LMFDB - Embedded newform 7840.2.a.cd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7840,2,Mod(1,7840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7840, 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("7840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7840 = 2^{5} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7840.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:	$62.6027151847$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.170145936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 22x^{2} - 9 $ x^6 - 10x^4 + 22x^2 - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.65843$ of defining polynomial
Character	$\chi$	$=$	7840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.52994 q^{3} -1.00000 q^{5} -0.659278 q^{9} +O(q^{10})$	$q-1.52994 q^{3} -1.00000 q^{5} -0.659278 q^{9} +1.30724 q^{11} -5.47520 q^{13} +1.52994 q^{15} -6.13448 q^{17} -7.06950 q^{19} -5.09415 q^{23} +1.00000 q^{25} +5.59848 q^{27} -4.47520 q^{29} -4.87145 q^{31} -2.00000 q^{33} -0.659278 q^{37} +8.37674 q^{39} +1.00000 q^{41} -9.10377 q^{43} +0.659278 q^{45} +1.30724 q^{47} +9.38540 q^{51} +7.47520 q^{53} -1.30724 q^{55} +10.8159 q^{57} -7.93133 q^{59} -13.7938 q^{61} +5.47520 q^{65} +7.20430 q^{67} +7.79376 q^{69} -8.82214 q^{71} +4.81592 q^{73} -1.52994 q^{75} -0.802911 q^{79} -6.58752 q^{81} -14.7781 q^{83} +6.13448 q^{85} +6.84680 q^{87} -4.34072 q^{89} +7.45304 q^{93} +7.06950 q^{95} +18.2690 q^{97} -0.861835 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{5} + 10 q^{9}+O(q^{10}) $ 6 * q - 6 * q^5 + 10 * q^9	$ 6 q - 6 q^{5} + 10 q^{9} - 2 q^{13} + 8 q^{17} + 6 q^{25} + 4 q^{29} - 12 q^{33} + 10 q^{37} + 6 q^{41} - 10 q^{45} + 14 q^{53} + 48 q^{57} - 24 q^{61} + 2 q^{65} - 12 q^{69} + 12 q^{73} + 78 q^{81} - 8 q^{85} - 40 q^{89} - 28 q^{93} + 20 q^{97}+O(q^{100}) $ 6 * q - 6 * q^5 + 10 * q^9 - 2 * q^13 + 8 * q^17 + 6 * q^25 + 4 * q^29 - 12 * q^33 + 10 * q^37 + 6 * q^41 - 10 * q^45 + 14 * q^53 + 48 * q^57 - 24 * q^61 + 2 * q^65 - 12 * q^69 + 12 * q^73 + 78 * q^81 - 8 * q^85 - 40 * q^89 - 28 * q^93 + 20 * q^97

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
$3$	−1.52994	−0.883312	−0.441656	−	0.897184i	$-0.645609\pi$
$3$	−1.52994	−0.883312	−0.441656	+	0.897184i	$0.645609\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.659278	−0.219759
$10$	0	0
$11$	1.30724	0.394147	0.197074	−	0.980389i	$-0.436856\pi$
$11$	1.30724	0.394147	0.197074	+	0.980389i	$0.436856\pi$
$12$	0	0
$13$	−5.47520	−1.51855	−0.759274	−	0.650771i	$-0.774446\pi$
$13$	−5.47520	−1.51855	−0.759274	+	0.650771i	$0.774446\pi$
$14$	0	0
$15$	1.52994	0.395029
$16$	0	0
$17$	−6.13448	−1.48783	−0.743915	−	0.668274i	$-0.767033\pi$
$17$	−6.13448	−1.48783	−0.743915	+	0.668274i	$0.767033\pi$
$18$	0	0
$19$	−7.06950	−1.62185	−0.810927	−	0.585147i	$-0.801037\pi$
$19$	−7.06950	−1.62185	−0.810927	+	0.585147i	$0.801037\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.09415	−1.06220	−0.531102	−	0.847308i	$-0.678222\pi$
$23$	−5.09415	−1.06220	−0.531102	+	0.847308i	$0.678222\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.59848	1.07743
$28$	0	0
$29$	−4.47520	−0.831024	−0.415512	−	0.909588i	$-0.636398\pi$
$29$	−4.47520	−0.831024	−0.415512	+	0.909588i	$0.636398\pi$
$30$	0	0
$31$	−4.87145	−0.874939	−0.437469	−	0.899233i	$-0.644125\pi$
$31$	−4.87145	−0.874939	−0.437469	+	0.899233i	$0.644125\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.659278	−0.108385	−0.0541923	−	0.998531i	$-0.517258\pi$
$37$	−0.659278	−0.108385	−0.0541923	+	0.998531i	$0.517258\pi$
$38$	0	0
$39$	8.37674	1.34135
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	−9.10377	−1.38831	−0.694156	−	0.719825i	$-0.744222\pi$
$43$	−9.10377	−1.38831	−0.694156	+	0.719825i	$0.744222\pi$
$44$	0	0
$45$	0.659278	0.0982794
$46$	0	0
$47$	1.30724	0.190680	0.0953402	−	0.995445i	$-0.469606\pi$
$47$	1.30724	0.190680	0.0953402	+	0.995445i	$0.469606\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	9.38540	1.31422
$52$	0	0
$53$	7.47520	1.02680	0.513399	−	0.858150i	$-0.328386\pi$
$53$	7.47520	1.02680	0.513399	+	0.858150i	$0.328386\pi$
$54$	0	0
$55$	−1.30724	−0.176268
$56$	0	0
$57$	10.8159	1.43260
$58$	0	0
$59$	−7.93133	−1.03257	−0.516286	−	0.856416i	$-0.672686\pi$
$59$	−7.93133	−1.03257	−0.516286	+	0.856416i	$0.672686\pi$
$60$	0	0
$61$	−13.7938	−1.76611	−0.883055	−	0.469270i	$-0.844517\pi$
$61$	−13.7938	−1.76611	−0.883055	+	0.469270i	$0.844517\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.47520	0.679115
$66$	0	0
$67$	7.20430	0.880146	0.440073	−	0.897962i	$-0.354953\pi$
$67$	7.20430	0.880146	0.440073	+	0.897962i	$0.354953\pi$
