LMFDB - Embedded newform 2704.2.f.m.337.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2704,2,Mod(337,2704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2704.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$21.5915487066$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 338)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.4
Root			$1.24698i$ of defining polynomial
Character	$\chi$	$=$	2704.337
Dual form			2704.2.f.m.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35690 q^{3} +0.890084i q^{5} -4.49396i q^{7} +2.55496 q^{9} +2.69202i q^{11} -2.09783i q^{15} -3.58211 q^{17} +2.93900i q^{19} +10.5918i q^{21} -6.09783 q^{23} +4.20775 q^{25} +1.04892 q^{27} +2.98792 q^{29} +2.39612i q^{31} -6.34481i q^{33} +4.00000 q^{35} +1.50604i q^{37} -3.65279i q^{41} +0.170915 q^{43} +2.27413i q^{45} +5.20775i q^{47} -13.1957 q^{49} +8.44265 q^{51} +6.09783 q^{53} -2.39612 q^{55} -6.92692i q^{57} +3.07069i q^{59} -13.9758 q^{61} -11.4819i q^{63} -11.0707i q^{67} +14.3720 q^{69} +10.0978i q^{71} -10.9487i q^{73} -9.91723 q^{75} +12.0978 q^{77} -2.81163 q^{79} -10.1371 q^{81} -2.93900i q^{83} -3.18837i q^{85} -7.04221 q^{87} -12.1806i q^{89} -5.64742i q^{93} -2.61596 q^{95} +12.9661i q^{97} +6.87800i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 16 q^{9} - 10 q^{17} - 10 q^{25} - 12 q^{27} - 20 q^{29} + 24 q^{35} + 22 q^{43} - 6 q^{49} - 32 q^{51} - 32 q^{55} - 8 q^{61} + 28 q^{69} - 46 q^{75} + 36 q^{77} + 36 q^{79} - 50 q^{81}+ \cdots - 36 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 + 16 * q^9 - 10 * q^17 - 10 * q^25 - 12 * q^27 - 20 * q^29 + 24 * q^35 + 22 * q^43 - 6 * q^49 - 32 * q^51 - 32 * q^55 - 8 * q^61 + 28 * q^69 - 46 * q^75 + 36 * q^77 + 36 * q^79 - 50 * q^81 + 20 * q^87 - 36 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$.

$n$	$677$	$1185$	$2367$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	−2.35690	−1.36075	−0.680377	−	0.732862i	$-0.738184\pi$
$3$	−2.35690	−1.36075	−0.680377	+	0.732862i	$0.738184\pi$
$4$	0	0
$5$	0.890084i	0.398058i	0.979994	+	0.199029i	$0.0637787\pi$
$5$	0.890084i	0.398058i	−0.979994	+	0.199029i	$0.936221\pi$
$6$	0	0
$7$	− 4.49396i	− 1.69856i	−0.527945	−	0.849278i	$-0.677037\pi$
$7$	− 4.49396i	− 1.69856i	0.527945	−	0.849278i	$-0.322963\pi$
$8$	0	0
$9$	2.55496	0.851653
$10$	0	0
$11$	2.69202i	0.811675i	0.913945	+	0.405838i	$0.133020\pi$
$11$	2.69202i	0.811675i	−0.913945	+	0.405838i	$0.866980\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 2.09783i	− 0.541659i
$16$	0	0
$17$	−3.58211	−0.868788	−0.434394	−	0.900723i	$-0.643037\pi$
$17$	−3.58211	−0.868788	−0.434394	+	0.900723i	$0.643037\pi$
$18$	0	0
$19$	2.93900i	0.674253i	0.941459	+	0.337127i	$0.109455\pi$
$19$	2.93900i	0.674253i	−0.941459	+	0.337127i	$0.890545\pi$
$20$	0	0
$21$	10.5918i	2.31132i
$22$	0	0
$23$	−6.09783	−1.27149	−0.635743	−	0.771901i	$-0.719306\pi$
$23$	−6.09783	−1.27149	−0.635743	+	0.771901i	$0.719306\pi$
$24$	0	0
$25$	4.20775	0.841550
$26$	0	0
$27$	1.04892	0.201864
$28$	0	0
$29$	2.98792	0.554843	0.277421	−	0.960748i	$-0.410520\pi$
$29$	2.98792	0.554843	0.277421	+	0.960748i	$0.410520\pi$
