929865i	$-0.619924\pi$
$277$	−12.2462	−0.735804	−0.367902	+	0.929865i	$0.619924\pi$
$278$	0	0
$279$	18.2462	1.09237
$280$	0	0
$281$	−9.68466	−0.577738	−0.288869	−	0.957369i	$-0.593279\pi$
$281$	−9.68466	−0.577738	−0.288869	+	0.957369i	$0.593279\pi$
$282$	0	0
$283$	10.5616	0.627819	0.313910	−	0.949453i	$-0.398361\pi$
$283$	10.5616	0.627819	0.313910	+	0.949453i	$0.398361\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	3.80776	0.223986
$290$	0	0
$291$	40.8078	2.39219
$292$	0	0
$293$	−2.31534	−0.135264	−0.0676318	−	0.997710i	$-0.521544\pi$
$293$	−2.31534	−0.135264	−0.0676318	+	0.997710i	$0.521544\pi$
$294$	0	0
$295$	−10.2462	−0.596557
$296$	0	0
$297$	3.68466	0.213806
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	34.2462	1.96739
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−3.19224	−0.182191	−0.0910953	−	0.995842i	$-0.529037\pi$
$307$	−3.19224	−0.182191	−0.0910953	+	0.995842i	$0.529037\pi$
$308$	0	0
$309$	50.4233	2.86848
$310$	0	0
$311$	−4.49242	−0.254742	−0.127371	−	0.991855i	$-0.540654\pi$
$311$	−4.49242	−0.254742	−0.127371	+	0.991855i	$0.540654\pi$
$312$	0	0
$313$	11.9309	0.674373	0.337186	−	0.941438i	$-0.390525\pi$
$313$	11.9309	0.674373	0.337186	+	0.941438i	$0.390525\pi$
$314$	0	0
$315$	3.56155	0.200671
$316$	0	0
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	−14.5616	−0.815290
$320$	0	0
$321$	17.6155	0.983203
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0.561553	0.0311493
$326$	0	0
$327$	−27.6847	−1.53097
$328$	0	0
$329$	11.6847	0.644196
$330$	0	0
$331$	8.49242	0.466786	0.233393	−	0.972383i	$-0.425017\pi$
$331$	8.49242	0.466786	0.233393	+	0.972383i	$0.425017\pi$
$332$	0	0
$333$	−25.3693	−1.39023
$334$	0	0
$335$	−6.24621	−0.341267
$336$	0	0
$337$	34.9848	1.90575	0.952873	−	0.303370i	$-0.0981117\pi$
$337$	34.9848	1.90575	0.952873	+	0.303370i	$0.0981117\pi$
$338$	0	0
$339$	13.7538	0.747003
$340$	0	0
$341$	−13.1231	−0.710656
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.2462	−1.19424	−0.597120	−	0.802152i	$-0.703688\pi$
$347$	−22.2462	−1.19424	−0.597120	+	0.802152i	$0.703688\pi$
$348$	0	0
$349$	−5.36932	−0.287413	−0.143706	−	0.989620i	$-0.545902\pi$
$349$	−5.36932	−0.287413	−0.143706	+	0.989620i	$0.545902\pi$
$350$	0	0
$351$	−0.807764	−0.0431153
$352$	0	0
$353$	18.3153	0.974827	0.487414	−	0.873171i	$-0.337940\pi$
$353$	18.3153	0.974827	0.487414	+	0.873171i	$0.337940\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	0	0
$357$	11.6847	0.618418
$358$	0	0
$359$	−18.7386	−0.988987	−0.494494	−	0.869181i	$-0.664646\pi$
$359$	−18.7386	−0.988987	−0.494494	+	0.869181i	$0.664646\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	11.3693	0.596734
$364$	0	0
$365$	−16.2462	−0.850366
$366$	0	0
$367$	3.05398	0.159416	0.0797081	−	0.996818i	$-0.474601\pi$
$367$	3.05398	0.159416	0.0797081	+	0.996818i	$0.474601\pi$
$368$	0	0
$369$	7.12311	0.370814
$370$	0	0
$371$	4.87689	0.253196
$372$	0	0
$373$	26.4924	1.37173	0.685863	−	0.727731i	$-0.259425\pi$
$373$	26.4924	1.37173	0.685863	+	0.727731i	$0.259425\pi$
$374$	0	0
$375$	2.56155	0.132278
$376$	0	0
$377$	3.19224	0.164409
$378$	0	0