$68$	0	0
$69$	7.79376	0.938258
$70$	0	0
$71$	−8.82214	−1.04700	−0.523498	−	0.852027i	$-0.675373\pi$
$71$	−8.82214	−1.04700	−0.523498	+	0.852027i	$0.675373\pi$
$72$	0	0
$73$	4.81592	0.563661	0.281831	−	0.959464i	$-0.409058\pi$
$73$	4.81592	0.563661	0.281831	+	0.959464i	$0.409058\pi$
$74$	0	0
$75$	−1.52994	−0.176662
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.802911	−0.0903345	−0.0451673	−	0.998979i	$-0.514382\pi$
$79$	−0.802911	−0.0903345	−0.0451673	+	0.998979i	$0.514382\pi$
$80$	0	0
$81$	−6.58752	−0.731946
$82$	0	0
$83$	−14.7781	−1.62211	−0.811055	−	0.584969i	$-0.801107\pi$
$83$	−14.7781	−1.62211	−0.811055	+	0.584969i	$0.801107\pi$
$84$	0	0
$85$	6.13448	0.665378
$86$	0	0
$87$	6.84680	0.734054
$88$	0	0
$89$	−4.34072	−0.460116	−0.230058	−	0.973177i	$-0.573892\pi$
$89$	−4.34072	−0.460116	−0.230058	+	0.973177i	$0.573892\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.45304	0.772844
$94$	0	0
$95$	7.06950	0.725315
$96$	0	0
$97$	18.2690	1.85493	0.927466	−	0.373908i	$-0.121982\pi$
$97$	18.2690	1.85493	0.927466	+	0.373908i	$0.121982\pi$
$98$	0	0
$99$	−0.861835	−0.0866176
$100$	0	0
$101$	6.47520	0.644307	0.322153	−	0.946688i	$-0.395593\pi$
$101$	6.47520	0.644307	0.322153	+	0.946688i	$0.395593\pi$
$102$	0	0
$103$	17.4805	1.72241	0.861203	−	0.508261i	$-0.169712\pi$
$103$	17.4805	1.72241	0.861203	+	0.508261i	$0.169712\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.97535	0.190964	0.0954820	−	0.995431i	$-0.469561\pi$
$107$	1.97535	0.190964	0.0954820	+	0.995431i	$0.469561\pi$
$108$	0	0
$109$	0.843355	0.0807787	0.0403894	−	0.999184i	$-0.487140\pi$
$109$	0.843355	0.0807787	0.0403894	+	0.999184i	$0.487140\pi$
$110$	0	0
$111$	1.00866	0.0957375
$112$	0	0
$113$	3.86552	0.363638	0.181819	−	0.983332i	$-0.441802\pi$
$113$	3.86552	0.363638	0.181819	+	0.983332i	$0.441802\pi$
$114$	0	0
$115$	5.09415	0.475032
$116$	0	0
$117$	3.60968	0.333715
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.29113	−0.844648
$122$	0	0
$123$	−1.52994	−0.137950
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.0415	−0.891038	−0.445519	−	0.895273i	$-0.646981\pi$
$127$	−10.0415	−0.891038	−0.445519	+	0.895273i	$0.646981\pi$
$128$	0	0
$129$	13.9282	1.22631
$130$	0	0
$131$	−20.2298	−1.76749	−0.883743	−	0.467973i	$-0.844984\pi$
$131$	−20.2298	−1.76749	−0.883743	+	0.467973i	$0.844984\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.59848	−0.481841
$136$	0	0
$137$	20.2690	1.73169	0.865847	−	0.500309i	$-0.166780\pi$
$137$	20.2690	1.73169	0.865847	+	0.500309i	$0.166780\pi$
$138$	0	0
$139$	7.57383	0.642404	0.321202	−	0.947011i	$-0.395913\pi$
$139$	7.57383	0.642404	0.321202	+	0.947011i	$0.395913\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	−7.15740	−0.598532
$144$	0	0
$145$	4.47520	0.371645
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.7442	−1.37174	−0.685868	−	0.727726i	$-0.740577\pi$
$149$	−16.7442	−1.37174	−0.685868	+	0.727726i	$0.740577\pi$
$150$	0	0
$151$	−2.25697	−0.183670	−0.0918349	−	0.995774i	$-0.529273\pi$
$151$	−2.25697	−0.183670	−0.0918349	+	0.995774i	$0.529273\pi$
$152$	0	0
$153$	4.04433	0.326965
$154$	0	0
$155$	4.87145	0.391284
$156$	0	0
$157$	−5.60968	−0.447701	−0.223851	−	0.974623i	$-0.571863\pi$
$157$	−5.60968	−0.447701	−0.223851	+	0.974623i	$0.571863\pi$
$158$	0	0
$159$	−11.4366	−0.906983
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−24.3273	−1.90546	−0.952731	−	0.303815i	$-0.901739\pi$
$163$	−24.3273	−1.90546	−0.952731	+	0.303815i	$0.901739\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	−21.8476	−1.69062	−0.845310	−	0.534276i	$-0.820584\pi$
$167$	−21.8476	−1.69062	−0.845310	+	0.534276i	$0.820584\pi$
$168$	0	0
$169$	16.9778	1.30599
$170$	0	0
$171$	4.66077	0.356418
$172$	0	0
$173$	−7.34072	−0.558105	−0.279052	−	0.960276i	$-0.590020\pi$
$173$	−7.34072	−0.558105	−0.279052	+	0.960276i	$0.590020\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.1345	0.912083
$178$	0	0
$179$	−9.23857	−0.690523	−0.345262	−	0.938506i	$-0.612210\pi$
$179$	−9.23857	−0.690523	−0.345262	+	0.938506i	$0.612210\pi$
$180$	0	0
$181$	−2.84335	−0.211345	−0.105672	−	0.994401i	$-0.533700\pi$
$181$	−2.84335	−0.211345	−0.105672	+	0.994401i	$0.533700\pi$
$182$	0	0
$183$	21.1036	1.56003
$184$	0	0
$185$	0.659278	0.0484711
$186$	0	0
$187$	−8.01923	−0.586424
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.8028	−0.926377	−0.463189	−	0.886260i	$-0.653295\pi$
$191$	−12.8028	−0.926377	−0.463189	+	0.886260i	$0.653295\pi$
$192$	0	0
$193$	11.3186	0.814728	0.407364	−	0.913266i	$-0.366448\pi$
$193$	11.3186	0.814728	0.407364	+	0.913266i	$0.366448\pi$
$194$	0	0
$195$	−8.37674	−0.599871