$30$	0	0
$31$	2.39612i	0.430357i	0.976575	+	0.215178i	$0.0690333\pi$
$31$	2.39612i	0.430357i	−0.976575	+	0.215178i	$0.930967\pi$
$32$	0	0
$33$	− 6.34481i	− 1.10449i
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	1.50604i	0.247592i	0.992308	+	0.123796i	$0.0395068\pi$
$37$	1.50604i	0.247592i	−0.992308	+	0.123796i	$0.960493\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.65279i	− 0.570470i	−0.958458	−	0.285235i	$-0.907928\pi$
$41$	− 3.65279i	− 0.570470i	0.958458	−	0.285235i	$-0.0920717\pi$
$42$	0	0
$43$	0.170915	0.0260643	0.0130322	−	0.999915i	$-0.495852\pi$
$43$	0.170915	0.0260643	0.0130322	+	0.999915i	$0.495852\pi$
$44$	0	0
$45$	2.27413i	0.339007i
$46$	0	0
$47$	5.20775i	0.759629i	0.925063	+	0.379814i	$0.124012\pi$
$47$	5.20775i	0.759629i	−0.925063	+	0.379814i	$0.875988\pi$
$48$	0	0
$49$	−13.1957	−1.88510
$50$	0	0
$51$	8.44265	1.18221
$52$	0	0
$53$	6.09783	0.837602	0.418801	−	0.908078i	$-0.362450\pi$
$53$	6.09783	0.837602	0.418801	+	0.908078i	$0.362450\pi$
$54$	0	0
$55$	−2.39612	−0.323093
$56$	0	0
$57$	− 6.92692i	− 0.917493i
$58$	0	0
$59$	3.07069i	0.399769i	0.979819	+	0.199885i	$0.0640568\pi$
$59$	3.07069i	0.399769i	−0.979819	+	0.199885i	$0.935943\pi$
$60$	0	0
$61$	−13.9758	−1.78942	−0.894711	−	0.446645i	$-0.852619\pi$
$61$	−13.9758	−1.78942	−0.894711	+	0.446645i	$0.852619\pi$
$62$	0	0
$63$	− 11.4819i	− 1.44658i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 11.0707i	− 1.35250i	−0.736672	−	0.676250i	$-0.763604\pi$
$67$	− 11.0707i	− 1.35250i	0.736672	−	0.676250i	$-0.236396\pi$
$68$	0	0
$69$	14.3720	1.73018
$70$	0	0
$71$	10.0978i	1.19839i	0.800602	+	0.599196i	$0.204513\pi$
$71$	10.0978i	1.19839i	−0.800602	+	0.599196i	$0.795487\pi$
$72$	0	0
$73$	− 10.9487i	− 1.28145i	−0.767772	−	0.640724i	$-0.778634\pi$
$73$	− 10.9487i	− 1.28145i	0.767772	−	0.640724i	$-0.221366\pi$
$74$	0	0
$75$	−9.91723	−1.14514
$76$	0	0
$77$	12.0978	1.37868
$78$	0	0
$79$	−2.81163	−0.316333	−0.158166	−	0.987412i	$-0.550558\pi$
$79$	−2.81163	−0.316333	−0.158166	+	0.987412i	$0.550558\pi$
$80$	0	0
$81$	−10.1371	−1.12634
$82$	0	0
$83$	− 2.93900i	− 0.322597i	−0.986906	−	0.161299i	$-0.948432\pi$
$83$	− 2.93900i	− 0.322597i	0.986906	−	0.161299i	$-0.0515682\pi$
$84$	0	0
$85$	− 3.18837i	− 0.345828i
$86$	0	0
$87$	−7.04221	−0.755004
$88$	0	0
$89$	− 12.1806i	− 1.29114i	−0.763700	−	0.645571i	$-0.776620\pi$
$89$	− 12.1806i	− 1.29114i	0.763700	−	0.645571i	$-0.223380\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 5.64742i	− 0.585610i
$94$	0	0
$95$	−2.61596	−0.268392
$96$	0	0
$97$	12.9661i	1.31651i	0.752794	+	0.658256i	$0.228706\pi$
$97$	12.9661i	1.31651i	−0.752794	+	0.658256i	$0.771294\pi$
$98$	0	0
$99$	6.87800i	0.691265i
$100$	0	0
$101$	10.1957	1.01451	0.507254	−	0.861797i	$-0.330661\pi$
$101$	10.1957	1.01451	0.507254	+	0.861797i	$0.330661\pi$
$102$	0	0
$103$	10.6703	1.05137	0.525686	−	0.850679i	$-0.323809\pi$
$103$	10.6703	1.05137	0.525686	+	0.850679i	$0.323809\pi$
$104$	0	0
$105$	−9.42758	−0.920038
$106$	0	0
$107$	20.5623	1.98783	0.993914	−	0.110158i	$-0.0351358\pi$
$107$	20.5623	1.98783	0.993914	+	0.110158i	$0.0351358\pi$
$108$	0	0
