LMFDB - Embedded newform 7120.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7120,2,Mod(1,7120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7120, 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("7120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7120 = 2^{4} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7120.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:	$56.8534862392$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 8x^{5} + 6x^{4} + 19x^{3} - 10x^{2} - 12x + 6 $ x^7 - x^6 - 8x^5 + 6x^4 + 19x^3 - 10x^2 - 12*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.89340$ of defining polynomial
Character	$\chi$	$=$	7120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.172659 q^{3} +1.00000 q^{5} +2.74591 q^{7} -2.97019 q^{9} +O(q^{10})$	$q-0.172659 q^{3} +1.00000 q^{5} +2.74591 q^{7} -2.97019 q^{9} +0.429723 q^{11} -6.91659 q^{13} -0.172659 q^{15} +5.29153 q^{17} +1.63823 q^{19} -0.474108 q^{21} +2.26995 q^{23} +1.00000 q^{25} +1.03081 q^{27} +3.93691 q^{29} +2.28028 q^{31} -0.0741958 q^{33} +2.74591 q^{35} -7.74257 q^{37} +1.19421 q^{39} -5.18947 q^{41} +6.95847 q^{43} -2.97019 q^{45} -8.34712 q^{47} +0.540044 q^{49} -0.913632 q^{51} +5.51251 q^{53} +0.429723 q^{55} -0.282856 q^{57} +6.35400 q^{59} +9.00292 q^{61} -8.15588 q^{63} -6.91659 q^{65} +8.39214 q^{67} -0.391929 q^{69} -12.7822 q^{71} +5.08311 q^{73} -0.172659 q^{75} +1.17998 q^{77} +1.33370 q^{79} +8.73259 q^{81} +6.64852 q^{83} +5.29153 q^{85} -0.679745 q^{87} +1.00000 q^{89} -18.9924 q^{91} -0.393712 q^{93} +1.63823 q^{95} +0.828069 q^{97} -1.27636 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9}+O(q^{10}) $ 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9	$ 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9} + 10 q^{11} - 7 q^{13} + 8 q^{15} - 13 q^{17} + 7 q^{19} + 16 q^{21} + 13 q^{23} + 7 q^{25} + 23 q^{27} - 4 q^{29} - q^{31} - 6 q^{33} + 16 q^{35} - 5 q^{37} + 13 q^{39} + 5 q^{41} + 31 q^{43} + 11 q^{45} + 14 q^{47} + 19 q^{49} + q^{51} - 13 q^{53} + 10 q^{55} + 21 q^{57} + 14 q^{59} + 3 q^{61} + 54 q^{63} - 7 q^{65} - q^{67} + 31 q^{69} + 8 q^{71} + 9 q^{73} + 8 q^{75} + 42 q^{77} - 9 q^{79} + 35 q^{81} + 42 q^{83} - 13 q^{85} - 6 q^{87} + 7 q^{89} - 31 q^{91} + 24 q^{93} + 7 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) $ 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9 + 10 * q^11 - 7 * q^13 + 8 * q^15 - 13 * q^17 + 7 * q^19 + 16 * q^21 + 13 * q^23 + 7 * q^25 + 23 * q^27 - 4 * q^29 - q^31 - 6 * q^33 + 16 * q^35 - 5 * q^37 + 13 * q^39 + 5 * q^41 + 31 * q^43 + 11 * q^45 + 14 * q^47 + 19 * q^49 + q^51 - 13 * q^53 + 10 * q^55 + 21 * q^57 + 14 * q^59 + 3 * q^61 + 54 * q^63 - 7 * q^65 - q^67 + 31 * q^69 + 8 * q^71 + 9 * q^73 + 8 * q^75 + 42 * q^77 - 9 * q^79 + 35 * q^81 + 42 * q^83 - 13 * q^85 - 6 * q^87 + 7 * q^89 - 31 * q^91 + 24 * q^93 + 7 * q^95 - 7 * q^97 - 5 * 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
$3$	−0.172659	−0.0996849	−0.0498425	−	0.998757i	$-0.515872\pi$
$3$	−0.172659	−0.0996849	−0.0498425	+	0.998757i	$0.515872\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.74591	1.03786	0.518929	−	0.854817i	$-0.326331\pi$
$7$	2.74591	1.03786	0.518929	+	0.854817i	$0.326331\pi$
$8$	0	0
$9$	−2.97019	−0.990063
$10$	0	0
$11$	0.429723	0.129566	0.0647832	−	0.997899i	$-0.479364\pi$
$11$	0.429723	0.129566	0.0647832	+	0.997899i	$0.479364\pi$
$12$	0	0
$13$	−6.91659	−1.91832	−0.959159	−	0.282868i	$-0.908714\pi$
$13$	−6.91659	−1.91832	−0.959159	+	0.282868i	$0.908714\pi$
$14$	0	0
$15$	−0.172659	−0.0445805
$16$	0	0
$17$	5.29153	1.28338	0.641692	−	0.766963i	$-0.278233\pi$
$17$	5.29153	1.28338	0.641692	+	0.766963i	$0.278233\pi$
$18$	0	0
$19$	1.63823	0.375835	0.187918	−	0.982185i	$-0.439826\pi$
$19$	1.63823	0.375835	0.187918	+	0.982185i	$0.439826\pi$
$20$	0	0
$21$	−0.474108	−0.103459
$22$	0	0
$23$	2.26995	0.473318	0.236659	−	0.971593i	$-0.423948\pi$
$23$	2.26995	0.473318	0.236659	+	0.971593i	$0.423948\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.03081	0.198379
$28$	0	0
$29$	3.93691	0.731066	0.365533	−	0.930798i	$-0.380887\pi$
$29$	3.93691	0.731066	0.365533	+	0.930798i	$0.380887\pi$
$30$	0	0
$31$	2.28028	0.409550	0.204775	−	0.978809i	$-0.434354\pi$
$31$	2.28028	0.409550	0.204775	+	0.978809i	$0.434354\pi$
$32$	0	0
$33$	−0.0741958	−0.0129158
$34$	0	0
$35$	2.74591	0.464144
$36$	0	0
$37$	−7.74257	−1.27287	−0.636435	−	0.771330i	$-0.719592\pi$
$37$	−7.74257	−1.27287	−0.636435	+	0.771330i	$0.719592\pi$
$38$	0	0
$39$	1.19421	0.191227
$40$	0	0
$41$	−5.18947	−0.810459	−0.405230	−	0.914215i	$-0.632808\pi$
$41$	−5.18947	−0.810459	−0.405230	+	0.914215i	$0.632808\pi$
$42$	0	0
$43$	6.95847	1.06116	0.530578	−	0.847636i	$-0.321975\pi$
$43$	6.95847	1.06116	0.530578	+	0.847636i	$0.321975\pi$
$44$	0	0
$45$	−2.97019	−0.442770
$46$	0	0
$47$	−8.34712	−1.21755	−0.608776	−	0.793342i	$-0.708339\pi$