$196$	0	0
$197$	17.6097	1.25464	0.627319	−	0.778762i	$-0.284152\pi$
$197$	17.6097	1.25464	0.627319	+	0.778762i	$0.284152\pi$
$198$	0	0
$199$	15.8627	1.12447	0.562237	−	0.826976i	$-0.309941\pi$
$199$	15.8627	1.12447	0.562237	+	0.826976i	$0.309941\pi$
$200$	0	0
$201$	−11.0222	−0.777444
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.00000	−0.0698430
$206$	0	0
$207$	3.35847	0.233429
$208$	0	0
$209$	−9.24153	−0.639250
$210$	0	0
$211$	7.96031	0.548010	0.274005	−	0.961728i	$-0.411651\pi$
$211$	7.96031	0.548010	0.274005	+	0.961728i	$0.411651\pi$
$212$	0	0
$213$	13.4974	0.924824
$214$	0	0
$215$	9.10377	0.620872
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−7.36808	−0.497889
$220$	0	0
$221$	33.5875	2.25934
$222$	0	0
$223$	−12.8028	−0.857338	−0.428669	−	0.903462i	$-0.641017\pi$
$223$	−12.8028	−0.857338	−0.428669	+	0.903462i	$0.641017\pi$
$224$	0	0
$225$	−0.659278	−0.0439519
$226$	0	0
$227$	−2.34487	−0.155635	−0.0778173	−	0.996968i	$-0.524795\pi$
$227$	−2.34487	−0.155635	−0.0778173	+	0.996968i	$0.524795\pi$
$228$	0	0
$229$	−16.9504	−1.12011	−0.560057	−	0.828454i	$-0.689221\pi$
$229$	−16.9504	−1.12011	−0.560057	+	0.828454i	$0.689221\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.50263	0.294977	0.147489	−	0.989064i	$-0.452881\pi$
$233$	4.50263	0.294977	0.147489	+	0.989064i	$0.452881\pi$
$234$	0	0
$235$	−1.30724	−0.0852749
$236$	0	0
$237$	1.22841	0.0797936
$238$	0	0
$239$	−3.86279	−0.249863	−0.124932	−	0.992165i	$-0.539871\pi$
$239$	−3.86279	−0.249863	−0.124932	+	0.992165i	$0.539871\pi$
$240$	0	0
$241$	−13.6097	−0.876677	−0.438338	−	0.898810i	$-0.644433\pi$
$241$	−13.6097	−0.876677	−0.438338	+	0.898810i	$0.644433\pi$
$242$	0	0
$243$	−6.71693	−0.430891
$244$	0	0
$245$	0	0
$246$	0	0
$247$	38.7069	2.46286
$248$	0	0
$249$	22.6097	1.43283
$250$	0	0
$251$	4.72463	0.298216	0.149108	−	0.988821i	$-0.452360\pi$
$251$	4.72463	0.298216	0.149108	+	0.988821i	$0.452360\pi$
$252$	0	0
$253$	−6.65928	−0.418665
$254$	0	0
$255$	−9.38540	−0.587736
$256$	0	0
$257$	−0.268960	−0.0167773	−0.00838864	−	0.999965i	$-0.502670\pi$
$257$	−0.268960	−0.0167773	−0.00838864	+	0.999965i	$0.502670\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.95040	0.182625
$262$	0	0
$263$	−7.56181	−0.466281	−0.233141	−	0.972443i	$-0.574900\pi$
$263$	−7.56181	−0.466281	−0.233141	+	0.972443i	$0.574900\pi$
$264$	0	0
$265$	−7.47520	−0.459198
$266$	0	0
$267$	6.64105	0.406426
$268$	0	0
$269$	12.1619	0.741525	0.370762	−	0.928728i	$-0.379096\pi$
$269$	12.1619	0.741525	0.370762	+	0.928728i	$0.379096\pi$
$270$	0	0
$271$	−6.32551	−0.384248	−0.192124	−	0.981371i	$-0.561538\pi$
$271$	−6.32551	−0.384248	−0.192124	+	0.981371i	$0.561538\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.30724	0.0788295
$276$	0	0
$277$	−22.4034	−1.34609	−0.673046	−	0.739600i	$-0.735014\pi$
$277$	−22.4034	−1.34609	−0.673046	+	0.739600i	$0.735014\pi$
$278$	0	0
$279$	3.21164	0.192276
$280$	0	0
$281$	6.39032	0.381214	0.190607	−	0.981666i	$-0.438954\pi$
$281$	6.39032	0.381214	0.190607	+	0.981666i	$0.438954\pi$
$282$	0	0
$283$	−4.06854	−0.241850	−0.120925	−	0.992662i	$-0.538586\pi$
$283$	−4.06854	−0.241850	−0.120925	+	0.992662i	$0.538586\pi$
$284$	0	0
$285$	−10.8159	−0.640680
$286$	0	0
$287$	0	0
$288$	0	0
$289$	20.6318	1.21364
$290$	0	0
$291$	−27.9504	−1.63848
$292$	0	0
$293$	5.97784	0.349229	0.174614	−	0.984637i	$-0.444132\pi$
$293$	5.97784	0.349229	0.174614	+	0.984637i	$0.444132\pi$
$294$	0	0
$295$	7.93133	0.461780
$296$	0	0
$297$	7.31856	0.424666
$298$	0	0
$299$	27.8915	1.61301
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−9.90668	−0.569124
$304$	0	0
$305$	13.7938	0.789828
$306$	0	0
$307$	−21.3433	−1.21813	−0.609063	−	0.793122i	$-0.708454\pi$
$307$	−21.3433	−1.21813	−0.609063	+	0.793122i	$0.708454\pi$
$308$	0	0
$309$	−26.7442	−1.52142
$310$	0	0
$311$	8.82214	0.500258	0.250129	−	0.968213i	$-0.419527\pi$
$311$	8.82214	0.500258	0.250129	+	0.968213i	$0.419527\pi$
$312$	0	0
$313$	−10.6371	−0.601245	−0.300623	−	0.953743i	$-0.597194\pi$
$313$	−10.6371	−0.601245	−0.300623	+	0.953743i	$0.597194\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.1345	0.569209	0.284605	−	0.958645i	$-0.408138\pi$
$317$	10.1345	0.569209	0.284605	+	0.958645i	$0.408138\pi$
$318$	0	0
$319$	−5.85016	−0.327546
$320$	0	0
$321$	−3.02216	−0.168681
$322$	0	0
$323$	43.3677	2.41304
$324$	0	0
$325$	−5.47520	−0.303710
$326$	0	0
$327$	−1.29028	−0.0713529
$328$	0	0
$329$	0	0
$330$	0	0
$331$	23.3776	1.28495	0.642474	−	0.766308i	$-0.277908\pi$
$331$	23.3776	1.28495	0.642474	+	0.766308i	$0.277908\pi$
$332$	0	0
$333$	0.434648	0.0238186
$334$	0	0
$335$	−7.20430	−0.393613