$109$	2.71379i	0.259934i	0.991518	+	0.129967i	$0.0414872\pi$
$109$	2.71379i	0.259934i	−0.991518	+	0.129967i	$0.958513\pi$
$110$	0	0
$111$	− 3.54958i	− 0.336911i
$112$	0	0
$113$	20.6504	1.94263	0.971313	−	0.237804i	$-0.0764277\pi$
$113$	20.6504	1.94263	0.971313	+	0.237804i	$0.0764277\pi$
$114$	0	0
$115$	− 5.42758i	− 0.506125i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	16.0978i	1.47569i
$120$	0	0
$121$	3.75302	0.341184
$122$	0	0
$123$	8.60925i	0.776270i
$124$	0	0
$125$	8.19567i	0.733043i
$126$	0	0
$127$	12.7681	1.13298	0.566492	−	0.824067i	$-0.308300\pi$
$127$	12.7681	1.13298	0.566492	+	0.824067i	$0.308300\pi$
$128$	0	0
$129$	−0.402829	−0.0354671
$130$	0	0
$131$	5.15883	0.450729	0.225365	−	0.974274i	$-0.427643\pi$
$131$	5.15883	0.450729	0.225365	+	0.974274i	$0.427643\pi$
$132$	0	0
$133$	13.2078	1.14526
$134$	0	0
$135$	0.933624i	0.0803536i
$136$	0	0
$137$	3.30127i	0.282047i	0.990006	+	0.141023i	$0.0450393\pi$
$137$	3.30127i	0.282047i	−0.990006	+	0.141023i	$0.954961\pi$
$138$	0	0
$139$	6.49157	0.550607	0.275304	−	0.961357i	$-0.411222\pi$
$139$	6.49157	0.550607	0.275304	+	0.961357i	$0.411222\pi$
$140$	0	0
$141$	− 12.2741i	− 1.03367i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.65950i	0.220859i
$146$	0	0
$147$	31.1008	2.56515
$148$	0	0
$149$	5.03146i	0.412193i	0.978532	+	0.206097i	$0.0660761\pi$
$149$	5.03146i	0.412193i	−0.978532	+	0.206097i	$0.933924\pi$
$150$	0	0
$151$	− 1.72587i	− 0.140450i	−0.997531	−	0.0702248i	$-0.977628\pi$
$151$	− 1.72587i	− 0.140450i	0.997531	−	0.0702248i	$-0.0223717\pi$
$152$	0	0
$153$	−9.15213	−0.739906
$154$	0	0
$155$	−2.13275	−0.171307
$156$	0	0
$157$	15.5060	1.23752	0.618758	−	0.785581i	$-0.287636\pi$
$157$	15.5060	1.23752	0.618758	+	0.785581i	$0.287636\pi$
$158$	0	0
$159$	−14.3720	−1.13977
$160$	0	0
$161$	27.4034i	2.15969i
$162$	0	0
$163$	− 19.7235i	− 1.54486i	−0.635099	−	0.772431i	$-0.719041\pi$
$163$	− 19.7235i	− 1.54486i	0.635099	−	0.772431i	$-0.280959\pi$
$164$	0	0
$165$	5.64742	0.439651
$166$	0	0
$167$	− 14.0000i	− 1.08335i	−0.840587	−	0.541676i	$-0.817790\pi$
$167$	− 14.0000i	− 1.08335i	0.840587	−	0.541676i	$-0.182210\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	7.50902i	0.574229i
$172$	0	0
$173$	−1.20775	−0.0918236	−0.0459118	−	0.998945i	$-0.514619\pi$
$173$	−1.20775	−0.0918236	−0.0459118	+	0.998945i	$0.514619\pi$
$174$	0	0
$175$	− 18.9095i	− 1.42942i
$176$	0	0
$177$	− 7.23729i	− 0.543988i
$178$	0	0
$179$	16.5157	1.23444	0.617222	−	0.786789i	$-0.288258\pi$
$179$	16.5157	1.23444	0.617222	+	0.786789i	$0.288258\pi$
$180$	0	0
$181$	−6.37196	−0.473624	−0.236812	−	0.971555i	$-0.576103\pi$
$181$	−6.37196	−0.473624	−0.236812	+	0.971555i	$0.576103\pi$
$182$	0	0
$183$	32.9396	2.43496
$184$	0	0
$185$	−1.34050	−0.0985557
$186$	0	0
$187$	− 9.64310i	− 0.705174i
$188$	0	0
$189$	− 4.71379i	− 0.342878i
$190$	0	0
$191$	2.49396	0.180457	0.0902283	−	0.995921i	$-0.471240\pi$
$191$	2.49396	0.180457	0.0902283	+	0.995921i	$0.471240\pi$
$192$	0	0
$193$	4.00538i	0.288313i	0.989555	+	0.144157i	$0.0460469\pi$