$379$	−30.2462	−1.55364	−0.776822	−	0.629721i	$-0.783169\pi$
$379$	−30.2462	−1.55364	−0.776822	+	0.629721i	$0.783169\pi$
$380$	0	0
$381$	26.2462	1.34463
$382$	0	0
$383$	−2.24621	−0.114776	−0.0573880	−	0.998352i	$-0.518277\pi$
$383$	−2.24621	−0.114776	−0.0573880	+	0.998352i	$0.518277\pi$
$384$	0	0
$385$	−2.56155	−0.130549
$386$	0	0
$387$	−14.2462	−0.724176
$388$	0	0
$389$	−23.4384	−1.18838	−0.594188	−	0.804326i	$-0.702527\pi$
$389$	−23.4384	−1.18838	−0.594188	+	0.804326i	$0.702527\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	5.75379	0.290240
$394$	0	0
$395$	12.8078	0.644429
$396$	0	0
$397$	18.1771	0.912282	0.456141	−	0.889908i	$-0.349231\pi$
$397$	18.1771	0.912282	0.456141	+	0.889908i	$0.349231\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.68466	0.283878	0.141939	−	0.989875i	$-0.454666\pi$
$401$	5.68466	0.283878	0.141939	+	0.989875i	$0.454666\pi$
$402$	0	0
$403$	2.87689	0.143308
$404$	0	0
$405$	7.00000	0.347833
$406$	0	0
$407$	18.2462	0.904431
$408$	0	0
$409$	−36.7386	−1.81661	−0.908304	−	0.418310i	$-0.862623\pi$
$409$	−36.7386	−1.81661	−0.908304	+	0.418310i	$0.862623\pi$
$410$	0	0
$411$	41.6155	2.05274
$412$	0	0
$413$	−10.2462	−0.504183
$414$	0	0
$415$	−6.24621	−0.306614
$416$	0	0
$417$	27.8617	1.36440
$418$	0	0
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	−3.93087	−0.191579	−0.0957894	−	0.995402i	$-0.530538\pi$
$421$	−3.93087	−0.191579	−0.0957894	+	0.995402i	$0.530538\pi$
$422$	0	0
$423$	−41.6155	−2.02342
$424$	0	0
$425$	4.56155	0.221268
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	3.68466	0.177897
$430$	0	0
$431$	8.31534	0.400536	0.200268	−	0.979741i	$-0.435819\pi$
$431$	8.31534	0.400536	0.200268	+	0.979741i	$0.435819\pi$
$432$	0	0
$433$	−4.24621	−0.204060	−0.102030	−	0.994781i	$-0.532534\pi$
$433$	−4.24621	−0.204060	−0.102030	+	0.994781i	$0.532534\pi$
$434$	0	0
$435$	14.5616	0.698173
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−2.87689	−0.137307	−0.0686533	−	0.997641i	$-0.521870\pi$
$439$	−2.87689	−0.137307	−0.0686533	+	0.997641i	$0.521870\pi$
$440$	0	0
$441$	3.56155	0.169598
$442$	0	0
$443$	8.49242	0.403487	0.201744	−	0.979438i	$-0.435339\pi$
$443$	8.49242	0.403487	0.201744	+	0.979438i	$0.435339\pi$
$444$	0	0
$445$	−12.2462	−0.580526
$446$	0	0
$447$	21.1231	0.999089
$448$	0	0
$449$	14.3153	0.675583	0.337791	−	0.941221i	$-0.390320\pi$
$449$	14.3153	0.675583	0.337791	+	0.941221i	$0.390320\pi$
$450$	0	0
$451$	−5.12311	−0.241238
$452$	0	0
$453$	0.807764	0.0379521
$454$	0	0
$455$	0.561553	0.0263260
$456$	0	0
$457$	−18.4924	−0.865039	−0.432520	−	0.901625i	$-0.642375\pi$
$457$	−18.4924	−0.865039	−0.432520	+	0.901625i	$0.642375\pi$
$458$	0	0
$459$	−6.56155	−0.306267
$460$	0	0
$461$	9.36932	0.436373	0.218186	−	0.975907i	$-0.429986\pi$
$461$	9.36932	0.436373	0.218186	+	0.975907i	$0.429986\pi$
$462$	0	0
$463$	21.1231	0.981674	0.490837	−	0.871251i	$-0.336691\pi$
$463$	21.1231	0.981674	0.490837	+	0.871251i	$0.336691\pi$
$464$	0	0
$465$	13.1231	0.608569
$466$	0	0
$467$	−1.93087	−0.0893500	−0.0446750	−	0.999002i	$-0.514225\pi$
$467$	−1.93087	−0.0893500	−0.0446750	+	0.999002i	$0.514225\pi$