$336$	0	0
$337$	1.18408	0.0645008	0.0322504	−	0.999480i	$-0.489733\pi$
$337$	1.18408	0.0645008	0.0322504	+	0.999480i	$0.489733\pi$
$338$	0	0
$339$	−5.91402	−0.321205
$340$	0	0
$341$	−6.36815	−0.344855
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−7.79376	−0.419602
$346$	0	0
$347$	22.3520	1.19992	0.599958	−	0.800031i	$-0.295184\pi$
$347$	22.3520	1.19992	0.599958	+	0.800031i	$0.295184\pi$
$348$	0	0
$349$	16.4752	0.881897	0.440949	−	0.897532i	$-0.354642\pi$
$349$	16.4752	0.881897	0.440949	+	0.897532i	$0.354642\pi$
$350$	0	0
$351$	−30.6528	−1.63613
$352$	0	0
$353$	−24.4034	−1.29886	−0.649432	−	0.760420i	$-0.724993\pi$
$353$	−24.4034	−1.29886	−0.649432	+	0.760420i	$0.724993\pi$
$354$	0	0
$355$	8.82214	0.468231
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.8907	−0.680344	−0.340172	−	0.940363i	$-0.610485\pi$
$359$	−12.8907	−0.680344	−0.340172	+	0.940363i	$0.610485\pi$
$360$	0	0
$361$	30.9778	1.63041
$362$	0	0
$363$	14.2149	0.746088
$364$	0	0
$365$	−4.81592	−0.252077
$366$	0	0
$367$	34.8262	1.81791	0.908957	−	0.416890i	$-0.136880\pi$
$367$	34.8262	1.81791	0.908957	+	0.416890i	$0.136880\pi$
$368$	0	0
$369$	−0.659278	−0.0343207
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6.95040	0.359878	0.179939	−	0.983678i	$-0.442410\pi$
$373$	6.95040	0.359878	0.179939	+	0.983678i	$0.442410\pi$
$374$	0	0
$375$	1.52994	0.0790059
$376$	0	0
$377$	24.5026	1.26195
$378$	0	0
$379$	−11.4656	−0.588948	−0.294474	−	0.955659i	$-0.595144\pi$
$379$	−11.4656	−0.588948	−0.294474	+	0.955659i	$0.595144\pi$
$380$	0	0
$381$	15.3629	0.787065
$382$	0	0
$383$	27.0766	1.38355	0.691775	−	0.722114i	$-0.256829\pi$
$383$	27.0766	1.38355	0.691775	+	0.722114i	$0.256829\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.00192	0.305095
$388$	0	0
$389$	−7.58752	−0.384702	−0.192351	−	0.981326i	$-0.561611\pi$
$389$	−7.58752	−0.384702	−0.192351	+	0.981326i	$0.561611\pi$
$390$	0	0
$391$	31.2500	1.58038
$392$	0	0
$393$	30.9504	1.56124
$394$	0	0
$395$	0.802911	0.0403988
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.9504	0.696650	0.348325	−	0.937374i	$-0.386751\pi$
$401$	13.9504	0.696650	0.348325	+	0.937374i	$0.386751\pi$
$402$	0	0
$403$	26.6722	1.32864
$404$	0	0
$405$	6.58752	0.327336
$406$	0	0
$407$	−0.861835	−0.0427196
$408$	0	0
$409$	−8.34072	−0.412422	−0.206211	−	0.978508i	$-0.566113\pi$
$409$	−8.34072	−0.412422	−0.206211	+	0.978508i	$0.566113\pi$
$410$	0	0
$411$	−31.0103	−1.52963
$412$	0	0
$413$	0	0
$414$	0	0
$415$	14.7781	0.725430
$416$	0	0
$417$	−11.5875	−0.567443
$418$	0	0
$419$	1.75264	0.0856223	0.0428111	−	0.999083i	$-0.486369\pi$
$419$	1.75264	0.0856223	0.0428111	+	0.999083i	$0.486369\pi$
$420$	0	0
$421$	−1.89295	−0.0922568	−0.0461284	−	0.998936i	$-0.514688\pi$
$421$	−1.89295	−0.0922568	−0.0461284	+	0.998936i	$0.514688\pi$
$422$	0	0
$423$	−0.861835	−0.0419038
$424$	0	0
$425$	−6.13448	−0.297566
$426$	0	0
$427$	0	0
$428$	0	0
$429$	10.9504	0.528690
$430$	0	0
$431$	27.9504	1.34633	0.673163	−	0.739494i	$-0.264935\pi$
$431$	27.9504	1.34633	0.673163	+	0.739494i	$0.264935\pi$
$432$	0	0
$433$	20.4034	0.980527	0.490263	−	0.871574i	$-0.336901\pi$
$433$	20.4034	0.980527	0.490263	+	0.871574i	$0.336901\pi$
$434$	0	0
$435$	−6.84680	−0.328279
$436$	0	0
$437$	36.0131	1.72274
$438$	0	0
$439$	−7.12842	−0.340221	−0.170111	−	0.985425i	$-0.554412\pi$
$439$	−7.12842	−0.340221	−0.170111	+	0.985425i	$0.554412\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11.7182	−0.556751	−0.278375	−	0.960472i	$-0.589796\pi$
$443$	−11.7182	−0.556751	−0.278375	+	0.960472i	$0.589796\pi$
$444$	0	0
$445$	4.34072	0.205770
$446$	0	0
$447$	25.6176	1.21167
$448$	0	0
$449$	37.2690	1.75883	0.879415	−	0.476055i	$-0.157934\pi$
$449$	37.2690	1.75883	0.879415	+	0.476055i	$0.157934\pi$
$450$	0	0
$451$	1.30724	0.0615555
$452$	0	0
$453$	3.45304	0.162238
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.3186	1.18435	0.592176	−	0.805809i	$-0.298269\pi$
$457$	25.3186	1.18435	0.592176	+	0.805809i	$0.298269\pi$
$458$	0	0
$459$	−34.3438	−1.60303
$460$	0	0
$461$	−26.2690	−1.22347	−0.611734	−	0.791064i	$-0.709528\pi$
$461$	−26.2690	−1.22347	−0.611734	+	0.791064i	$0.709528\pi$
$462$	0	0
$463$	−24.4621	−1.13685	−0.568425	−	0.822735i	$-0.692447\pi$
$463$	−24.4621	−1.13685	−0.568425	+	0.822735i	$0.692447\pi$
$464$	0	0
$465$	−7.45304	−0.345626
$466$	0	0
$467$	38.6600	1.78897	0.894487	−	0.447095i	$-0.147541\pi$
$467$	38.6600	1.78897	0.894487	+	0.447095i	$0.147541\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	8.58249	0.395460
$472$	0	0
$473$	−11.9008	−0.547200
$474$	0	0
$475$	−7.06950	−0.324371
$476$	0	0
$477$	−4.92824	−0.225649