$47$	−8.34712	−1.21755	−0.608776	+	0.793342i	$0.708339\pi$
$48$	0	0
$49$	0.540044	0.0771492
$50$	0	0
$51$	−0.913632	−0.127934
$52$	0	0
$53$	5.51251	0.757201	0.378600	−	0.925560i	$-0.376405\pi$
$53$	5.51251	0.757201	0.378600	+	0.925560i	$0.376405\pi$
$54$	0	0
$55$	0.429723	0.0579439
$56$	0	0
$57$	−0.282856	−0.0374651
$58$	0	0
$59$	6.35400	0.827220	0.413610	−	0.910454i	$-0.364268\pi$
$59$	6.35400	0.827220	0.413610	+	0.910454i	$0.364268\pi$
$60$	0	0
$61$	9.00292	1.15271	0.576353	−	0.817201i	$-0.304475\pi$
$61$	9.00292	1.15271	0.576353	+	0.817201i	$0.304475\pi$
$62$	0	0
$63$	−8.15588	−1.02754
$64$	0	0
$65$	−6.91659	−0.857898
$66$	0	0
$67$	8.39214	1.02526	0.512632	−	0.858609i	$-0.328671\pi$
$67$	8.39214	1.02526	0.512632	+	0.858609i	$0.328671\pi$
$68$	0	0
$69$	−0.391929	−0.0471827
$70$	0	0
$71$	−12.7822	−1.51696	−0.758482	−	0.651695i	$-0.774058\pi$
$71$	−12.7822	−1.51696	−0.758482	+	0.651695i	$0.774058\pi$
$72$	0	0
$73$	5.08311	0.594933	0.297466	−	0.954732i	$-0.403858\pi$
$73$	5.08311	0.594933	0.297466	+	0.954732i	$0.403858\pi$
$74$	0	0
$75$	−0.172659	−0.0199370
$76$	0	0
$77$	1.17998	0.134472
$78$	0	0
$79$	1.33370	0.150053	0.0750266	−	0.997182i	$-0.476096\pi$
$79$	1.33370	0.150053	0.0750266	+	0.997182i	$0.476096\pi$
$80$	0	0
$81$	8.73259	0.970287
$82$	0	0
$83$	6.64852	0.729770	0.364885	−	0.931053i	$-0.381108\pi$
$83$	6.64852	0.729770	0.364885	+	0.931053i	$0.381108\pi$
$84$	0	0
$85$	5.29153	0.573947
$86$	0	0
$87$	−0.679745	−0.0728763
$88$	0	0
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−18.9924	−1.99094
$92$	0	0
$93$	−0.393712	−0.0408260
$94$	0	0
$95$	1.63823	0.168079
$96$	0	0
$97$	0.828069	0.0840776	0.0420388	−	0.999116i	$-0.486615\pi$
$97$	0.828069	0.0840776	0.0420388	+	0.999116i	$0.486615\pi$
$98$	0	0
$99$	−1.27636	−0.128279
$100$	0	0
$101$	−1.43539	−0.142826	−0.0714132	−	0.997447i	$-0.522751\pi$
$101$	−1.43539	−0.142826	−0.0714132	+	0.997447i	$0.522751\pi$
$102$	0	0
$103$	6.02978	0.594132	0.297066	−	0.954857i	$-0.403992\pi$
$103$	6.02978	0.594132	0.297066	+	0.954857i	$0.403992\pi$
$104$	0	0
$105$	−0.474108	−0.0462682
$106$	0	0
$107$	11.0707	1.07024	0.535121	−	0.844775i	$-0.320266\pi$
$107$	11.0707	1.07024	0.535121	+	0.844775i	$0.320266\pi$
$108$	0	0
$109$	8.99213	0.861290	0.430645	−	0.902521i	$-0.358286\pi$
$109$	8.99213	0.861290	0.430645	+	0.902521i	$0.358286\pi$
$110$	0	0
$111$	1.33683	0.126886
$112$	0	0
$113$	−5.54942	−0.522045	−0.261023	−	0.965333i	$-0.584060\pi$
$113$	−5.54942	−0.522045	−0.261023	+	0.965333i	$0.584060\pi$
$114$	0	0
$115$	2.26995	0.211674
$116$	0	0
$117$	20.5436	1.89926
$118$	0	0
$119$	14.5301	1.33197
$120$	0	0
$121$	−10.8153	−0.983213
$122$	0	0
$123$	0.896011	0.0807906
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.6399	1.47655	0.738277	−	0.674497i	$-0.235640\pi$
$127$	16.6399	1.47655	0.738277	+	0.674497i	$0.235640\pi$
$128$	0	0
$129$	−1.20144	−0.105781
$130$	0	0
$131$	−0.840072	−0.0733974	−0.0366987	−	0.999326i	$-0.511684\pi$
$131$	−0.840072	−0.0733974	−0.0366987	+	0.999326i	$0.511684\pi$
$132$	0	0
$133$	4.49843	0.390064
$134$	0	0
$135$	1.03081	0.0887179
$136$	0	0
$137$	−15.0870	−1.28897	−0.644485	−	0.764617i	$-0.722928\pi$
$137$	−15.0870	−1.28897	−0.644485	+	0.764617i	$0.722928\pi$
$138$	0	0
$139$	4.93675	0.418730	0.209365	−	0.977838i	$-0.432860\pi$
$139$	4.93675	0.418730	0.209365	+	0.977838i	$0.432860\pi$
$140$	0	0
$141$	1.44121	0.121372
$142$	0	0
$143$	−2.97222	−0.248550
$144$	0	0
$145$	3.93691	0.326943
$146$	0	0
$147$	−0.0932437	−0.00769061
$148$	0	0
$149$	22.4744	1.84117	0.920587	−	0.390537i	$-0.127711\pi$
$149$	22.4744	1.84117	0.920587	+	0.390537i	$0.127711\pi$
$150$	0	0
$151$	−8.83321	−0.718837	−0.359418	−	0.933177i	$-0.617025\pi$
$151$	−8.83321	−0.718837	−0.359418	+	0.933177i	$0.617025\pi$
$152$	0	0
$153$	−15.7168	−1.27063
$154$	0	0
$155$	2.28028	0.183156
$156$	0	0
$157$	−4.37859	−0.349450	−0.174725	−	0.984617i	$-0.555904\pi$
$157$	−4.37859	−0.349450	−0.174725	+	0.984617i	$0.555904\pi$
$158$	0	0
$159$	−0.951786	−0.0754815
$160$	0	0
$161$	6.23310	0.491237
$162$	0	0
$163$	17.3126	1.35603	0.678014	−	0.735049i	$-0.262841\pi$
$163$	17.3126	1.35603	0.678014	+	0.735049i	$0.262841\pi$
$164$	0	0
$165$	−0.0741958	−0.00577613
$166$	0	0
$167$	−4.89427	−0.378730	−0.189365	−	0.981907i	$-0.560643\pi$
$167$	−4.89427	−0.378730	−0.189365	+	0.981907i	$0.560643\pi$
$168$	0	0
$169$	34.8393	2.67994
$170$	0	0
$171$	−4.86585	−0.372101
$172$	0	0
$173$	−2.08921	−0.158840	−0.0794198	−	0.996841i	$-0.525307\pi$
$173$	−2.08921	−0.158840	−0.0794198	+	0.996841i	$0.525307\pi$
$174$	0	0
$175$	2.74591	0.207572
$176$	0	0
$177$	−1.09708	−0.0824614
$178$	0	0
$179$	17.4654	1.30542	0.652712	−	0.757606i	$-0.273631\pi$
$179$	17.4654	1.30542	0.652712	+	0.757606i	$0.273631\pi$
$180$	0	0
$181$	12.0475	0.895485	0.447743	−	0.894162i	$-0.352228\pi$
$181$	12.0475	0.895485	0.447743	+	0.894162i	$0.352228\pi$