$193$	4.00538i	0.288313i	−0.989555	+	0.144157i	$0.953953\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.6340i	1.68385i	0.539592	+	0.841927i	$0.318578\pi$
$197$	23.6340i	1.68385i	−0.539592	+	0.841927i	$0.681422\pi$
$198$	0	0
$199$	16.5375	1.17231	0.586156	−	0.810198i	$-0.300641\pi$
$199$	16.5375	1.17231	0.586156	+	0.810198i	$0.300641\pi$
$200$	0	0
$201$	26.0925i	1.84042i
$202$	0	0
$203$	− 13.4276i	− 0.942432i
$204$	0	0
$205$	3.25129	0.227080
$206$	0	0
$207$	−15.5797	−1.08286
$208$	0	0
$209$	−7.91185	−0.547274
$210$	0	0
$211$	1.66056	0.114318	0.0571589	−	0.998365i	$-0.481796\pi$
$211$	1.66056	0.114318	0.0571589	+	0.998365i	$0.481796\pi$
$212$	0	0
$213$	− 23.7995i	− 1.63072i
$214$	0	0
$215$	0.152129i	0.0103751i
$216$	0	0
$217$	10.7681	0.730985
$218$	0	0
$219$	25.8049i	1.74374i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0.792249i	0.0530529i	0.999648	+	0.0265265i	$0.00844463\pi$
$223$	0.792249i	0.0530529i	−0.999648	+	0.0265265i	$0.991555\pi$
$224$	0	0
$225$	10.7506	0.716708
$226$	0	0
$227$	21.7603i	1.44428i	0.691745	+	0.722141i	$0.256842\pi$
$227$	21.7603i	1.44428i	−0.691745	+	0.722141i	$0.743158\pi$
$228$	0	0
$229$	1.97584i	0.130567i	0.997867	+	0.0652835i	$0.0207952\pi$
$229$	1.97584i	0.130567i	−0.997867	+	0.0652835i	$0.979205\pi$
$230$	0	0
$231$	−28.5133	−1.87604
$232$	0	0
$233$	−18.2349	−1.19461	−0.597304	−	0.802015i	$-0.703761\pi$
$233$	−18.2349	−1.19461	−0.597304	+	0.802015i	$0.703761\pi$
$234$	0	0
$235$	−4.63533	−0.302376
$236$	0	0
$237$	6.62671	0.430451
$238$	0	0
$239$	22.0737i	1.42783i	0.700234	+	0.713914i	$0.253079\pi$
$239$	22.0737i	1.42783i	−0.700234	+	0.713914i	$0.746921\pi$
$240$	0	0
$241$	− 6.98792i	− 0.450131i	−0.974344	−	0.225066i	$-0.927740\pi$
$241$	− 6.98792i	− 0.450131i	0.974344	−	0.225066i	$-0.0722597\pi$
$242$	0	0
$243$	20.7453	1.33081
$244$	0	0
$245$	− 11.7453i	− 0.750377i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.92692i	0.438976i
$250$	0	0
$251$	−12.5593	−0.792734	−0.396367	−	0.918092i	$-0.629729\pi$
$251$	−12.5593	−0.792734	−0.396367	+	0.918092i	$0.629729\pi$
$252$	0	0
$253$	− 16.4155i	− 1.03203i
$254$	0	0
$255$	7.51466i	0.470587i
$256$	0	0
$257$	16.9933	1.06001	0.530006	−	0.847994i	$-0.322190\pi$
$257$	16.9933	1.06001	0.530006	+	0.847994i	$0.322190\pi$
$258$	0	0
$259$	6.76809	0.420548
$260$	0	0
$261$	7.63401	0.472533
$262$	0	0
$263$	4.39612	0.271077	0.135538	−	0.990772i	$-0.456724\pi$
$263$	4.39612	0.271077	0.135538	+	0.990772i	$0.456724\pi$
$264$	0	0
$265$	5.42758i	0.333414i
$266$	0	0
$267$	28.7084i	1.75693i
$268$	0	0
$269$	15.5603	0.948730	0.474365	−	0.880328i	$-0.342678\pi$
$269$	15.5603	0.948730	0.474365	+	0.880328i	$0.342678\pi$
$270$	0	0
$271$	− 21.9952i	− 1.33611i	−0.744110	−	0.668057i	$-0.767126\pi$
$271$	− 21.9952i	− 1.33611i	0.744110	−	0.668057i	$-0.232874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	11.3274i	0.683065i
$276$	0	0
$277$	1.87800	0.112838	0.0564191	−	0.998407i	$-0.482032\pi$
$277$	1.87800	0.112838	0.0564191	+	0.998407i	$0.482032\pi$
$278$	0	0
$279$	6.12200i	0.366514i
$280$	0	0
$281$	− 9.20536i	− 0.549146i	−0.961566	−	0.274573i	$-0.911464\pi$
$281$	− 9.20536i	− 0.549146i	0.961566	−	0.274573i	$-0.0885364\pi$
$282$	0	0