$478$	0	0
$479$	−10.9912	−0.502202	−0.251101	−	0.967961i	$-0.580793\pi$
$479$	−10.9912	−0.502202	−0.251101	+	0.967961i	$0.580793\pi$
$480$	0	0
$481$	3.60968	0.164587
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−18.2690	−0.829551
$486$	0	0
$487$	−33.3552	−1.51147	−0.755734	−	0.654878i	$-0.772720\pi$
$487$	−33.3552	−1.51147	−0.755734	+	0.654878i	$0.772720\pi$
$488$	0	0
$489$	37.2194	1.68312
$490$	0	0
$491$	1.24832	0.0563357	0.0281678	−	0.999603i	$-0.491033\pi$
$491$	1.24832	0.0563357	0.0281678	+	0.999603i	$0.491033\pi$
$492$	0	0
$493$	27.4530	1.23642
$494$	0	0
$495$	0.861835	0.0387366
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−35.6760	−1.59708	−0.798539	−	0.601943i	$-0.794394\pi$
$499$	−35.6760	−1.59708	−0.798539	+	0.601943i	$0.794394\pi$
$500$	0	0
$501$	33.4256	1.49335
$502$	0	0
$503$	28.8292	1.28543	0.642716	−	0.766105i	$-0.277808\pi$
$503$	28.8292	1.28543	0.642716	+	0.766105i	$0.277808\pi$
$504$	0	0
$505$	−6.47520	−0.288143
$506$	0	0
$507$	−25.9751	−1.15359
$508$	0	0
$509$	20.7442	0.919469	0.459734	−	0.888056i	$-0.347945\pi$
$509$	20.7442	0.919469	0.459734	+	0.888056i	$0.347945\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−39.5785	−1.74743
$514$	0	0
$515$	−17.4805	−0.770283
$516$	0	0
$517$	1.70887	0.0751562
$518$	0	0
$519$	11.2309	0.492981
$520$	0	0
$521$	−9.02743	−0.395499	−0.197749	−	0.980253i	$-0.563363\pi$
$521$	−9.02743	−0.395499	−0.197749	+	0.980253i	$0.563363\pi$
$522$	0	0
$523$	3.77490	0.165065	0.0825323	−	0.996588i	$-0.473699\pi$
$523$	3.77490	0.165065	0.0825323	+	0.996588i	$0.473699\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	29.8838	1.30176
$528$	0	0
$529$	2.95040	0.128278
$530$	0	0
$531$	5.22896	0.226917
$532$	0	0
$533$	−5.47520	−0.237157
$534$	0	0
$535$	−1.97535	−0.0854017
$536$	0	0
$537$	14.1345	0.609948
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−34.7442	−1.49377	−0.746884	−	0.664954i	$-0.768451\pi$
$541$	−34.7442	−1.49377	−0.746884	+	0.664954i	$0.768451\pi$
$542$	0	0
$543$	4.35017	0.186684
$544$	0	0
$545$	−0.843355	−0.0361254
$546$	0	0
$547$	−10.4400	−0.446382	−0.223191	−	0.974775i	$-0.571647\pi$
$547$	−10.4400	−0.446382	−0.223191	+	0.974775i	$0.571647\pi$
$548$	0	0
$549$	9.09393	0.388119
$550$	0	0
$551$	31.6374	1.34780
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.00866	−0.0428151
$556$	0	0
$557$	21.8786	0.927028	0.463514	−	0.886090i	$-0.346588\pi$
$557$	21.8786	0.927028	0.463514	+	0.886090i	$0.346588\pi$
$558$	0	0
$559$	49.8450	2.10822
$560$	0	0
$561$	12.2690	0.517996
$562$	0	0
$563$	9.81878	0.413812	0.206906	−	0.978361i	$-0.433661\pi$
$563$	9.81878	0.413812	0.206906	+	0.978361i	$0.433661\pi$
$564$	0	0
$565$	−3.86552	−0.162624
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.9282	1.12889	0.564445	−	0.825471i	$-0.309090\pi$
$569$	26.9282	1.12889	0.564445	+	0.825471i	$0.309090\pi$
$570$	0	0
$571$	3.41739	0.143013	0.0715066	−	0.997440i	$-0.477219\pi$
$571$	3.41739	0.143013	0.0715066	+	0.997440i	$0.477219\pi$
$572$	0	0
$573$	19.5875	0.818280
$574$	0	0
$575$	−5.09415	−0.212441
$576$	0	0
$577$	34.8512	1.45087	0.725437	−	0.688288i	$-0.241637\pi$
$577$	34.8512	1.45087	0.725437	+	0.688288i	$0.241637\pi$
$578$	0	0
$579$	−17.3167	−0.719659
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.77188	0.404710
$584$	0	0
$585$	−3.60968	−0.149242
$586$	0	0
$587$	15.4752	0.638730	0.319365	−	0.947632i	$-0.396530\pi$
$587$	15.4752	0.638730	0.319365	+	0.947632i	$0.396530\pi$
$588$	0	0
$589$	34.4387	1.41902
$590$	0	0
$591$	−26.9418	−1.10824
$592$	0	0
$593$	−12.4034	−0.509348	−0.254674	−	0.967027i	$-0.581968\pi$
$593$	−12.4034	−0.509348	−0.254674	+	0.967027i	$0.581968\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−24.2690	−0.993262
$598$	0	0
$599$	−32.7340	−1.33747	−0.668737	−	0.743499i	$-0.733165\pi$
$599$	−32.7340	−1.33747	−0.668737	+	0.743499i	$0.733165\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	−4.74964	−0.193420
$604$	0	0
$605$	9.29113	0.377738
$606$	0	0
$607$	−35.2137	−1.42928	−0.714639	−	0.699493i	$-0.753409\pi$
$607$	−35.2137	−1.42928	−0.714639	+	0.699493i	$0.753409\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.15740	−0.289557
$612$	0	0
$613$	16.6946	0.674287	0.337144	−	0.941453i	$-0.390539\pi$
$613$	16.6946	0.674287	0.337144	+	0.941453i	$0.390539\pi$
$614$	0	0
$615$	1.52994	0.0616932
$616$	0	0
$617$	31.9557	1.28649	0.643243	−	0.765662i	$-0.277588\pi$
$617$	31.9557	1.28649	0.643243	+	0.765662i	$0.277588\pi$
$618$	0	0
$619$	−47.4652	−1.90779	−0.953894	−	0.300143i	$-0.902966\pi$
$619$	−47.4652	−1.90779	−0.953894	+	0.300143i	$0.902966\pi$
$620$	0	0
$621$	−28.5195	−1.14445