$468$	0	0
$469$	−6.24621	−0.288423
$470$	0	0
$471$	26.8769	1.23842
$472$	0	0
$473$	10.2462	0.471121
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−17.3693	−0.795286
$478$	0	0
$479$	14.7386	0.673425	0.336713	−	0.941607i	$-0.390685\pi$
$479$	14.7386	0.673425	0.336713	+	0.941607i	$0.390685\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.9309	0.723384
$486$	0	0
$487$	9.61553	0.435721	0.217861	−	0.975980i	$-0.430092\pi$
$487$	9.61553	0.435721	0.217861	+	0.975980i	$0.430092\pi$
$488$	0	0
$489$	43.8617	1.98350
$490$	0	0
$491$	6.06913	0.273896	0.136948	−	0.990578i	$-0.456271\pi$
$491$	6.06913	0.273896	0.136948	+	0.990578i	$0.456271\pi$
$492$	0	0
$493$	25.9309	1.16787
$494$	0	0
$495$	9.12311	0.410053
$496$	0	0
$497$	4.00000	0.179425
$498$	0	0
$499$	−23.0540	−1.03204	−0.516019	−	0.856577i	$-0.672587\pi$
$499$	−23.0540	−1.03204	−0.516019	+	0.856577i	$0.672587\pi$
$500$	0	0
$501$	−22.5616	−1.00798
$502$	0	0
$503$	32.1771	1.43471	0.717353	−	0.696710i	$-0.245354\pi$
$503$	32.1771	1.43471	0.717353	+	0.696710i	$0.245354\pi$
$504$	0	0
$505$	13.3693	0.594927
$506$	0	0
$507$	32.4924	1.44304
$508$	0	0
$509$	−44.7386	−1.98301	−0.991503	−	0.130087i	$-0.958474\pi$
$509$	−44.7386	−1.98301	−0.991503	+	0.130087i	$0.958474\pi$
$510$	0	0
$511$	−16.2462	−0.718690
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19.6847	0.867410
$516$	0	0
$517$	29.9309	1.31636
$518$	0	0
$519$	32.1771	1.41242
$520$	0	0
$521$	−17.8617	−0.782537	−0.391269	−	0.920277i	$-0.627964\pi$
$521$	−17.8617	−0.782537	−0.391269	+	0.920277i	$0.627964\pi$
$522$	0	0
$523$	−36.9848	−1.61723	−0.808617	−	0.588335i	$-0.799784\pi$
$523$	−36.9848	−1.61723	−0.808617	+	0.588335i	$0.799784\pi$
$524$	0	0
$525$	2.56155	0.111795
$526$	0	0
$527$	23.3693	1.01798
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	36.4924	1.58364
$532$	0	0
$533$	1.12311	0.0486471
$534$	0	0
$535$	6.87689	0.297314
$536$	0	0
$537$	−36.4924	−1.57476
$538$	0	0
$539$	−2.56155	−0.110334
$540$	0	0
$541$	−22.8078	−0.980582	−0.490291	−	0.871559i	$-0.663110\pi$
$541$	−22.8078	−0.980582	−0.490291	+	0.871559i	$0.663110\pi$
$542$	0	0
$543$	−32.9848	−1.41552
$544$	0	0
$545$	−10.8078	−0.462954
$546$	0	0
$547$	−33.1231	−1.41624	−0.708121	−	0.706091i	$-0.750457\pi$
$547$	−33.1231	−1.41624	−0.708121	+	0.706091i	$0.750457\pi$
$548$	0	0
$549$	7.12311	0.304007
$550$	0	0
$551$	0	0
$552$	0	0
$553$	12.8078	0.544642
$554$	0	0
$555$	−18.2462	−0.774509
$556$	0	0
$557$	−4.87689	−0.206641	−0.103320	−	0.994648i	$-0.532947\pi$
$557$	−4.87689	−0.206641	−0.103320	+	0.994648i	$0.532947\pi$
$558$	0	0
$559$	−2.24621	−0.0950046
$560$	0	0
$561$	29.9309	1.26368
$562$	0	0
$563$	−28.9848	−1.22157	−0.610783	−	0.791798i	$-0.709145\pi$
$563$	−28.9848	−1.22157	−0.610783	+	0.791798i	$0.709145\pi$
$564$	0	0
$565$	5.36932	0.225889
$566$	0	0
$567$	7.00000	0.293972
$568$	0	0
$569$	15.7538	0.660433	0.330217	−	0.943905i	$-0.392878\pi$
$569$	15.7538	0.660433	0.330217	+	0.943905i	$0.392878\pi$
$570$	0	0
$571$	7.50758	0.314182	0.157091	−	0.987584i	$-0.449788\pi$
$571$	7.50758	0.314182	0.157091	+	0.987584i	$0.449788\pi$