$283$	4.70841	0.279886	0.139943	−	0.990160i	$-0.455308\pi$
$283$	4.70841	0.279886	0.139943	+	0.990160i	$0.455308\pi$
$284$	0	0
$285$	6.16554	0.365215
$286$	0	0
$287$	−16.4155	−0.968976
$288$	0	0
$289$	−4.16852	−0.245207
$290$	0	0
$291$	− 30.5599i	− 1.79145i
$292$	0	0
$293$	13.7017i	0.800462i	0.916414	+	0.400231i	$0.131070\pi$
$293$	13.7017i	0.800462i	−0.916414	+	0.400231i	$0.868930\pi$
$294$	0	0
$295$	−2.73317	−0.159131
$296$	0	0
$297$	2.82371i	0.163848i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 0.768086i	− 0.0442717i
$302$	0	0
$303$	−24.0301	−1.38049
$304$	0	0
$305$	− 12.4397i	− 0.712293i
$306$	0	0
$307$	8.03252i	0.458440i	0.973375	+	0.229220i	$0.0736176\pi$
$307$	8.03252i	0.458440i	−0.973375	+	0.229220i	$0.926382\pi$
$308$	0	0
$309$	−25.1487	−1.43066
$310$	0	0
$311$	4.09783	0.232367	0.116183	−	0.993228i	$-0.462934\pi$
$311$	4.09783	0.232367	0.116183	+	0.993228i	$0.462934\pi$
$312$	0	0
$313$	4.37435	0.247253	0.123627	−	0.992329i	$-0.460548\pi$
$313$	4.37435	0.247253	0.123627	+	0.992329i	$0.460548\pi$
$314$	0	0
$315$	10.2198	0.575822
$316$	0	0
$317$	− 20.3612i	− 1.14360i	−0.820393	−	0.571800i	$-0.806245\pi$
$317$	− 20.3612i	− 1.14360i	0.820393	−	0.571800i	$-0.193755\pi$
$318$	0	0
$319$	8.04354i	0.450352i
$320$	0	0
$321$	−48.4631	−2.70495
$322$	0	0
$323$	− 10.5278i	− 0.585783i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 6.39612i	− 0.353706i
$328$	0	0
$329$	23.4034	1.29027
$330$	0	0
$331$	− 6.13275i	− 0.337087i	−0.985694	−	0.168543i	$-0.946094\pi$
$331$	− 6.13275i	− 0.337087i	0.985694	−	0.168543i	$-0.0539063\pi$
$332$	0	0
$333$	3.84787i	0.210862i
$334$	0	0
$335$	9.85384	0.538373
$336$	0	0
$337$	27.8485	1.51700	0.758501	−	0.651672i	$-0.225932\pi$
$337$	27.8485	1.51700	0.758501	+	0.651672i	$0.225932\pi$
$338$	0	0
$339$	−48.6708	−2.64344
$340$	0	0
$341$	−6.45042	−0.349310
$342$	0	0
$343$	27.8431i	1.50339i
$344$	0	0
$345$	12.7922i	0.688712i
$346$	0	0
$347$	−4.43967	−0.238334	−0.119167	−	0.992874i	$-0.538022\pi$
$347$	−4.43967	−0.238334	−0.119167	+	0.992874i	$0.538022\pi$
$348$	0	0
$349$	19.9215i	1.06638i	0.845997	+	0.533188i	$0.179006\pi$
$349$	19.9215i	1.06638i	−0.845997	+	0.533188i	$0.820994\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 30.5894i	− 1.62811i	−0.580788	−	0.814055i	$-0.697256\pi$
$353$	− 30.5894i	− 1.62811i	0.580788	−	0.814055i	$-0.302744\pi$
$354$	0	0
$355$	−8.98792	−0.477029
$356$	0	0
$357$	− 37.9409i	− 2.00805i
$358$	0	0
$359$	− 21.6039i	− 1.14021i	−0.821572	−	0.570104i	$-0.806903\pi$
$359$	− 21.6039i	− 1.14021i	0.821572	−	0.570104i	$-0.193097\pi$
$360$	0	0
$361$	10.3623	0.545383
$362$	0	0
$363$	−8.84548	−0.464267
$364$	0	0
$365$	9.74525	0.510090
$366$	0	0
$367$	−18.7681	−0.979686	−0.489843	−	0.871811i	$-0.662946\pi$
$367$	−18.7681	−0.979686	−0.489843	+	0.871811i	$0.662946\pi$
$368$	0	0
$369$	− 9.33273i	− 0.485843i
$370$	0	0
$371$	− 27.4034i	− 1.42271i
$372$	0	0
$373$	−9.42758	−0.488142	−0.244071	−	0.969757i	$-0.578483\pi$
$373$	−9.42758	−0.488142	−0.244071	+	0.969757i	$0.578483\pi$
$374$	0	0
$375$	− 19.3163i	− 0.997491i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 32.0103i	− 1.64426i	−0.569302	−	0.822129i	$-0.692786\pi$