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	14.1390	0.564657
$628$	0	0
$629$	4.04433	0.161258
$630$	0	0
$631$	23.6123	0.939991	0.469995	−	0.882669i	$-0.344256\pi$
$631$	23.6123	0.939991	0.469995	+	0.882669i	$0.344256\pi$
$632$	0	0
$633$	−12.1788	−0.484064
$634$	0	0
$635$	10.0415	0.398484
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.81625	0.230087
$640$	0	0
$641$	36.9008	1.45749	0.728747	−	0.684783i	$-0.240103\pi$
$641$	36.9008	1.45749	0.728747	+	0.684783i	$0.240103\pi$
$642$	0	0
$643$	18.0897	0.713388	0.356694	−	0.934221i	$-0.383904\pi$
$643$	18.0897	0.713388	0.356694	+	0.934221i	$0.383904\pi$
$644$	0	0
$645$	−13.9282	−0.548424
$646$	0	0
$647$	−16.7365	−0.657981	−0.328990	−	0.944333i	$-0.606708\pi$
$647$	−16.7365	−0.657981	−0.328990	+	0.944333i	$0.606708\pi$
$648$	0	0
$649$	−10.3682	−0.406986
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.0627	−0.745982	−0.372991	−	0.927835i	$-0.621668\pi$
$653$	−19.0627	−0.745982	−0.372991	+	0.927835i	$0.621668\pi$
$654$	0	0
$655$	20.2298	0.790443
$656$	0	0
$657$	−3.17503	−0.123870
$658$	0	0
$659$	−23.7940	−0.926883	−0.463441	−	0.886128i	$-0.653386\pi$
$659$	−23.7940	−0.926883	−0.463441	+	0.886128i	$0.653386\pi$
$660$	0	0
$661$	−12.7885	−0.497415	−0.248707	−	0.968579i	$-0.580006\pi$
$661$	−12.7885	−0.497415	−0.248707	+	0.968579i	$0.580006\pi$
$662$	0	0
$663$	−51.3869	−1.99570
$664$	0	0
$665$	0	0
$666$	0	0
$667$	22.7974	0.882718
$668$	0	0
$669$	19.5875	0.757297
$670$	0	0
$671$	−18.0317	−0.696108
$672$	0	0
$673$	2.68144	0.103362	0.0516810	−	0.998664i	$-0.483542\pi$
$673$	2.68144	0.103362	0.0516810	+	0.998664i	$0.483542\pi$
$674$	0	0
$675$	5.59848	0.215486
$676$	0	0
$677$	−46.2821	−1.77876	−0.889382	−	0.457164i	$-0.848865\pi$
$677$	−46.2821	−1.77876	−0.889382	+	0.457164i	$0.848865\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	3.58752	0.137474
$682$	0	0
$683$	−11.0032	−0.421027	−0.210514	−	0.977591i	$-0.567514\pi$
$683$	−11.0032	−0.421027	−0.210514	+	0.977591i	$0.567514\pi$
$684$	0	0
$685$	−20.2690	−0.774437
$686$	0	0
$687$	25.9331	0.989411
$688$	0	0
$689$	−40.9282	−1.55924
$690$	0	0
$691$	22.5457	0.857678	0.428839	−	0.903381i	$-0.358923\pi$
$691$	22.5457	0.857678	0.428839	+	0.903381i	$0.358923\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.57383	−0.287292
$696$	0	0
$697$	−6.13448	−0.232360
$698$	0	0
$699$	−6.88877	−0.260557
$700$	0	0
$701$	−29.6946	−1.12155	−0.560774	−	0.827969i	$-0.689496\pi$
$701$	−29.6946	−1.12155	−0.560774	+	0.827969i	$0.689496\pi$
$702$	0	0
$703$	4.66077	0.175784
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−9.83809	−0.369477	−0.184739	−	0.982788i	$-0.559144\pi$
$709$	−9.83809	−0.369477	−0.184739	+	0.982788i	$0.559144\pi$
$710$	0	0
$711$	0.529342	0.0198519
$712$	0	0
$713$	24.8159	0.929364
$714$	0	0
$715$	7.15740	0.267672
$716$	0	0
$717$	5.90985	0.220707
$718$	0	0
$719$	−20.5284	−0.765579	−0.382790	−	0.923836i	$-0.625037\pi$
$719$	−20.5284	−0.765579	−0.382790	+	0.923836i	$0.625037\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	20.8220	0.774379
$724$	0	0
$725$	−4.47520	−0.166205
$726$	0	0
$727$	−22.0234	−0.816804	−0.408402	−	0.912802i	$-0.633914\pi$
$727$	−22.0234	−0.816804	−0.408402	+	0.912802i	$0.633914\pi$
$728$	0	0
$729$	30.0391	1.11256
$730$	0	0
$731$	55.8469	2.06557
$732$	0	0
$733$	−6.65928	−0.245966	−0.122983	−	0.992409i	$-0.539246\pi$
$733$	−6.65928	−0.245966	−0.122983	+	0.992409i	$0.539246\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.41775	0.346907
$738$	0	0
$739$	20.0780	0.738582	0.369291	−	0.929314i	$-0.379600\pi$
$739$	20.0780	0.738582	0.369291	+	0.929314i	$0.379600\pi$
$740$	0	0
$741$	−59.2194	−2.17548
$742$	0	0
$743$	−39.6098	−1.45314	−0.726571	−	0.687092i	$-0.758887\pi$
$743$	−39.6098	−1.45314	−0.726571	+	0.687092i	$0.758887\pi$
$744$	0	0
$745$	16.7442	0.613459
$746$	0	0
$747$	9.74290	0.356474
$748$	0	0
$749$	0	0
$750$	0	0
$751$	14.1390	0.515939	0.257970	−	0.966153i	$-0.416947\pi$
$751$	14.1390	0.515939	0.257970	+	0.966153i	$0.416947\pi$
$752$	0	0
$753$	−7.22841	−0.263418
$754$	0	0
$755$	2.25697	0.0821397
$756$	0	0
$757$	−49.9361	−1.81496	−0.907479	−	0.420097i	$-0.861996\pi$
$757$	−49.9361	−1.81496	−0.907479	+	0.420097i	$0.861996\pi$
$758$	0	0
$759$	10.1883	0.369812
$760$	0	0
$761$	−22.6593	−0.821398	−0.410699	−	0.911771i	$-0.634715\pi$
$761$	−22.6593	−0.821398	−0.410699	+	0.911771i	$0.634715\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.04433	−0.146223
$766$	0	0
$767$	43.4257	1.56801
$768$	0	0
$769$	−7.97784	−0.287688	−0.143844	−	0.989600i	$-0.545946\pi$
$769$	−7.97784	−0.287688	−0.143844	+	0.989600i	$0.545946\pi$
$770$	0	0