$572$	0	0
$573$	−68.0388	−2.84236
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.1922	0.549200	0.274600	−	0.961559i	$-0.411455\pi$
$577$	13.1922	0.549200	0.274600	+	0.961559i	$0.411455\pi$
$578$	0	0
$579$	41.6155	1.72948
$580$	0	0
$581$	−6.24621	−0.259137
$582$	0	0
$583$	12.4924	0.517383
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−32.4924	−1.34111	−0.670553	−	0.741862i	$-0.733943\pi$
$587$	−32.4924	−1.34111	−0.670553	+	0.741862i	$0.733943\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	63.3693	2.60667
$592$	0	0
$593$	17.6847	0.726222	0.363111	−	0.931746i	$-0.381715\pi$
$593$	17.6847	0.726222	0.363111	+	0.931746i	$0.381715\pi$
$594$	0	0
$595$	4.56155	0.187005
$596$	0	0
$597$	40.9848	1.67740
$598$	0	0
$599$	44.8078	1.83080	0.915398	−	0.402550i	$-0.131876\pi$
$599$	44.8078	1.83080	0.915398	+	0.402550i	$0.131876\pi$
$600$	0	0
$601$	−2.49242	−0.101668	−0.0508340	−	0.998707i	$-0.516188\pi$
$601$	−2.49242	−0.101668	−0.0508340	+	0.998707i	$0.516188\pi$
$602$	0	0
$603$	22.2462	0.905936
$604$	0	0
$605$	4.43845	0.180449
$606$	0	0
$607$	−4.31534	−0.175154	−0.0875772	−	0.996158i	$-0.527912\pi$
$607$	−4.31534	−0.175154	−0.0875772	+	0.996158i	$0.527912\pi$
$608$	0	0
$609$	14.5616	0.590064
$610$	0	0
$611$	−6.56155	−0.265452
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	5.12311	0.206584
$616$	0	0
$617$	42.9848	1.73050	0.865252	−	0.501337i	$-0.167158\pi$
$617$	42.9848	1.73050	0.865252	+	0.501337i	$0.167158\pi$
$618$	0	0
$619$	32.9848	1.32577	0.662886	−	0.748720i	$-0.269331\pi$
$619$	32.9848	1.32577	0.662886	+	0.748720i	$0.269331\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.2462	−0.490634
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−32.4924	−1.29556
$630$	0	0
$631$	12.8078	0.509869	0.254935	−	0.966958i	$-0.417946\pi$
$631$	12.8078	0.509869	0.254935	+	0.966958i	$0.417946\pi$
$632$	0	0
$633$	−27.0540	−1.07530
$634$	0	0
$635$	10.2462	0.406608
$636$	0	0
$637$	0.561553	0.0222495
$638$	0	0
$639$	−14.2462	−0.563571
$640$	0	0
$641$	12.2462	0.483696	0.241848	−	0.970314i	$-0.422246\pi$
$641$	12.2462	0.483696	0.241848	+	0.970314i	$0.422246\pi$
$642$	0	0
$643$	−10.5616	−0.416507	−0.208253	−	0.978075i	$-0.566778\pi$
$643$	−10.5616	−0.416507	−0.208253	+	0.978075i	$0.566778\pi$
$644$	0	0
$645$	−10.2462	−0.403444
$646$	0	0
$647$	−50.2462	−1.97538	−0.987691	−	0.156416i	$-0.950006\pi$
$647$	−50.2462	−1.97538	−0.987691	+	0.156416i	$0.950006\pi$
$648$	0	0
$649$	−26.2462	−1.03025
$650$	0	0
$651$	13.1231	0.514335
$652$	0	0
$653$	−23.1231	−0.904877	−0.452439	−	0.891796i	$-0.649446\pi$
$653$	−23.1231	−0.904877	−0.452439	+	0.891796i	$0.649446\pi$
$654$	0	0
$655$	2.24621	0.0877667
$656$	0	0
$657$	57.8617	2.25740
$658$	0	0
$659$	23.6847	0.922623	0.461312	−	0.887238i	$-0.347379\pi$
$659$	23.6847	0.922623	0.461312	+	0.887238i	$0.347379\pi$
$660$	0	0
$661$	−8.87689	−0.345271	−0.172636	−	0.984986i	$-0.555228\pi$
$661$	−8.87689	−0.345271	−0.172636	+	0.984986i	$0.555228\pi$
$662$	0	0
$663$	−6.56155	−0.254830
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−43.0540	−1.66456
$670$	0	0
$671$	−5.12311	−0.197775