$379$	− 32.0103i	− 1.64426i	0.569302	−	0.822129i	$-0.307214\pi$
$380$	0	0
$381$	−30.0930	−1.54171
$382$	0	0
$383$	− 15.9517i	− 0.815092i	−0.913185	−	0.407546i	$-0.866385\pi$
$383$	− 15.9517i	− 0.815092i	0.913185	−	0.407546i	$-0.133615\pi$
$384$	0	0
$385$	10.7681i	0.548792i
$386$	0	0
$387$	0.436681	0.0221978
$388$	0	0
$389$	18.8659	0.956540	0.478270	−	0.878213i	$-0.341264\pi$
$389$	18.8659	0.956540	0.478270	+	0.878213i	$0.341264\pi$
$390$	0	0
$391$	21.8431	1.10465
$392$	0	0
$393$	−12.1588	−0.613332
$394$	0	0
$395$	− 2.50258i	− 0.125919i
$396$	0	0
$397$	− 37.6969i	− 1.89195i	−0.324233	−	0.945977i	$-0.605106\pi$
$397$	− 37.6969i	− 1.89195i	0.324233	−	0.945977i	$-0.394894\pi$
$398$	0	0
$399$	−31.1293	−1.55841
$400$	0	0
$401$	3.22952i	0.161275i	0.996744	+	0.0806373i	$0.0256955\pi$
$401$	3.22952i	0.161275i	−0.996744	+	0.0806373i	$0.974304\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 9.02284i	− 0.448348i
$406$	0	0
$407$	−4.05429	−0.200964
$408$	0	0
$409$	− 3.54527i	− 0.175302i	−0.996151	−	0.0876511i	$-0.972064\pi$
$409$	− 3.54527i	− 0.175302i	0.996151	−	0.0876511i	$-0.0279361\pi$
$410$	0	0
$411$	− 7.78076i	− 0.383797i
$412$	0	0
$413$	13.7995	0.679031
$414$	0	0
$415$	2.61596	0.128412
$416$	0	0
$417$	−15.2999	−0.749242
$418$	0	0
$419$	14.4155	0.704243	0.352122	−	0.935954i	$-0.385460\pi$
$419$	14.4155	0.704243	0.352122	+	0.935954i	$0.385460\pi$
$420$	0	0
$421$	10.0978i	0.492138i	0.969252	+	0.246069i	$0.0791390\pi$
$421$	10.0978i	0.492138i	−0.969252	+	0.246069i	$0.920861\pi$
$422$	0	0
$423$	13.3056i	0.646940i
$424$	0	0
$425$	−15.0726	−0.731129
$426$	0	0
$427$	62.8068i	3.03944i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.2198i	0.973955i	0.873414	+	0.486978i	$0.161901\pi$
$431$	20.2198i	0.973955i	−0.873414	+	0.486978i	$0.838099\pi$
$432$	0	0
$433$	−36.4849	−1.75335	−0.876675	−	0.481083i	$-0.840244\pi$
$433$	−36.4849	−1.75335	−0.876675	+	0.481083i	$0.840244\pi$
$434$	0	0
$435$	− 6.26816i	− 0.300535i
$436$	0	0
$437$	− 17.9215i	− 0.857304i
$438$	0	0
$439$	−28.3612	−1.35361	−0.676803	−	0.736164i	$-0.736635\pi$
$439$	−28.3612	−1.35361	−0.676803	+	0.736164i	$0.736635\pi$
$440$	0	0
$441$	−33.7144	−1.60545
$442$	0	0
$443$	−2.80061	−0.133061	−0.0665305	−	0.997784i	$-0.521193\pi$
$443$	−2.80061	−0.133061	−0.0665305	+	0.997784i	$0.521193\pi$
$444$	0	0
$445$	10.8418	0.513949
$446$	0	0
$447$	− 11.8586i	− 0.560894i
$448$	0	0
$449$	13.2760i	0.626535i	0.949665	+	0.313268i	$0.101424\pi$
$449$	13.2760i	0.626535i	−0.949665	+	0.313268i	$0.898576\pi$
$450$	0	0
$451$	9.83340	0.463037
$452$	0	0
$453$	4.06770i	0.191117i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 13.6474i	− 0.638399i	−0.947688	−	0.319200i	$-0.896586\pi$
$457$	− 13.6474i	− 0.638399i	0.947688	−	0.319200i	$-0.103414\pi$
$458$	0	0
$459$	−3.75733	−0.175377
$460$	0	0
$461$	12.1655i	0.566606i	0.959031	+	0.283303i	$0.0914301\pi$
$461$	12.1655i	0.566606i	−0.959031	+	0.283303i	$0.908570\pi$
$462$	0	0
$463$	− 4.24996i	− 0.197513i	−0.995112	−	0.0987563i	$-0.968514\pi$
$463$	− 4.24996i	− 0.197513i	0.995112	−	0.0987563i	$-0.0314864\pi$