$771$	0.411494	0.0148196
$772$	0	0
$773$	−26.0575	−0.937221	−0.468611	−	0.883405i	$-0.655245\pi$
$773$	−26.0575	−0.937221	−0.468611	+	0.883405i	$0.655245\pi$
$774$	0	0
$775$	−4.87145	−0.174988
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.06950	−0.253291
$780$	0	0
$781$	−11.5327	−0.412671
$782$	0	0
$783$	−25.0543	−0.895369
$784$	0	0
$785$	5.60968	0.200218
$786$	0	0
$787$	41.1567	1.46708	0.733538	−	0.679648i	$-0.237867\pi$
$787$	41.1567	1.46708	0.733538	+	0.679648i	$0.237867\pi$
$788$	0	0
$789$	11.5691	0.411872
$790$	0	0
$791$	0	0
$792$	0	0
$793$	75.5236	2.68192
$794$	0	0
$795$	11.4366	0.405615
$796$	0	0
$797$	45.8565	1.62432	0.812160	−	0.583435i	$-0.198292\pi$
$797$	45.8565	1.62432	0.812160	+	0.583435i	$0.198292\pi$
$798$	0	0
$799$	−8.01923	−0.283700
$800$	0	0
$801$	2.86174	0.101115
$802$	0	0
$803$	6.29556	0.222166
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−18.6070	−0.654998
$808$	0	0
$809$	−54.7573	−1.92516	−0.962582	−	0.270991i	$-0.912649\pi$
$809$	−54.7573	−1.92516	−0.962582	+	0.270991i	$0.912649\pi$
$810$	0	0
$811$	27.8915	0.979404	0.489702	−	0.871890i	$-0.337106\pi$
$811$	27.8915	0.979404	0.489702	+	0.871890i	$0.337106\pi$
$812$	0	0
$813$	9.67767	0.339411
$814$	0	0
$815$	24.3273	0.852148
$816$	0	0
$817$	64.3591	2.25164
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.1698	0.564329	0.282164	−	0.959366i	$-0.408948\pi$
$821$	16.1698	0.564329	0.282164	+	0.959366i	$0.408948\pi$
$822$	0	0
$823$	−57.1072	−1.99063	−0.995317	−	0.0966693i	$-0.969181\pi$
$823$	−57.1072	−1.99063	−0.995317	+	0.0966693i	$0.969181\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	19.5617	0.680226	0.340113	−	0.940384i	$-0.389535\pi$
$827$	19.5617	0.680226	0.340113	+	0.940384i	$0.389535\pi$
$828$	0	0
$829$	34.2690	1.19021	0.595105	−	0.803648i	$-0.297110\pi$
$829$	34.2690	1.19021	0.595105	+	0.803648i	$0.297110\pi$
$830$	0	0
$831$	34.2760	1.18902
$832$	0	0
$833$	0	0
$834$	0	0
$835$	21.8476	0.756069
$836$	0	0
$837$	−27.2727	−0.942684
$838$	0	0
$839$	14.4666	0.499441	0.249721	−	0.968318i	$-0.419661\pi$
$839$	14.4666	0.499441	0.249721	+	0.968318i	$0.419661\pi$
$840$	0	0
$841$	−8.97257	−0.309399
$842$	0	0
$843$	−9.77681	−0.336731
$844$	0	0
$845$	−16.9778	−0.584055
$846$	0	0
$847$	0	0
$848$	0	0
$849$	6.22463	0.213629
$850$	0	0
$851$	3.35847	0.115127
$852$	0	0
$853$	−29.3760	−1.00582	−0.502908	−	0.864340i	$-0.667736\pi$
$853$	−29.3760	−1.00582	−0.502908	+	0.864340i	$0.667736\pi$
$854$	0	0
$855$	−4.66077	−0.159395
$856$	0	0
$857$	−25.2637	−0.862991	−0.431496	−	0.902115i	$-0.642014\pi$
$857$	−25.2637	−0.862991	−0.431496	+	0.902115i	$0.642014\pi$
$858$	0	0
$859$	44.8557	1.53046	0.765228	−	0.643759i	$-0.222626\pi$
$859$	44.8557	1.53046	0.765228	+	0.643759i	$0.222626\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.6011	−1.31400	−0.656999	−	0.753892i	$-0.728174\pi$
$863$	−38.6011	−1.31400	−0.656999	+	0.753892i	$0.728174\pi$
$864$	0	0
$865$	7.34072	0.249592
$866$	0	0
$867$	−31.5655	−1.07202
$868$	0	0
$869$	−1.04960	−0.0356051
$870$	0	0
$871$	−39.4450	−1.33654
$872$	0	0
$873$	−12.0443	−0.407639
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−16.8381	−0.568582	−0.284291	−	0.958738i	$-0.591758\pi$
$877$	−16.8381	−0.568582	−0.284291	+	0.958738i	$0.591758\pi$
$878$	0	0
$879$	−9.14574	−0.308478
$880$	0	0
$881$	−0.0495963	−0.00167094	−0.000835472	−	1.00000i	$-0.500266\pi$
$881$	−0.0495963	−0.00167094	−0.000835472		1.00000i	$0.500266\pi$
$882$	0	0
$883$	−7.57383	−0.254880	−0.127440	−	0.991846i	$-0.540676\pi$
$883$	−7.57383	−0.254880	−0.127440	+	0.991846i	$0.540676\pi$
$884$	0	0
$885$	−12.1345	−0.407896
$886$	0	0
$887$	32.0649	1.07663	0.538317	−	0.842742i	$-0.319060\pi$
$887$	32.0649	1.07663	0.538317	+	0.842742i	$0.319060\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−8.61146	−0.288495
$892$	0	0
$893$	−9.24153	−0.309256
$894$	0	0
$895$	9.23857	0.308811
$896$	0	0
$897$	−42.6724	−1.42479
$898$	0	0
$899$	21.8007	0.727095
$900$	0	0
$901$	−45.8565	−1.52770
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.84335	0.0945163
$906$	0	0
$907$	5.48063	0.181981	0.0909907	−	0.995852i	$-0.470997\pi$
$907$	5.48063	0.181981	0.0909907	+	0.995852i	$0.470997\pi$
$908$	0	0
$909$	−4.26896	−0.141592
$910$	0	0
$911$	19.0404	0.630837	0.315418	−	0.948953i	$-0.397855\pi$
$911$	19.0404	0.630837	0.315418	+	0.948953i	$0.397855\pi$
$912$	0	0
$913$	−19.3186	−0.639351
$914$	0	0
$915$	−21.1036	−0.697665
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3.29954	0.108842	0.0544209	−	0.998518i	$-0.482669\pi$
$919$	3.29954	0.108842	0.0544209	+	0.998518i	$0.482669\pi$
$920$	0	0
$921$	32.6540	1.07599
$922$	0	0