$672$	0	0
$673$	−24.2462	−0.934623	−0.467311	−	0.884093i	$-0.654777\pi$
$673$	−24.2462	−0.934623	−0.467311	+	0.884093i	$0.654777\pi$
$674$	0	0
$675$	−1.43845	−0.0553659
$676$	0	0
$677$	−13.1922	−0.507019	−0.253509	−	0.967333i	$-0.581585\pi$
$677$	−13.1922	−0.507019	−0.253509	+	0.967333i	$0.581585\pi$
$678$	0	0
$679$	15.9309	0.611371
$680$	0	0
$681$	−34.4233	−1.31910
$682$	0	0
$683$	−6.87689	−0.263137	−0.131569	−	0.991307i	$-0.542001\pi$
$683$	−6.87689	−0.263137	−0.131569	+	0.991307i	$0.542001\pi$
$684$	0	0
$685$	16.2462	0.620736
$686$	0	0
$687$	0.630683	0.0240621
$688$	0	0
$689$	−2.73863	−0.104334
$690$	0	0
$691$	−28.4924	−1.08390	−0.541951	−	0.840410i	$-0.682314\pi$
$691$	−28.4924	−1.08390	−0.541951	+	0.840410i	$0.682314\pi$
$692$	0	0
$693$	9.12311	0.346558
$694$	0	0
$695$	10.8769	0.412584
$696$	0	0
$697$	9.12311	0.345562
$698$	0	0
$699$	−64.9848	−2.45795
$700$	0	0
$701$	−30.8078	−1.16359	−0.581797	−	0.813334i	$-0.697650\pi$
$701$	−30.8078	−1.16359	−0.581797	+	0.813334i	$0.697650\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−29.9309	−1.12726
$706$	0	0
$707$	13.3693	0.502805
$708$	0	0
$709$	−11.3002	−0.424387	−0.212194	−	0.977228i	$-0.568061\pi$
$709$	−11.3002	−0.424387	−0.212194	+	0.977228i	$0.568061\pi$
$710$	0	0
$711$	−45.6155	−1.71072
$712$	0	0
$713$	0	0
$714$	0	0
$715$	1.43845	0.0537949
$716$	0	0
$717$	12.3153	0.459925
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	19.6847	0.733095
$722$	0	0
$723$	48.9848	1.82177
$724$	0	0
$725$	5.68466	0.211123
$726$	0	0
$727$	−32.9848	−1.22334	−0.611670	−	0.791113i	$-0.709502\pi$
$727$	−32.9848	−1.22334	−0.611670	+	0.791113i	$0.709502\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−18.2462	−0.674861
$732$	0	0
$733$	−22.1771	−0.819129	−0.409565	−	0.912281i	$-0.634319\pi$
$733$	−22.1771	−0.819129	−0.409565	+	0.912281i	$0.634319\pi$
$734$	0	0
$735$	2.56155	0.0944843
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	1.93087	0.0710282	0.0355141	−	0.999369i	$-0.488693\pi$
$739$	1.93087	0.0710282	0.0355141	+	0.999369i	$0.488693\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.7538	−0.504578	−0.252289	−	0.967652i	$-0.581183\pi$
$743$	−13.7538	−0.504578	−0.252289	+	0.967652i	$0.581183\pi$
$744$	0	0
$745$	8.24621	0.302118
$746$	0	0
$747$	22.2462	0.813946
$748$	0	0
$749$	6.87689	0.251276
$750$	0	0
$751$	33.9309	1.23816	0.619078	−	0.785330i	$-0.287507\pi$
$751$	33.9309	1.23816	0.619078	+	0.785330i	$0.287507\pi$
$752$	0	0
$753$	−59.8617	−2.18148
$754$	0	0
$755$	0.315342	0.0114765
$756$	0	0
$757$	−43.6155	−1.58523	−0.792617	−	0.609720i	$-0.791282\pi$
$757$	−43.6155	−1.58523	−0.792617	+	0.609720i	$0.791282\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.3693	−0.484637	−0.242319	−	0.970197i	$-0.577908\pi$
$761$	−13.3693	−0.484637	−0.242319	+	0.970197i	$0.577908\pi$
$762$	0	0
$763$	−10.8078	−0.391267
$764$	0	0
$765$	−16.2462	−0.587383
$766$	0	0
$767$	5.75379	0.207757
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	78.1080	2.81299
$772$	0	0
$773$	26.8078	0.964208	0.482104	−	0.876114i	$-0.339873\pi$
$773$	26.8078	0.964208	0.482104	+	0.876114i	$0.339873\pi$