$182$	0	0
$183$	−1.55444	−0.114907
$184$	0	0
$185$	−7.74257	−0.569245
$186$	0	0
$187$	2.27389	0.166283
$188$	0	0
$189$	2.83051	0.205890
$190$	0	0
$191$	7.90966	0.572323	0.286161	−	0.958181i	$-0.407621\pi$
$191$	7.90966	0.572323	0.286161	+	0.958181i	$0.407621\pi$
$192$	0	0
$193$	2.39653	0.172506	0.0862530	−	0.996273i	$-0.472511\pi$
$193$	2.39653	0.172506	0.0862530	+	0.996273i	$0.472511\pi$
$194$	0	0
$195$	1.19421	0.0855195
$196$	0	0
$197$	−16.9355	−1.20660	−0.603302	−	0.797513i	$-0.706149\pi$
$197$	−16.9355	−1.20660	−0.603302	+	0.797513i	$0.706149\pi$
$198$	0	0
$199$	25.0231	1.77384	0.886919	−	0.461926i	$-0.152841\pi$
$199$	25.0231	1.77384	0.886919	+	0.461926i	$0.152841\pi$
$200$	0	0
$201$	−1.44898	−0.102203
$202$	0	0
$203$	10.8104	0.758743
$204$	0	0
$205$	−5.18947	−0.362448
$206$	0	0
$207$	−6.74219	−0.468615
$208$	0	0
$209$	0.703985	0.0486957
$210$	0	0
$211$	−8.90186	−0.612829	−0.306414	−	0.951898i	$-0.599129\pi$
$211$	−8.90186	−0.612829	−0.306414	+	0.951898i	$0.599129\pi$
$212$	0	0
$213$	2.20696	0.151218
$214$	0	0
$215$	6.95847	0.474564
$216$	0	0
$217$	6.26145	0.425055
$218$	0	0
$219$	−0.877646	−0.0593058
$220$	0	0
$221$	−36.5993	−2.46194
$222$	0	0
$223$	−28.1197	−1.88303	−0.941516	−	0.336968i	$-0.890599\pi$
$223$	−28.1197	−1.88303	−0.941516	+	0.336968i	$0.890599\pi$
$224$	0	0
$225$	−2.97019	−0.198013
$226$	0	0
$227$	16.0209	1.06334	0.531671	−	0.846951i	$-0.321564\pi$
$227$	16.0209	1.06334	0.531671	+	0.846951i	$0.321564\pi$
$228$	0	0
$229$	6.71542	0.443767	0.221884	−	0.975073i	$-0.428779\pi$
$229$	6.71542	0.443767	0.221884	+	0.975073i	$0.428779\pi$
$230$	0	0
$231$	−0.203735	−0.0134048
$232$	0	0
$233$	−9.20337	−0.602933	−0.301466	−	0.953477i	$-0.597476\pi$
$233$	−9.20337	−0.602933	−0.301466	+	0.953477i	$0.597476\pi$
$234$	0	0
$235$	−8.34712	−0.544506
$236$	0	0
$237$	−0.230276	−0.0149580
$238$	0	0
$239$	1.01200	0.0654610	0.0327305	−	0.999464i	$-0.489580\pi$
$239$	1.01200	0.0654610	0.0327305	+	0.999464i	$0.489580\pi$
$240$	0	0
$241$	6.14399	0.395769	0.197885	−	0.980225i	$-0.436593\pi$
$241$	6.14399	0.395769	0.197885	+	0.980225i	$0.436593\pi$
$242$	0	0
$243$	−4.60019	−0.295102
$244$	0	0
$245$	0.540044	0.0345022
$246$	0	0
$247$	−11.3310	−0.720972
$248$	0	0
$249$	−1.14793	−0.0727471
$250$	0	0
$251$	−15.4001	−0.972049	−0.486024	−	0.873945i	$-0.661553\pi$
$251$	−15.4001	−0.972049	−0.486024	+	0.873945i	$0.661553\pi$
$252$	0	0
$253$	0.975453	0.0613262
$254$	0	0
$255$	−0.913632	−0.0572138
$256$	0	0
$257$	1.91078	0.119191	0.0595957	−	0.998223i	$-0.481019\pi$
$257$	1.91078	0.119191	0.0595957	+	0.998223i	$0.481019\pi$
$258$	0	0
$259$	−21.2604	−1.32106
$260$	0	0
$261$	−11.6934	−0.723802
$262$	0	0
$263$	21.3509	1.31655	0.658277	−	0.752776i	$-0.271286\pi$
$263$	21.3509	1.31655	0.658277	+	0.752776i	$0.271286\pi$
$264$	0	0
$265$	5.51251	0.338631
$266$	0	0
$267$	−0.172659	−0.0105666
$268$	0	0
$269$	−3.71865	−0.226730	−0.113365	−	0.993553i	$-0.536163\pi$
$269$	−3.71865	−0.226730	−0.113365	+	0.993553i	$0.536163\pi$
$270$	0	0
$271$	29.0720	1.76600	0.882999	−	0.469375i	$-0.155521\pi$
$271$	29.0720	1.76600	0.882999	+	0.469375i	$0.155521\pi$
$272$	0	0
$273$	3.27921	0.198467
$274$	0	0
$275$	0.429723	0.0259133
$276$	0	0
$277$	−22.1766	−1.33246	−0.666230	−	0.745746i	$-0.732093\pi$
$277$	−22.1766	−1.33246	−0.666230	+	0.745746i	$0.732093\pi$
$278$	0	0
$279$	−6.77286	−0.405480
$280$	0	0
$281$	−27.5045	−1.64078	−0.820391	−	0.571803i	$-0.806244\pi$
$281$	−27.5045	−1.64078	−0.820391	+	0.571803i	$0.806244\pi$
$282$	0	0
$283$	11.7574	0.698902	0.349451	−	0.936955i	$-0.386368\pi$
$283$	11.7574	0.698902	0.349451	+	0.936955i	$0.386368\pi$
$284$	0	0
$285$	−0.282856	−0.0167549
$286$	0	0
$287$	−14.2498	−0.841141
$288$	0	0
$289$	11.0002	0.647073
$290$	0	0
$291$	−0.142974	−0.00838127
$292$	0	0
$293$	−12.7263	−0.743476	−0.371738	−	0.928338i	$-0.621238\pi$
$293$	−12.7263	−0.743476	−0.371738	+	0.928338i	$0.621238\pi$
$294$	0	0
$295$	6.35400	0.369944
$296$	0	0
$297$	0.442963	0.0257033
$298$	0	0
$299$	−15.7004	−0.907975
$300$	0	0
$301$	19.1074	1.10133
$302$	0	0
$303$	0.247833	0.0142376
$304$	0	0
$305$	9.00292	0.515506
$306$	0	0
$307$	4.59039	0.261987	0.130994	−	0.991383i	$-0.458183\pi$
$307$	4.59039	0.261987	0.130994	+	0.991383i	$0.458183\pi$
$308$	0	0
$309$	−1.04110	−0.0592260
$310$	0	0
$311$	15.6135	0.885361	0.442680	−	0.896679i	$-0.354028\pi$
$311$	15.6135	0.885361	0.442680	+	0.896679i	$0.354028\pi$
$312$	0	0
$313$	22.3598	1.26385	0.631925	−	0.775029i	$-0.282265\pi$
$313$	22.3598	1.26385	0.631925	+	0.775029i	$0.282265\pi$
$314$	0	0
$315$	−8.15588	−0.459532
$316$	0	0
$317$	19.3949	1.08932	0.544662	−	0.838655i	$-0.316658\pi$
$317$	19.3949	1.08932	0.544662	+	0.838655i	$0.316658\pi$
$318$	0	0
$319$	1.69178	0.0947217
$320$	0	0
$321$	−1.91146	−0.106687
$322$	0	0
$323$	8.66873	0.482341
$324$	0	0
$325$	−6.91659	−0.383664
$326$	0	0