$923$	48.3030	1.58991
$924$	0	0
$925$	−0.659278	−0.0216769
$926$	0	0
$927$	−11.5245	−0.378515
$928$	0	0
$929$	37.9061	1.24366	0.621829	−	0.783153i	$-0.286390\pi$
$929$	37.9061	1.24366	0.621829	+	0.783153i	$0.286390\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−13.4974	−0.441884
$934$	0	0
$935$	8.01923	0.262257
$936$	0	0
$937$	13.3186	0.435098	0.217549	−	0.976049i	$-0.430194\pi$
$937$	13.3186	0.435098	0.217549	+	0.976049i	$0.430194\pi$
$938$	0	0
$939$	16.2742	0.531087
$940$	0	0
$941$	11.5875	0.377742	0.188871	−	0.982002i	$-0.439517\pi$
$941$	11.5875	0.377742	0.188871	+	0.982002i	$0.439517\pi$
$942$	0	0
$943$	−5.09415	−0.165888
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.7161	−1.06313	−0.531565	−	0.847018i	$-0.678396\pi$
$947$	−32.7161	−1.06313	−0.531565	+	0.847018i	$0.678396\pi$
$948$	0	0
$949$	−26.3682	−0.855946
$950$	0	0
$951$	−15.5052	−0.502789
$952$	0	0
$953$	27.7310	0.898296	0.449148	−	0.893457i	$-0.351728\pi$
$953$	27.7310	0.898296	0.449148	+	0.893457i	$0.351728\pi$
$954$	0	0
$955$	12.8028	0.414288
$956$	0	0
$957$	8.95040	0.289325
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−7.26896	−0.234483
$962$	0	0
$963$	−1.30230	−0.0419661
$964$	0	0
$965$	−11.3186	−0.364357
$966$	0	0
$967$	10.4699	0.336690	0.168345	−	0.985728i	$-0.446158\pi$
$967$	10.4699	0.336690	0.168345	+	0.985728i	$0.446158\pi$
$968$	0	0
$969$	−66.3501	−2.13147
$970$	0	0
$971$	8.76322	0.281225	0.140613	−	0.990065i	$-0.455093\pi$
$971$	8.76322	0.281225	0.140613	+	0.990065i	$0.455093\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	8.37674	0.268270
$976$	0	0
$977$	−50.2690	−1.60825	−0.804123	−	0.594463i	$-0.797365\pi$
$977$	−50.2690	−1.60825	−0.804123	+	0.594463i	$0.797365\pi$
$978$	0	0
$979$	−5.67436	−0.181353
$980$	0	0
$981$	−0.556006	−0.0177519
$982$	0	0
$983$	1.76466	0.0562840	0.0281420	−	0.999604i	$-0.491041\pi$
$983$	1.76466	0.0562840	0.0281420	+	0.999604i	$0.491041\pi$
$984$	0	0
$985$	−17.6097	−0.561091
$986$	0	0
$987$	0	0
$988$	0	0
$989$	46.3760	1.47467
$990$	0	0
$991$	−17.3467	−0.551035	−0.275518	−	0.961296i	$-0.588849\pi$
$991$	−17.3467	−0.551035	−0.275518	+	0.961296i	$0.588849\pi$
$992$	0	0
$993$	−35.7663	−1.13501
$994$	0	0
$995$	−15.8627	−0.502880
$996$	0	0
$997$	−28.4478	−0.900950	−0.450475	−	0.892789i	$-0.648745\pi$
$997$	−28.4478	−0.900950	−0.450475	+	0.892789i	$0.648745\pi$
$998$	0	0
$999$	−3.69096	−0.116777

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7840.2.a.cd.1.2		6
4.3	odd	2	inner	7840.2.a.cd.1.5		6
7.3	odd	6		1120.2.q.k.961.2	yes	12
7.5	odd	6		1120.2.q.k.641.2	✓	12
7.6	odd	2		7840.2.a.ce.1.5		6
28.3	even	6		1120.2.q.k.961.5	yes	12
28.19	even	6		1120.2.q.k.641.5	yes	12
28.27	even	2		7840.2.a.ce.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.q.k.641.2	✓	12	7.5	odd	6
1120.2.q.k.641.5	yes	12	28.19	even	6
1120.2.q.k.961.2	yes	12	7.3	odd	6
1120.2.q.k.961.5	yes	12	28.3	even	6
7840.2.a.cd.1.2		6	1.1	even	1	trivial
7840.2.a.cd.1.5		6	4.3	odd	2	inner
7840.2.a.ce.1.2		6	28.27	even	2
7840.2.a.ce.1.5		6	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7840 = 2^{5} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7840.a (trivial)

\(f(q)\)	\(=\)	\(q-1.52994 q^{3} -1.00000 q^{5} -0.659278 q^{9} +O(q^{10})\)	\(q-1.52994 q^{3} -1.00000 q^{5} -0.659278 q^{9} +1.30724 q^{11} -5.47520 q^{13} +1.52994 q^{15} -6.13448 q^{17} -7.06950 q^{19} -5.09415 q^{23} +1.00000 q^{25} +5.59848 q^{27} -4.47520 q^{29} -4.87145 q^{31} -2.00000 q^{33} -0.659278 q^{37} +8.37674 q^{39} +1.00000 q^{41} -9.10377 q^{43} +0.659278 q^{45} +1.30724 q^{47} +9.38540 q^{51} +7.47520 q^{53} -1.30724 q^{55} +10.8159 q^{57} -7.93133 q^{59} -13.7938 q^{61} +5.47520 q^{65} +7.20430 q^{67} +7.79376 q^{69} -8.82214 q^{71} +4.81592 q^{73} -1.52994 q^{75} -0.802911 q^{79} -6.58752 q^{81} -14.7781 q^{83} +6.13448 q^{85} +6.84680 q^{87} -4.34072 q^{89} +7.45304 q^{93} +7.06950 q^{95} +18.2690 q^{97} -0.861835 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{5} + 10 q^{9}+O(q^{10}) \) 6 * q - 6 * q^5 + 10 * q^9	\( 6 q - 6 q^{5} + 10 q^{9} - 2 q^{13} + 8 q^{17} + 6 q^{25} + 4 q^{29} - 12 q^{33} + 10 q^{37} + 6 q^{41} - 10 q^{45} + 14 q^{53} + 48 q^{57} - 24 q^{61} + 2 q^{65} - 12 q^{69} + 12 q^{73} + 78 q^{81} - 8 q^{85} - 40 q^{89} - 28 q^{93} + 20 q^{97}+O(q^{100}) \) 6 * q - 6 * q^5 + 10 * q^9 - 2 * q^13 + 8 * q^17 + 6 * q^25 + 4 * q^29 - 12 * q^33 + 10 * q^37 + 6 * q^41 - 10 * q^45 + 14 * q^53 + 48 * q^57 - 24 * q^61 + 2 * q^65 - 12 * q^69 + 12 * q^73 + 78 * q^81 - 8 * q^85 - 40 * q^89 - 28 * q^93 + 20 * q^97

Introduction

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