$464$	0	0
$465$	5.02667	0.233106
$466$	0	0
$467$	8.21552	0.380169	0.190084	−	0.981768i	$-0.439124\pi$
$467$	8.21552	0.380169	0.190084	+	0.981768i	$0.439124\pi$
$468$	0	0
$469$	−49.7512	−2.29730
$470$	0	0
$471$	−36.5461	−1.68396
$472$	0	0
$473$	0.460107i	0.0211558i
$474$	0	0
$475$	12.3666i	0.567418i
$476$	0	0
$477$	15.5797	0.713346
$478$	0	0
$479$	31.0267i	1.41764i	0.705387	+	0.708822i	$0.250773\pi$
$479$	31.0267i	1.41764i	−0.705387	+	0.708822i	$0.749227\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 64.5870i	− 2.93881i
$484$	0	0
$485$	−11.5410	−0.524048
$486$	0	0
$487$	10.9987i	0.498397i	0.968452	+	0.249199i	$0.0801672\pi$
$487$	10.9987i	0.498397i	−0.968452	+	0.249199i	$0.919833\pi$
$488$	0	0
$489$	46.4862i	2.10218i
$490$	0	0
$491$	21.2336	0.958258	0.479129	−	0.877745i	$-0.340953\pi$
$491$	21.2336	0.958258	0.479129	+	0.877745i	$0.340953\pi$
$492$	0	0
$493$	−10.7030	−0.482041
$494$	0	0
$495$	−6.12200	−0.275163
$496$	0	0
$497$	45.3793	2.03554
$498$	0	0
$499$	16.8635i	0.754915i	0.926027	+	0.377458i	$0.123202\pi$
$499$	16.8635i	0.754915i	−0.926027	+	0.377458i	$0.876798\pi$
$500$	0	0
$501$	32.9965i	1.47418i
$502$	0	0
$503$	−20.3806	−0.908725	−0.454363	−	0.890817i	$-0.650133\pi$
$503$	−20.3806	−0.908725	−0.454363	+	0.890817i	$0.650133\pi$
$504$	0	0
$505$	9.07500i	0.403832i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.9215i	0.971655i	0.874055	+	0.485828i	$0.161482\pi$
$509$	21.9215i	0.971655i	−0.874055	+	0.485828i	$0.838518\pi$
$510$	0	0
$511$	−49.2030	−2.17661
$512$	0	0
$513$	3.08277i	0.136108i
$514$	0	0
$515$	9.49742i	0.418506i
$516$	0	0
$517$	−14.0194	−0.616572
$518$	0	0
$519$	2.84654	0.124949
$520$	0	0
$521$	26.1564	1.14593	0.572967	−	0.819578i	$-0.305792\pi$
$521$	26.1564	1.14593	0.572967	+	0.819578i	$0.305792\pi$
$522$	0	0
$523$	−4.91185	−0.214780	−0.107390	−	0.994217i	$-0.534249\pi$
$523$	−4.91185	−0.214780	−0.107390	+	0.994217i	$0.534249\pi$
$524$	0	0
$525$	44.5676i	1.94509i
$526$	0	0
$527$	− 8.58317i	− 0.373889i
$528$	0	0
$529$	14.1836	0.616678
$530$	0	0
$531$	7.84548i	0.340465i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	18.3021i	0.791270i
$536$	0	0
$537$	−38.9259	−1.67977
$538$	0	0
$539$	− 35.5230i	− 1.53009i
$540$	0	0
$541$	− 0.459042i	− 0.0197358i	−0.999951	−	0.00986789i	$-0.996859\pi$
$541$	− 0.459042i	− 0.0197358i	0.999951	−	0.00986789i	$-0.00314110\pi$
$542$	0	0
$543$	15.0180	0.644486
$544$	0	0
$545$	−2.41550	−0.103469
$546$	0	0
$547$	20.3327	0.869365	0.434682	−	0.900584i	$-0.356861\pi$
$547$	20.3327	0.869365	0.434682	+	0.900584i	$0.356861\pi$
$548$	0	0
$549$	−35.7077	−1.52397
$550$	0	0
$551$	8.78150i	0.374104i
$552$	0	0
$553$	12.6353i	0.537309i
$554$	0	0
$555$	3.15942	0.134110
$556$	0	0
$557$	43.9469i	1.86209i	0.364905	+	0.931045i	$0.381101\pi$
$557$	43.9469i	1.86209i	−0.364905	+	0.931045i	$0.618899\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	22.7278i	0.959568i
$562$	0	0
$563$	43.5991	1.83748	0.918741	−	0.394860i	$-0.129207\pi$
$563$	43.5991	1.83748	0.918741	+	0.394860i	$0.129207\pi$
$564$	0	0
$565$	18.3806i	0.773277i
$566$	0	0
$567$	45.5555i	1.91315i
$568$	0	0