$327$	−1.55258	−0.0858576
$328$	0	0
$329$	−22.9205	−1.26365
$330$	0	0
$331$	−25.2472	−1.38771	−0.693855	−	0.720115i	$-0.744089\pi$
$331$	−25.2472	−1.38771	−0.693855	+	0.720115i	$0.744089\pi$
$332$	0	0
$333$	22.9969	1.26022
$334$	0	0
$335$	8.39214	0.458512
$336$	0	0
$337$	4.05916	0.221116	0.110558	−	0.993870i	$-0.464736\pi$
$337$	4.05916	0.221116	0.110558	+	0.993870i	$0.464736\pi$
$338$	0	0
$339$	0.958159	0.0520400
$340$	0	0
$341$	0.979889	0.0530640
$342$	0	0
$343$	−17.7385	−0.957788
$344$	0	0
$345$	−0.391929	−0.0211007
$346$	0	0
$347$	−25.2031	−1.35297	−0.676487	−	0.736454i	$-0.736499\pi$
$347$	−25.2031	−1.35297	−0.676487	+	0.736454i	$0.736499\pi$
$348$	0	0
$349$	−11.8501	−0.634321	−0.317161	−	0.948372i	$-0.602729\pi$
$349$	−11.8501	−0.634321	−0.317161	+	0.948372i	$0.602729\pi$
$350$	0	0
$351$	−7.12969	−0.380555
$352$	0	0
$353$	13.9770	0.743921	0.371961	−	0.928249i	$-0.378686\pi$
$353$	13.9770	0.743921	0.371961	+	0.928249i	$0.378686\pi$
$354$	0	0
$355$	−12.7822	−0.678406
$356$	0	0
$357$	−2.50875	−0.132777
$358$	0	0
$359$	−12.6401	−0.667120	−0.333560	−	0.942729i	$-0.608250\pi$
$359$	−12.6401	−0.667120	−0.333560	+	0.942729i	$0.608250\pi$
$360$	0	0
$361$	−16.3162	−0.858748
$362$	0	0
$363$	1.86737	0.0980115
$364$	0	0
$365$	5.08311	0.266062
$366$	0	0
$367$	18.5135	0.966398	0.483199	−	0.875510i	$-0.339475\pi$
$367$	18.5135	0.966398	0.483199	+	0.875510i	$0.339475\pi$
$368$	0	0
$369$	15.4137	0.802405
$370$	0	0
$371$	15.1369	0.785867
$372$	0	0
$373$	29.1116	1.50734	0.753670	−	0.657253i	$-0.228282\pi$
$373$	29.1116	1.50734	0.753670	+	0.657253i	$0.228282\pi$
$374$	0	0
$375$	−0.172659	−0.00891609
$376$	0	0
$377$	−27.2300	−1.40242
$378$	0	0
$379$	−16.6575	−0.855637	−0.427818	−	0.903865i	$-0.640718\pi$
$379$	−16.6575	−0.855637	−0.427818	+	0.903865i	$0.640718\pi$
$380$	0	0
$381$	−2.87304	−0.147190
$382$	0	0
$383$	−6.49778	−0.332021	−0.166010	−	0.986124i	$-0.553089\pi$
$383$	−6.49778	−0.332021	−0.166010	+	0.986124i	$0.553089\pi$
$384$	0	0
$385$	1.17998	0.0601375
$386$	0	0
$387$	−20.6680	−1.05061
$388$	0	0
$389$	26.9389	1.36585	0.682927	−	0.730486i	$-0.260706\pi$
$389$	26.9389	1.36585	0.682927	+	0.730486i	$0.260706\pi$
$390$	0	0
$391$	12.0115	0.607449
$392$	0	0
$393$	0.145046	0.00731662
$394$	0	0
$395$	1.33370	0.0671058
$396$	0	0
$397$	26.0878	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$397$	26.0878	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$398$	0	0
$399$	−0.776697	−0.0388835
$400$	0	0
$401$	37.7689	1.88609	0.943045	−	0.332666i	$-0.107948\pi$
$401$	37.7689	1.88609	0.943045	+	0.332666i	$0.107948\pi$
$402$	0	0
$403$	−15.7718	−0.785647
$404$	0	0
$405$	8.73259	0.433926
$406$	0	0
$407$	−3.32716	−0.164921
$408$	0	0
$409$	−32.8776	−1.62569	−0.812846	−	0.582479i	$-0.802083\pi$
$409$	−32.8776	−1.62569	−0.812846	+	0.582479i	$0.802083\pi$
$410$	0	0
$411$	2.60491	0.128491
$412$	0	0
$413$	17.4475	0.858537
$414$	0	0
$415$	6.64852	0.326363
$416$	0	0
$417$	−0.852377	−0.0417411
$418$	0	0
$419$	17.5007	0.854964	0.427482	−	0.904024i	$-0.359401\pi$
$419$	17.5007	0.854964	0.427482	+	0.904024i	$0.359401\pi$
$420$	0	0
$421$	2.78590	0.135777	0.0678883	−	0.997693i	$-0.478374\pi$
$421$	2.78590	0.135777	0.0678883	+	0.997693i	$0.478374\pi$
$422$	0	0
$423$	24.7925	1.20545
$424$	0	0
$425$	5.29153	0.256677
$426$	0	0
$427$	24.7213	1.19635
$428$	0	0
$429$	0.513182	0.0247767
$430$	0	0
$431$	−14.3931	−0.693290	−0.346645	−	0.937996i	$-0.612679\pi$
$431$	−14.3931	−0.693290	−0.346645	+	0.937996i	$0.612679\pi$
$432$	0	0
$433$	6.91601	0.332362	0.166181	−	0.986095i	$-0.446856\pi$
$433$	6.91601	0.332362	0.166181	+	0.986095i	$0.446856\pi$
$434$	0	0
$435$	−0.679745	−0.0325913
$436$	0	0
$437$	3.71870	0.177890
$438$	0	0
$439$	−33.3144	−1.59001	−0.795005	−	0.606602i	$-0.792532\pi$
$439$	−33.3144	−1.59001	−0.795005	+	0.606602i	$0.792532\pi$
$440$	0	0
$441$	−1.60403	−0.0763826
$442$	0	0
$443$	17.6967	0.840794	0.420397	−	0.907340i	$-0.361891\pi$
$443$	17.6967	0.840794	0.420397	+	0.907340i	$0.361891\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	0	0
$447$	−3.88042	−0.183537
$448$	0	0
$449$	42.0230	1.98319	0.991593	−	0.129394i	$-0.0413032\pi$
$449$	42.0230	1.98319	0.991593	+	0.129394i	$0.0413032\pi$
$450$	0	0
$451$	−2.23004	−0.105008
$452$	0	0
$453$	1.52514	0.0716572
$454$	0	0
$455$	−18.9924	−0.890376
$456$	0	0
$457$	−16.9157	−0.791283	−0.395641	−	0.918405i	$-0.629478\pi$
$457$	−16.9157	−0.791283	−0.395641	+	0.918405i	$0.629478\pi$
$458$	0	0
$459$	5.45455	0.254597
$460$	0	0
$461$	−9.77419	−0.455229	−0.227615	−	0.973751i	$-0.573093\pi$
$461$	−9.77419	−0.455229	−0.227615	+	0.973751i	$0.573093\pi$
$462$	0	0
$463$	−40.4822	−1.88137	−0.940684	−	0.339283i	$-0.889816\pi$
$463$	−40.4822	−1.88137	−0.940684	+	0.339283i	$0.889816\pi$
$464$	0	0
$465$	−0.393712	−0.0182579
$466$	0	0
$467$	21.7726	1.00752	0.503759	−	0.863844i	$-0.331950\pi$
$467$	21.7726	1.00752	0.503759	+	0.863844i	$0.331950\pi$