$774$	0	0
$775$	5.12311	0.184027
$776$	0	0
$777$	−18.2462	−0.654579
$778$	0	0
$779$	0	0
$780$	0	0
$781$	10.2462	0.366638
$782$	0	0
$783$	−8.17708	−0.292225
$784$	0	0
$785$	10.4924	0.374491
$786$	0	0
$787$	−15.0540	−0.536616	−0.268308	−	0.963333i	$-0.586465\pi$
$787$	−15.0540	−0.536616	−0.268308	+	0.963333i	$0.586465\pi$
$788$	0	0
$789$	−18.8769	−0.672035
$790$	0	0
$791$	5.36932	0.190911
$792$	0	0
$793$	1.12311	0.0398827
$794$	0	0
$795$	−12.4924	−0.443061
$796$	0	0
$797$	39.9309	1.41442	0.707212	−	0.707002i	$-0.249953\pi$
$797$	39.9309	1.41442	0.707212	+	0.707002i	$0.249953\pi$
$798$	0	0
$799$	−53.3002	−1.88563
$800$	0	0
$801$	43.6155	1.54108
$802$	0	0
$803$	−41.6155	−1.46858
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−57.6155	−2.02816
$808$	0	0
$809$	−33.0540	−1.16212	−0.581058	−	0.813862i	$-0.697361\pi$
$809$	−33.0540	−1.16212	−0.581058	+	0.813862i	$0.697361\pi$
$810$	0	0
$811$	−27.8617	−0.978358	−0.489179	−	0.872183i	$-0.662704\pi$
$811$	−27.8617	−0.978358	−0.489179	+	0.872183i	$0.662704\pi$
$812$	0	0
$813$	18.8769	0.662042
$814$	0	0
$815$	17.1231	0.599796
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−40.4233	−1.41078	−0.705391	−	0.708818i	$-0.749229\pi$
$821$	−40.4233	−1.41078	−0.705391	+	0.708818i	$0.749229\pi$
$822$	0	0
$823$	−31.3693	−1.09347	−0.546733	−	0.837307i	$-0.684129\pi$
$823$	−31.3693	−1.09347	−0.546733	+	0.837307i	$0.684129\pi$
$824$	0	0
$825$	6.56155	0.228444
$826$	0	0
$827$	26.1080	0.907862	0.453931	−	0.891037i	$-0.350021\pi$
$827$	26.1080	0.907862	0.453931	+	0.891037i	$0.350021\pi$
$828$	0	0
$829$	28.2462	0.981031	0.490516	−	0.871432i	$-0.336808\pi$
$829$	28.2462	0.981031	0.490516	+	0.871432i	$0.336808\pi$
$830$	0	0
$831$	31.3693	1.08819
$832$	0	0
$833$	4.56155	0.158048
$834$	0	0
$835$	−8.80776	−0.304805
$836$	0	0
$837$	−7.36932	−0.254721
$838$	0	0
$839$	−48.9848	−1.69114	−0.845572	−	0.533861i	$-0.820741\pi$
$839$	−48.9848	−1.69114	−0.845572	+	0.533861i	$0.820741\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	0	0
$843$	24.8078	0.854425
$844$	0	0
$845$	12.6847	0.436366
$846$	0	0
$847$	4.43845	0.152507
$848$	0	0
$849$	−27.0540	−0.928490
$850$	0	0
$851$	0	0
$852$	0	0
$853$	44.2462	1.51496	0.757481	−	0.652858i	$-0.226430\pi$
$853$	44.2462	1.51496	0.757481	+	0.652858i	$0.226430\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.7386	0.981693	0.490847	−	0.871246i	$-0.336688\pi$
$857$	28.7386	0.981693	0.490847	+	0.871246i	$0.336688\pi$
$858$	0	0
$859$	43.2311	1.47502	0.737512	−	0.675334i	$-0.236000\pi$
$859$	43.2311	1.47502	0.737512	+	0.675334i	$0.236000\pi$
$860$	0	0
$861$	5.12311	0.174595
$862$	0	0
$863$	48.3542	1.64599	0.822997	−	0.568045i	$-0.192300\pi$
$863$	48.3542	1.64599	0.822997	+	0.568045i	$0.192300\pi$
$864$	0	0
$865$	12.5616	0.427106
$866$	0	0
$867$	−9.75379	−0.331256
$868$	0	0
$869$	32.8078	1.11293
$870$	0	0
$871$	3.50758	0.118850
$872$	0	0
$873$	−56.7386	−1.92031
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−18.6307	−0.629113	−0.314557	−	0.949239i	$-0.601856\pi$
$877$	−18.6307	−0.629113	−0.314557	+	0.949239i	$0.601856\pi$
$878$	0	0