$468$	0	0
$469$	23.0441	1.06408
$470$	0	0
$471$	0.756005	0.0348349
$472$	0	0
$473$	2.99022	0.137490
$474$	0	0
$475$	1.63823	0.0751671
$476$	0	0
$477$	−16.3732	−0.749677
$478$	0	0
$479$	14.3976	0.657842	0.328921	−	0.944357i	$-0.393315\pi$
$479$	14.3976	0.657842	0.328921	+	0.944357i	$0.393315\pi$
$480$	0	0
$481$	53.5522	2.44177
$482$	0	0
$483$	−1.07620	−0.0489689
$484$	0	0
$485$	0.828069	0.0376007
$486$	0	0
$487$	29.7669	1.34886	0.674432	−	0.738337i	$-0.264388\pi$
$487$	29.7669	1.34886	0.674432	+	0.738337i	$0.264388\pi$
$488$	0	0
$489$	−2.98918	−0.135176
$490$	0	0
$491$	32.2121	1.45371	0.726857	−	0.686789i	$-0.240980\pi$
$491$	32.2121	1.45371	0.726857	+	0.686789i	$0.240980\pi$
$492$	0	0
$493$	20.8323	0.938238
$494$	0	0
$495$	−1.27636	−0.0573681
$496$	0	0
$497$	−35.0987	−1.57439
$498$	0	0
$499$	−20.7011	−0.926707	−0.463353	−	0.886174i	$-0.653354\pi$
$499$	−20.7011	−0.926707	−0.463353	+	0.886174i	$0.653354\pi$
$500$	0	0
$501$	0.845041	0.0377536
$502$	0	0
$503$	−39.1454	−1.74541	−0.872703	−	0.488251i	$-0.837635\pi$
$503$	−39.1454	−1.74541	−0.872703	+	0.488251i	$0.837635\pi$
$504$	0	0
$505$	−1.43539	−0.0638739
$506$	0	0
$507$	−6.01533	−0.267150
$508$	0	0
$509$	−8.05887	−0.357203	−0.178602	−	0.983921i	$-0.557157\pi$
$509$	−8.05887	−0.357203	−0.178602	+	0.983921i	$0.557157\pi$
$510$	0	0
$511$	13.9578	0.617456
$512$	0	0
$513$	1.68870	0.0745580
$514$	0	0
$515$	6.02978	0.265704
$516$	0	0
$517$	−3.58695	−0.157754
$518$	0	0
$519$	0.360722	0.0158339
$520$	0	0
$521$	5.83455	0.255616	0.127808	−	0.991799i	$-0.459206\pi$
$521$	5.83455	0.255616	0.127808	+	0.991799i	$0.459206\pi$
$522$	0	0
$523$	33.8865	1.48175	0.740877	−	0.671640i	$-0.234410\pi$
$523$	33.8865	1.48175	0.740877	+	0.671640i	$0.234410\pi$
$524$	0	0
$525$	−0.474108	−0.0206918
$526$	0	0
$527$	12.0662	0.525610
$528$	0	0
$529$	−17.8473	−0.775970
$530$	0	0
$531$	−18.8726	−0.819000
$532$	0	0
$533$	35.8935	1.55472
$534$	0	0
$535$	11.0707	0.478627
$536$	0	0
$537$	−3.01556	−0.130131
$538$	0	0
$539$	0.232070	0.00999595
$540$	0	0
$541$	−22.7941	−0.979995	−0.489997	−	0.871724i	$-0.663002\pi$
$541$	−22.7941	−0.979995	−0.489997	+	0.871724i	$0.663002\pi$
$542$	0	0
$543$	−2.08012	−0.0892664
$544$	0	0
$545$	8.99213	0.385181
$546$	0	0
$547$	−21.1319	−0.903533	−0.451767	−	0.892136i	$-0.649206\pi$
$547$	−21.1319	−0.903533	−0.451767	+	0.892136i	$0.649206\pi$
$548$	0	0
$549$	−26.7404	−1.14125
$550$	0	0
$551$	6.44956	0.274761
$552$	0	0
$553$	3.66223	0.155734
$554$	0	0
$555$	1.33683	0.0567451
$556$	0	0
$557$	19.8251	0.840017	0.420008	−	0.907520i	$-0.362027\pi$
$557$	19.8251	0.840017	0.420008	+	0.907520i	$0.362027\pi$
$558$	0	0
$559$	−48.1289	−2.03564
$560$	0	0
$561$	−0.392609	−0.0165760
$562$	0	0
$563$	10.7352	0.452433	0.226217	−	0.974077i	$-0.427364\pi$
$563$	10.7352	0.452433	0.226217	+	0.974077i	$0.427364\pi$
$564$	0	0
$565$	−5.54942	−0.233466
$566$	0	0
$567$	23.9789	1.00702
$568$	0	0
$569$	19.1468	0.802677	0.401338	−	0.915930i	$-0.368545\pi$
$569$	19.1468	0.802677	0.401338	+	0.915930i	$0.368545\pi$
$570$	0	0
$571$	22.6449	0.947661	0.473831	−	0.880616i	$-0.342871\pi$
$571$	22.6449	0.947661	0.473831	+	0.880616i	$0.342871\pi$
$572$	0	0
$573$	−1.36568	−0.0570520
$574$	0	0
$575$	2.26995	0.0946637
$576$	0	0
$577$	−24.6727	−1.02714	−0.513570	−	0.858048i	$-0.671677\pi$
$577$	−24.6727	−1.02714	−0.513570	+	0.858048i	$0.671677\pi$
$578$	0	0
$579$	−0.413783	−0.0171963
$580$	0	0
$581$	18.2563	0.757398
$582$	0	0
$583$	2.36885	0.0981078
$584$	0	0
$585$	20.5436	0.849373
$586$	0	0
$587$	14.3041	0.590395	0.295198	−	0.955436i	$-0.404615\pi$
$587$	14.3041	0.590395	0.295198	+	0.955436i	$0.404615\pi$
$588$	0	0
$589$	3.73562	0.153923
$590$	0	0
$591$	2.92407	0.120280
$592$	0	0
$593$	31.7751	1.30485	0.652423	−	0.757855i	$-0.273752\pi$
$593$	31.7751	1.30485	0.652423	+	0.757855i	$0.273752\pi$
$594$	0	0
$595$	14.5301	0.595675
$596$	0	0
$597$	−4.32047	−0.176825
$598$	0	0
$599$	33.7567	1.37926	0.689631	−	0.724161i	$-0.257773\pi$
$599$	33.7567	1.37926	0.689631	+	0.724161i	$0.257773\pi$
$600$	0	0
$601$	−34.6759	−1.41446	−0.707229	−	0.706985i	$-0.750055\pi$
$601$	−34.6759	−1.41446	−0.707229	+	0.706985i	$0.750055\pi$
$602$	0	0
$603$	−24.9262	−1.01508
$604$	0	0
$605$	−10.8153	−0.439706
$606$	0	0
$607$	12.8661	0.522217	0.261109	−	0.965309i	$-0.415912\pi$
$607$	12.8661	0.522217	0.261109	+	0.965309i	$0.415912\pi$
$608$	0	0
$609$	−1.86652	−0.0756353
$610$	0	0
$611$	57.7336	2.33565
$612$	0	0
$613$	16.6348	0.671872	0.335936	−	0.941885i	$-0.390947\pi$
$613$	16.6348	0.671872	0.335936	+	0.941885i	$0.390947\pi$
$614$	0	0
$615$	0.896011	0.0361306
$616$	0	0
$617$	33.5170	1.34934	0.674671	−	0.738118i	$-0.264285\pi$
$617$	33.5170	1.34934	0.674671	+	0.738118i	$0.264285\pi$
$618$	0	0
$619$	−34.1671	−1.37329	−0.686646	−	0.726992i	$-0.740918\pi$
$619$	−34.1671	−1.37329	−0.686646	+	0.726992i	$0.740918\pi$
$620$	0	0
$621$	2.33989	0.0938966
$622$	0	0
$623$	2.74591	0.110013
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.121550	−0.00485422
$628$	0	0
$629$	−40.9700	−1.63358
$630$	0	0
$631$	2.76256	0.109976	0.0549878	−	0.998487i	$-0.482488\pi$
$631$	2.76256	0.109976	0.0549878	+	0.998487i	$0.482488\pi$
$632$	0	0
$633$	1.53699	0.0610898
$634$	0	0
$635$	16.6399	0.660335
$636$	0	0
$637$	−3.73527	−0.147997
$638$	0	0
$639$	37.9654	1.50189
$640$	0	0
$641$	46.4551	1.83487	0.917434	−	0.397889i	$-0.130257\pi$
$641$	46.4551	1.83487	0.917434	+	0.397889i	$0.130257\pi$
$642$	0	0
$643$	45.9246	1.81109	0.905545	−	0.424249i	$-0.139462\pi$
$643$	45.9246	1.81109	0.905545	+	0.424249i	$0.139462\pi$
$644$	0	0
$645$	−1.20144	−0.0473068
$646$	0	0
$647$	−14.4744	−0.569048	−0.284524	−	0.958669i	$-0.591836\pi$
$647$	−14.4744	−0.569048	−0.284524	+	0.958669i	$0.591836\pi$
$648$	0	0
$649$	2.73046	0.107180
$650$	0	0
$651$	−1.08110	−0.0423716
$652$	0	0
$653$	29.4446	1.15226	0.576129	−	0.817359i	$-0.304563\pi$
$653$	29.4446	1.15226	0.576129	+	0.817359i	$0.304563\pi$
$654$	0	0
$655$	−0.840072	−0.0328243
$656$	0	0
$657$	−15.0978	−0.589021
$658$	0	0
$659$	25.5581	0.995601	0.497800	−	0.867292i	$-0.334141\pi$
$659$	25.5581	0.995601	0.497800	+	0.867292i	$0.334141\pi$
$660$	0	0
$661$	−22.7900	−0.886428	−0.443214	−	0.896416i	$-0.646162\pi$
$661$	−22.7900	−0.886428	−0.443214	+	0.896416i	$0.646162\pi$
$662$	0	0
$663$	6.31922	0.245418
$664$	0	0
$665$	4.49843	0.174442
$666$	0	0
$667$	8.93661	0.346027
$668$	0	0
$669$	4.85512	0.187710
$670$	0	0
$671$	3.86877	0.149352
$672$	0	0
$673$	26.0465	1.00402	0.502009	−	0.864862i	$-0.332594\pi$
$673$	26.0465	1.00402	0.502009	+	0.864862i	$0.332594\pi$
$674$	0	0
$675$	1.03081	0.0396759
$676$	0	0
$677$	8.03655	0.308870	0.154435	−	0.988003i	$-0.450644\pi$
$677$	8.03655	0.308870	0.154435	+	0.988003i	$0.450644\pi$
$678$	0	0
$679$	2.27381	0.0872606
$680$	0	0
$681$	−2.76615	−0.105999
$682$	0	0
$683$	8.08860	0.309502	0.154751	−	0.987954i	$-0.450543\pi$
$683$	8.08860	0.309502	0.154751	+	0.987954i	$0.450543\pi$
$684$	0	0
$685$	−15.0870	−0.576445
$686$	0	0
$687$	−1.15948	−0.0442369
$688$	0	0
$689$	−38.1278	−1.45255
$690$	0	0
$691$	−27.2840	−1.03793	−0.518966	−	0.854795i	$-0.673683\pi$
$691$	−27.2840	−1.03793	−0.518966	+	0.854795i	$0.673683\pi$
$692$	0	0
$693$	−3.50477	−0.133135
$694$	0	0
$695$	4.93675	0.187262
$696$	0	0
$697$	−27.4602	−1.04013
$698$	0	0
$699$	1.58905	0.0601033
$700$	0	0
$701$	−9.03883	−0.341392	−0.170696	−	0.985324i	$-0.554602\pi$
$701$	−9.03883	−0.341392	−0.170696	+	0.985324i	$0.554602\pi$
$702$	0	0
$703$	−12.6841	−0.478390
$704$	0	0
$705$	1.44121	0.0542790
$706$	0	0
$707$	−3.94145	−0.148233
$708$	0	0
$709$	−46.5488	−1.74818	−0.874088	−	0.485767i	$-0.838540\pi$
$709$	−46.5488	−1.74818	−0.874088	+	0.485767i	$0.838540\pi$
$710$	0	0
$711$	−3.96135	−0.148562
$712$	0	0
$713$	5.17613	0.193848
$714$	0	0
$715$	−2.97222	−0.111155
$716$	0	0
$717$	−0.174732	−0.00652547
$718$	0	0
$719$	49.3092	1.83892	0.919462	−	0.393180i	$-0.128625\pi$
$719$	49.3092	1.83892	0.919462	+	0.393180i	$0.128625\pi$
$720$	0	0
$721$	16.5573	0.616625
$722$	0	0
$723$	−1.06082	−0.0394522
$724$	0	0
$725$	3.93691	0.146213
$726$	0	0
$727$	39.4930	1.46471	0.732357	−	0.680921i	$-0.238420\pi$
$727$	39.4930	1.46471	0.732357	+	0.680921i	$0.238420\pi$
$728$	0	0
$729$	−25.4035	−0.940870
$730$	0	0
$731$	36.8209	1.36187
$732$	0	0
$733$	−4.32917	−0.159902	−0.0799508	−	0.996799i	$-0.525476\pi$
$733$	−4.32917	−0.159902	−0.0799508	+	0.996799i	$0.525476\pi$
$734$	0	0
$735$	−0.0932437	−0.00343935
$736$	0	0
$737$	3.60630	0.132840
$738$	0	0
$739$	−2.52150	−0.0927549	−0.0463775	−	0.998924i	$-0.514768\pi$
$739$	−2.52150	−0.0927549	−0.0463775	+	0.998924i	$0.514768\pi$
$740$	0	0
$741$	1.95640	0.0718700
$742$	0	0
$743$	−43.3061	−1.58875	−0.794373	−	0.607431i	$-0.792200\pi$
$743$	−43.3061	−1.58875	−0.794373	+	0.607431i	$0.792200\pi$
$744$	0	0
$745$	22.4744	0.823398
$746$	0	0
$747$	−19.7474	−0.722518
$748$	0	0
$749$	30.3991	1.11076
$750$	0	0
$751$	−43.4241	−1.58457	−0.792284	−	0.610153i	$-0.791108\pi$
$751$	−43.4241	−1.58457	−0.792284	+	0.610153i	$0.791108\pi$
$752$	0	0
$753$	2.65898	0.0968986
$754$	0	0
$755$	−8.83321	−0.321474
$756$	0	0
$757$	46.8153	1.70153	0.850765	−	0.525547i	$-0.176139\pi$
$757$	46.8153	1.70153	0.850765	+	0.525547i	$0.176139\pi$
$758$	0	0
$759$	−0.168421	−0.00611330
$760$	0	0
$761$	−48.9144	−1.77315	−0.886573	−	0.462588i	$-0.846921\pi$
$761$	−48.9144	−1.77315	−0.886573	+	0.462588i	$0.846921\pi$
$762$	0	0
$763$	24.6916	0.893896
$764$	0	0
$765$	−15.7168	−0.568243
$766$	0	0
$767$	−43.9480	−1.58687
$768$	0	0
$769$	12.2118	0.440370	0.220185	−	0.975458i	$-0.429334\pi$
$769$	12.2118	0.440370	0.220185	+	0.975458i	$0.429334\pi$
$770$	0	0
$771$	−0.329914	−0.0118816
$772$	0	0
$773$	−23.1151	−0.831391	−0.415695	−	0.909504i	$-0.636462\pi$