$879$	5.93087	0.200043
$880$	0	0
$881$	−3.12311	−0.105220	−0.0526101	−	0.998615i	$-0.516754\pi$
$881$	−3.12311	−0.105220	−0.0526101	+	0.998615i	$0.516754\pi$
$882$	0	0
$883$	−0.138261	−0.00465284	−0.00232642	−	0.999997i	$-0.500741\pi$
$883$	−0.138261	−0.00465284	−0.00232642	+	0.999997i	$0.500741\pi$
$884$	0	0
$885$	26.2462	0.882257
$886$	0	0
$887$	−20.4924	−0.688068	−0.344034	−	0.938957i	$-0.611794\pi$
$887$	−20.4924	−0.688068	−0.344034	+	0.938957i	$0.611794\pi$
$888$	0	0
$889$	10.2462	0.343647
$890$	0	0
$891$	17.9309	0.600707
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−14.2462	−0.476198
$896$	0	0
$897$	0	0
$898$	0	0
$899$	29.1231	0.971310
$900$	0	0
$901$	−22.2462	−0.741129
$902$	0	0
$903$	−10.2462	−0.340973
$904$	0	0
$905$	−12.8769	−0.428042
$906$	0	0
$907$	−1.75379	−0.0582336	−0.0291168	−	0.999576i	$-0.509269\pi$
$907$	−1.75379	−0.0582336	−0.0291168	+	0.999576i	$0.509269\pi$
$908$	0	0
$909$	−47.6155	−1.57931
$910$	0	0
$911$	10.7386	0.355787	0.177893	−	0.984050i	$-0.443072\pi$
$911$	10.7386	0.355787	0.177893	+	0.984050i	$0.443072\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	5.12311	0.169365
$916$	0	0
$917$	2.24621	0.0741764
$918$	0	0
$919$	−31.0540	−1.02438	−0.512188	−	0.858873i	$-0.671165\pi$
$919$	−31.0540	−1.02438	−0.512188	+	0.858873i	$0.671165\pi$
$920$	0	0
$921$	8.17708	0.269444
$922$	0	0
$923$	−2.24621	−0.0739349
$924$	0	0
$925$	−7.12311	−0.234206
$926$	0	0
$927$	−70.1080	−2.30265
$928$	0	0
$929$	40.1080	1.31590	0.657950	−	0.753062i	$-0.271424\pi$
$929$	40.1080	1.31590	0.657950	+	0.753062i	$0.271424\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	11.5076	0.376741
$934$	0	0
$935$	11.6847	0.382129
$936$	0	0
$937$	42.3153	1.38238	0.691191	−	0.722672i	$-0.257086\pi$
$937$	42.3153	1.38238	0.691191	+	0.722672i	$0.257086\pi$
$938$	0	0
$939$	−30.5616	−0.997339
$940$	0	0
$941$	38.4924	1.25482	0.627409	−	0.778690i	$-0.284116\pi$
$941$	38.4924	1.25482	0.627409	+	0.778690i	$0.284116\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1.43845	−0.0467927
$946$	0	0
$947$	27.3693	0.889383	0.444692	−	0.895684i	$-0.353313\pi$
$947$	27.3693	0.889383	0.444692	+	0.895684i	$0.353313\pi$
$948$	0	0
$949$	9.12311	0.296149
$950$	0	0
$951$	25.6155	0.830640
$952$	0	0
$953$	12.2462	0.396694	0.198347	−	0.980132i	$-0.436443\pi$
$953$	12.2462	0.396694	0.198347	+	0.980132i	$0.436443\pi$
$954$	0	0
$955$	−26.5616	−0.859512
$956$	0	0
$957$	37.3002	1.20574
$958$	0	0
$959$	16.2462	0.524618
$960$	0	0
$961$	−4.75379	−0.153348
$962$	0	0
$963$	−24.4924	−0.789257
$964$	0	0
$965$	16.2462	0.522984
$966$	0	0
$967$	27.8617	0.895973	0.447987	−	0.894040i	$-0.352141\pi$
$967$	27.8617	0.895973	0.447987	+	0.894040i	$0.352141\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−59.8617	−1.92105	−0.960527	−	0.278186i	$-0.910267\pi$
$971$	−59.8617	−1.92105	−0.960527	+	0.278186i	$0.910267\pi$
$972$	0	0
$973$	10.8769	0.348697
$974$	0	0
$975$	−1.43845	−0.0460672
$976$	0	0
$977$	29.8617	0.955362	0.477681	−	0.878533i	$-0.341478\pi$
$977$	29.8617	0.955362	0.477681	+	0.878533i	$0.341478\pi$
$978$	0	0
$979$	−31.3693	−1.00257
$980$	0	0