$773$	−23.1151	−0.831391	−0.415695	+	0.909504i	$0.636462\pi$
$774$	0	0
$775$	2.28028	0.0819100
$776$	0	0
$777$	3.67081	0.131690
$778$	0	0
$779$	−8.50154	−0.304599
$780$	0	0
$781$	−5.49279	−0.196548
$782$	0	0
$783$	4.05821	0.145028
$784$	0	0
$785$	−4.37859	−0.156279
$786$	0	0
$787$	24.5229	0.874147	0.437073	−	0.899426i	$-0.356015\pi$
$787$	24.5229	0.874147	0.437073	+	0.899426i	$0.356015\pi$
$788$	0	0
$789$	−3.68644	−0.131241
$790$	0	0
$791$	−15.2382	−0.541809
$792$	0	0
$793$	−62.2696	−2.21126
$794$	0	0
$795$	−0.951786	−0.0337564
$796$	0	0
$797$	−46.4290	−1.64460	−0.822299	−	0.569055i	$-0.807309\pi$
$797$	−46.4290	−1.64460	−0.822299	+	0.569055i	$0.807309\pi$
$798$	0	0
$799$	−44.1690	−1.56259
$800$	0	0
$801$	−2.97019	−0.104946
$802$	0	0
$803$	2.18433	0.0770833
$804$	0	0
$805$	6.23310	0.219688
$806$	0	0
$807$	0.642060	0.0226016
$808$	0	0
$809$	−9.67473	−0.340145	−0.170073	−	0.985432i	$-0.554400\pi$
$809$	−9.67473	−0.340145	−0.170073	+	0.985432i	$0.554400\pi$
$810$	0	0
$811$	−27.8332	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$811$	−27.8332	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$812$	0	0
$813$	−5.01955	−0.176043
$814$	0	0
$815$	17.3126	0.606434
$816$	0	0
$817$	11.3996	0.398820
$818$	0	0
$819$	56.4109	1.97116
$820$	0	0
$821$	−37.2035	−1.29841	−0.649206	−	0.760612i	$-0.724899\pi$
$821$	−37.2035	−1.29841	−0.649206	+	0.760612i	$0.724899\pi$
$822$	0	0
$823$	25.9987	0.906257	0.453129	−	0.891445i	$-0.350308\pi$
$823$	25.9987	0.906257	0.453129	+	0.891445i	$0.350308\pi$
$824$	0	0
$825$	−0.0741958	−0.00258316
$826$	0	0
$827$	−10.2724	−0.357207	−0.178603	−	0.983921i	$-0.557158\pi$
$827$	−10.2724	−0.357207	−0.178603	+	0.983921i	$0.557158\pi$
$828$	0	0
$829$	13.8050	0.479466	0.239733	−	0.970839i	$-0.422940\pi$
$829$	13.8050	0.479466	0.239733	+	0.970839i	$0.422940\pi$
$830$	0	0
$831$	3.82899	0.132826
$832$	0	0
$833$	2.85766	0.0990120
$834$	0	0
$835$	−4.89427	−0.169373
$836$	0	0
$837$	2.35053	0.0812463
$838$	0	0
$839$	−36.6994	−1.26700	−0.633502	−	0.773741i	$-0.718383\pi$
$839$	−36.6994	−1.26700	−0.633502	+	0.773741i	$0.718383\pi$
$840$	0	0
$841$	−13.5007	−0.465542
$842$	0	0
$843$	4.74891	0.163561
$844$	0	0
$845$	34.8393	1.19851
$846$	0	0
$847$	−29.6980	−1.02043
$848$	0	0
$849$	−2.03002	−0.0696700
$850$	0	0
$851$	−17.5753	−0.602473
$852$	0	0
$853$	−42.5986	−1.45855	−0.729275	−	0.684221i	$-0.760142\pi$
$853$	−42.5986	−1.45855	−0.729275	+	0.684221i	$0.760142\pi$
$854$	0	0
$855$	−4.86585	−0.166408
$856$	0	0
$857$	53.8456	1.83933	0.919665	−	0.392704i	$-0.128460\pi$
$857$	53.8456	1.83933	0.919665	+	0.392704i	$0.128460\pi$
$858$	0	0
$859$	3.35768	0.114563	0.0572813	−	0.998358i	$-0.481757\pi$
$859$	3.35768	0.114563	0.0572813	+	0.998358i	$0.481757\pi$
$860$	0	0
$861$	2.46037	0.0838491
$862$	0	0
$863$	−58.0162	−1.97489	−0.987447	−	0.157949i	$-0.949512\pi$
$863$	−58.0162	−1.97489	−0.987447	+	0.157949i	$0.949512\pi$
$864$	0	0
$865$	−2.08921	−0.0710353
$866$	0	0
$867$	−1.89930	−0.0645034
$868$	0	0
$869$	0.573123	0.0194419
$870$	0	0
$871$	−58.0450	−1.96678
$872$	0	0
$873$	−2.45952	−0.0832421
$874$	0	0
$875$	2.74591	0.0928288
$876$	0	0
$877$	−33.8297	−1.14235	−0.571173	−	0.820830i	$-0.693512\pi$
$877$	−33.8297	−1.14235	−0.571173	+	0.820830i	$0.693512\pi$
$878$	0	0
$879$	2.19731	0.0741134
$880$	0	0
$881$	17.7123	0.596744	0.298372	−	0.954450i	$-0.403556\pi$
$881$	17.7123	0.596744	0.298372	+	0.954450i	$0.403556\pi$
$882$	0	0
$883$	46.8318	1.57602	0.788008	−	0.615665i	$-0.211112\pi$
$883$	46.8318	1.57602	0.788008	+	0.615665i	$0.211112\pi$
$884$	0	0
$885$	−1.09708	−0.0368779
$886$	0	0
$887$	19.8302	0.665832	0.332916	−	0.942957i	$-0.391967\pi$
$887$	19.8302	0.665832	0.332916	+	0.942957i	$0.391967\pi$
$888$	0	0
$889$	45.6918	1.53245
$890$	0	0
$891$	3.75260	0.125717
$892$	0	0
$893$	−13.6745	−0.457599
$894$	0	0
$895$	17.4654	0.583803
$896$	0	0
$897$	2.71081	0.0905114
$898$	0	0
$899$	8.97726	0.299408
$900$	0	0
$901$	29.1696	0.971779
$902$	0	0
$903$	−3.29906	−0.109786
$904$	0	0
$905$	12.0475	0.400473
$906$	0	0
$907$	4.41800	0.146697	0.0733487	−	0.997306i	$-0.476631\pi$
$907$	4.41800	0.146697	0.0733487	+	0.997306i	$0.476631\pi$
$908$	0	0
$909$	4.26337	0.141407
$910$	0	0
$911$	−43.9432	−1.45590	−0.727951	−	0.685629i	$-0.759527\pi$
$911$	−43.9432	−1.45590	−0.727951	+	0.685629i	$0.759527\pi$
$912$	0	0
$913$	2.85702	0.0945537
$914$	0	0
$915$	−1.55444	−0.0513882
$916$	0	0
$917$	−2.30676	−0.0761761
$918$	0	0
$919$	−54.2143	−1.78836	−0.894182	−	0.447704i	$-0.852242\pi$
$919$	−54.2143	−1.78836	−0.894182	+	0.447704i	$0.852242\pi$
$920$	0	0
$921$	−0.792573	−0.0261162
$922$	0	0
$923$	88.4090	2.91002
$924$	0	0
$925$	−7.74257	−0.254574
$926$	0	0
$927$	−17.9096	−0.588228
$928$	0	0
$929$	−0.0310742	−0.00101951	−0.000509756	−	1.00000i	$-0.500162\pi$
$929$	−0.0310742	−0.00101951	−0.000509756		1.00000i	$0.500162\pi$
$930$	0	0
$931$	0.884716	0.0289954
$932$	0	0
$933$	−2.69582	−0.0882571
$934$	0	0
$935$	2.27389	0.0743642
$936$	0	0
$937$	−18.6118	−0.608020	−0.304010	−	0.952669i	$-0.598326\pi$
$937$	−18.6118	−0.608020	−0.304010	+	0.952669i	$0.598326\pi$
$938$	0	0
$939$	−3.86063	−0.125987
$940$	0	0
$941$	4.95121	0.161405	0.0807025	−	0.996738i	$-0.474284\pi$
$941$	4.95121	0.161405	0.0807025	+	0.996738i	$0.474284\pi$
$942$	0	0
$943$	−11.7799	−0.383605
$944$	0	0
$945$	2.83051	0.0920766
$946$	0	0
$947$	−33.9363	−1.10278	−0.551391	−	0.834247i	$-0.685903\pi$
$947$	−33.9363	−1.10278	−0.551391	+	0.834247i	$0.685903\pi$
$948$	0	0
$949$	−35.1578	−1.14127
$950$	0	0
$951$	−3.34871	−0.108589
$952$	0	0
$953$	−5.44095	−0.176250	−0.0881249	−	0.996109i	$-0.528087\pi$
$953$	−5.44095	−0.176250	−0.0881249	+	0.996109i	$0.528087\pi$
$954$	0	0
$955$	7.90966	0.255951
$956$	0	0
$957$	−0.292102	−0.00944232
$958$	0	0
$959$	−41.4276	−1.33777
$960$	0	0
$961$	−25.8003	−0.832269
$962$	0	0
$963$	−32.8820	−1.05961
$964$	0	0
$965$	2.39653	0.0771470
$966$	0	0
$967$	−0.538092	−0.0173039	−0.00865193	−	0.999963i	$-0.502754\pi$
$967$	−0.538092	−0.0173039	−0.00865193	+	0.999963i	$0.502754\pi$
$968$	0	0
$969$	−1.49674	−0.0480821
$970$	0	0
$971$	6.65013	0.213413	0.106706	−	0.994291i	$-0.465969\pi$
$971$	6.65013	0.213413	0.106706	+	0.994291i	$0.465969\pi$
$972$	0	0
$973$	13.5559	0.434582
$974$	0	0
$975$	1.19421	0.0382455
$976$	0	0
$977$	−19.8824	−0.636094	−0.318047	−	0.948075i	$-0.603027\pi$
$977$	−19.8824	−0.636094	−0.318047	+	0.948075i	$0.603027\pi$
$978$	0	0
$979$	0.429723	0.0137340
$980$	0	0
$981$	−26.7083	−0.852731
$982$	0	0
$983$	13.2974	0.424121	0.212060	−	0.977257i	$-0.431983\pi$
$983$	13.2974	0.424121	0.212060	+	0.977257i	$0.431983\pi$
$984$	0	0
$985$	−16.9355	−0.539610
$986$	0	0
$987$	3.95743	0.125966
$988$	0	0
$989$	15.7954	0.502265
$990$	0	0
$991$	24.2737	0.771081	0.385540	−	0.922691i	$-0.374015\pi$
$991$	24.2737	0.771081	0.385540	+	0.922691i	$0.374015\pi$
$992$	0	0
$993$	4.35916	0.138334
$994$	0	0
$995$	25.0231	0.793284
$996$	0	0
$997$	−4.40285	−0.139440	−0.0697199	−	0.997567i	$-0.522211\pi$
$997$	−4.40285	−0.139440	−0.0697199	+	0.997567i	$0.522211\pi$
$998$	0	0
$999$	−7.98111	−0.252511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7120.2.a.bj.1.2		7
4.3	odd	2		445.2.a.f.1.5	✓	7
12.11	even	2		4005.2.a.o.1.3		7
20.19	odd	2		2225.2.a.k.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.f.1.5	✓	7	4.3	odd	2
2225.2.a.k.1.3		7	20.19	odd	2
4005.2.a.o.1.3		7	12.11	even	2
7120.2.a.bj.1.2		7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7120 = 2^{4} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7120.a (trivial)

\(f(q)\)	\(=\)	\(q-0.172659 q^{3} +1.00000 q^{5} +2.74591 q^{7} -2.97019 q^{9} +O(q^{10})\)	\(q-0.172659 q^{3} +1.00000 q^{5} +2.74591 q^{7} -2.97019 q^{9} +0.429723 q^{11} -6.91659 q^{13} -0.172659 q^{15} +5.29153 q^{17} +1.63823 q^{19} -0.474108 q^{21} +2.26995 q^{23} +1.00000 q^{25} +1.03081 q^{27} +3.93691 q^{29} +2.28028 q^{31} -0.0741958 q^{33} +2.74591 q^{35} -7.74257 q^{37} +1.19421 q^{39} -5.18947 q^{41} +6.95847 q^{43} -2.97019 q^{45} -8.34712 q^{47} +0.540044 q^{49} -0.913632 q^{51} +5.51251 q^{53} +0.429723 q^{55} -0.282856 q^{57} +6.35400 q^{59} +9.00292 q^{61} -8.15588 q^{63} -6.91659 q^{65} +8.39214 q^{67} -0.391929 q^{69} -12.7822 q^{71} +5.08311 q^{73} -0.172659 q^{75} +1.17998 q^{77} +1.33370 q^{79} +8.73259 q^{81} +6.64852 q^{83} +5.29153 q^{85} -0.679745 q^{87} +1.00000 q^{89} -18.9924 q^{91} -0.393712 q^{93} +1.63823 q^{95} +0.828069 q^{97} -1.27636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9}+O(q^{10}) \) 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9	\( 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9} + 10 q^{11} - 7 q^{13} + 8 q^{15} - 13 q^{17} + 7 q^{19} + 16 q^{21} + 13 q^{23} + 7 q^{25} + 23 q^{27} - 4 q^{29} - q^{31} - 6 q^{33} + 16 q^{35} - 5 q^{37} + 13 q^{39} + 5 q^{41} + 31 q^{43} + 11 q^{45} + 14 q^{47} + 19 q^{49} + q^{51} - 13 q^{53} + 10 q^{55} + 21 q^{57} + 14 q^{59} + 3 q^{61} + 54 q^{63} - 7 q^{65} - q^{67} + 31 q^{69} + 8 q^{71} + 9 q^{73} + 8 q^{75} + 42 q^{77} - 9 q^{79} + 35 q^{81} + 42 q^{83} - 13 q^{85} - 6 q^{87} + 7 q^{89} - 31 q^{91} + 24 q^{93} + 7 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) \) 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9 + 10 * q^11 - 7 * q^13 + 8 * q^15 - 13 * q^17 + 7 * q^19 + 16 * q^21 + 13 * q^23 + 7 * q^25 + 23 * q^27 - 4 * q^29 - q^31 - 6 * q^33 + 16 * q^35 - 5 * q^37 + 13 * q^39 + 5 * q^41 + 31 * q^43 + 11 * q^45 + 14 * q^47 + 19 * q^49 + q^51 - 13 * q^53 + 10 * q^55 + 21 * q^57 + 14 * q^59 + 3 * q^61 + 54 * q^63 - 7 * q^65 - q^67 + 31 * q^69 + 8 * q^71 + 9 * q^73 + 8 * q^75 + 42 * q^77 - 9 * q^79 + 35 * q^81 + 42 * q^83 - 13 * q^85 - 6 * q^87 + 7 * q^89 - 31 * q^91 + 24 * q^93 + 7 * q^95 - 7 * q^97 - 5 * q^99

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