$981$	38.4924	1.22897
$982$	0	0
$983$	−35.0540	−1.11805	−0.559024	−	0.829151i	$-0.688824\pi$
$983$	−35.0540	−1.11805	−0.559024	+	0.829151i	$0.688824\pi$
$984$	0	0
$985$	24.7386	0.788238
$986$	0	0
$987$	−29.9309	−0.952710
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−25.7538	−0.818096	−0.409048	−	0.912513i	$-0.634139\pi$
$991$	−25.7538	−0.818096	−0.409048	+	0.912513i	$0.634139\pi$
$992$	0	0
$993$	−21.7538	−0.690336
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	−2.31534	−0.0733276	−0.0366638	−	0.999328i	$-0.511673\pi$
$997$	−2.31534	−0.0733276	−0.0366638	+	0.999328i	$0.511673\pi$
$998$	0	0
$999$	10.2462	0.324176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.a.bc.1.1		2
4.3	odd	2		2240.2.a.bj.1.2		2
8.3	odd	2		1120.2.a.r.1.1	✓	2
8.5	even	2		1120.2.a.t.1.2	yes	2
40.19	odd	2		5600.2.a.bh.1.2		2
40.29	even	2		5600.2.a.ba.1.1		2
56.13	odd	2		7840.2.a.bc.1.1		2
56.27	even	2		7840.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.a.r.1.1	✓	2	8.3	odd	2
1120.2.a.t.1.2	yes	2	8.5	even	2
2240.2.a.bc.1.1		2	1.1	even	1	trivial
2240.2.a.bj.1.2		2	4.3	odd	2
5600.2.a.ba.1.1		2	40.29	even	2
5600.2.a.bh.1.2		2	40.19	odd	2
7840.2.a.bc.1.1		2	56.13	odd	2
7840.2.a.bh.1.2		2	56.27	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q-2.56155 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.56155 q^{9} -2.56155 q^{11} +0.561553 q^{13} +2.56155 q^{15} +4.56155 q^{17} +2.56155 q^{21} +1.00000 q^{25} -1.43845 q^{27} +5.68466 q^{29} +5.12311 q^{31} +6.56155 q^{33} +1.00000 q^{35} -7.12311 q^{37} -1.43845 q^{39} +2.00000 q^{41} -4.00000 q^{43} -3.56155 q^{45} -11.6847 q^{47} +1.00000 q^{49} -11.6847 q^{51} -4.87689 q^{53} +2.56155 q^{55} +10.2462 q^{59} +2.00000 q^{61} -3.56155 q^{63} -0.561553 q^{65} +6.24621 q^{67} -4.00000 q^{71} +16.2462 q^{73} -2.56155 q^{75} +2.56155 q^{77} -12.8078 q^{79} -7.00000 q^{81} +6.24621 q^{83} -4.56155 q^{85} -14.5616 q^{87} +12.2462 q^{89} -0.561553 q^{91} -13.1231 q^{93} -15.9309 q^{97} -9.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - 2 * q^5 - 2 * q^7 + 3 * q^9	\( 2 q - q^{3} - 2 q^{5} - 2 q^{7} + 3 q^{9} - q^{11} - 3 q^{13} + q^{15} + 5 q^{17} + q^{21} + 2 q^{25} - 7 q^{27} - q^{29} + 2 q^{31} + 9 q^{33} + 2 q^{35} - 6 q^{37} - 7 q^{39} + 4 q^{41} - 8 q^{43} - 3 q^{45} - 11 q^{47} + 2 q^{49} - 11 q^{51} - 18 q^{53} + q^{55} + 4 q^{59} + 4 q^{61} - 3 q^{63} + 3 q^{65} - 4 q^{67} - 8 q^{71} + 16 q^{73} - q^{75} + q^{77} - 5 q^{79} - 14 q^{81} - 4 q^{83} - 5 q^{85} - 25 q^{87} + 8 q^{89} + 3 q^{91} - 18 q^{93} - 3 q^{97} - 10 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 - 2 * q^7 + 3 * q^9 - q^11 - 3 * q^13 + q^15 + 5 * q^17 + q^21 + 2 * q^25 - 7 * q^27 - q^29 + 2 * q^31 + 9 * q^33 + 2 * q^35 - 6 * q^37 - 7 * q^39 + 4 * q^41 - 8 * q^43 - 3 * q^45 - 11 * q^47 + 2 * q^49 - 11 * q^51 - 18 * q^53 + q^55 + 4 * q^59 + 4 * q^61 - 3 * q^63 + 3 * q^65 - 4 * q^67 - 8 * q^71 + 16 * q^73 - q^75 + q^77 - 5 * q^79 - 14 * q^81 - 4 * q^83 - 5 * q^85 - 25 * q^87 + 8 * q^89 + 3 * q^91 - 18 * q^93 - 3 * q^97 - 10 * q^99

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