$569$	−8.28919	−0.347501	−0.173751	−	0.984790i	$-0.555589\pi$
$569$	−8.28919	−0.347501	−0.173751	+	0.984790i	$0.555589\pi$
$570$	0	0
$571$	−22.9836	−0.961834	−0.480917	−	0.876766i	$-0.659696\pi$
$571$	−22.9836	−0.961834	−0.480917	+	0.876766i	$0.659696\pi$
$572$	0	0
$573$	−5.87800	−0.245557
$574$	0	0
$575$	−25.6582	−1.07002
$576$	0	0
$577$	0.553630i	0.0230479i	0.999934	+	0.0115240i	$0.00366827\pi$
$577$	0.553630i	0.0230479i	−0.999934	+	0.0115240i	$0.996332\pi$
$578$	0	0
$579$	− 9.44026i	− 0.392324i
$580$	0	0
$581$	−13.2078	−0.547950
$582$	0	0
$583$	16.4155i	0.679861i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.0355i	0.661856i	0.943656	+	0.330928i	$0.107362\pi$
$587$	16.0355i	0.661856i	−0.943656	+	0.330928i	$0.892638\pi$
$588$	0	0
$589$	−7.04221	−0.290169
$590$	0	0
$591$	− 55.7029i	− 2.29131i
$592$	0	0
$593$	− 25.9976i	− 1.06759i	−0.845613	−	0.533797i	$-0.820765\pi$
$593$	− 25.9976i	− 1.06759i	0.845613	−	0.533797i	$-0.179235\pi$
$594$	0	0
$595$	−14.3284	−0.587408
$596$	0	0
$597$	−38.9772	−1.59523
$598$	0	0
$599$	16.2150	0.662529	0.331264	−	0.943538i	$-0.392525\pi$
$599$	16.2150	0.662529	0.331264	+	0.943538i	$0.392525\pi$
$600$	0	0
$601$	−29.5200	−1.20415	−0.602074	−	0.798440i	$-0.705659\pi$
$601$	−29.5200	−1.20415	−0.602074	+	0.798440i	$0.705659\pi$
$602$	0	0
$603$	− 28.2851i	− 1.15186i
$604$	0	0
$605$	3.34050i	0.135811i
$606$	0	0
$607$	−37.4228	−1.51894	−0.759472	−	0.650540i	$-0.774543\pi$
$607$	−37.4228	−1.51894	−0.759472	+	0.650540i	$0.774543\pi$
$608$	0	0
$609$	31.6474i	1.28242i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 38.0737i	− 1.53778i	−0.639380	−	0.768891i	$-0.720809\pi$
$613$	− 38.0737i	− 1.53778i	0.639380	−	0.768891i	$-0.279191\pi$
$614$	0	0
$615$	−7.66296	−0.309000
$616$	0	0
$617$	− 41.6383i	− 1.67630i	−0.545443	−	0.838148i	$-0.683639\pi$
$617$	− 41.6383i	− 1.67630i	0.545443	−	0.838148i	$-0.316361\pi$
$618$	0	0
$619$	1.67158i	0.0671864i	0.999436	+	0.0335932i	$0.0106951\pi$
$619$	1.67158i	0.0671864i	−0.999436	+	0.0335932i	$0.989305\pi$
$620$	0	0
$621$	−6.39612	−0.256668
$622$	0	0
$623$	−54.7391	−2.19308
$624$	0	0
$625$	13.7439	0.549757
$626$	0	0
$627$	18.6474	0.744706
$628$	0	0
$629$	− 5.39480i	− 0.215105i
$630$	0	0
$631$	− 8.70304i	− 0.346462i	−0.984881	−	0.173231i	$-0.944579\pi$
$631$	− 8.70304i	− 0.346462i	0.984881	−	0.173231i	$-0.0554208\pi$
$632$	0	0
$633$	−3.91377	−0.155559
$634$	0	0
$635$	11.3647i	0.450993i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	25.7995i	1.02061i
$640$	0	0
$641$	19.9075	0.786301	0.393150	−	0.919474i	$-0.371385\pi$
$641$	19.9075	0.786301	0.393150	+	0.919474i	$0.371385\pi$
$642$	0	0
$643$	27.0756i	1.06776i	0.845561	+	0.533879i	$0.179266\pi$
$643$	27.0756i	1.06776i	−0.845561	+	0.533879i	$0.820734\pi$
$644$	0	0
$645$	− 0.358552i	− 0.0141180i
$646$	0	0
$647$	9.43237	0.370825	0.185412	−	0.982661i	$-0.440638\pi$
$647$	9.43237	0.370825	0.185412	+	0.982661i	$0.440638\pi$
$648$	0	0
$649$	−8.26636	−0.324483
$650$	0	0
$651$	−25.3793	−0.994692
$652$	0	0
$653$	−37.6292	−1.47255	−0.736273	−	0.676685i	$-0.763416\pi$
$653$	−37.6292	−1.47255	−0.736273	+	0.676685i	$0.763416\pi$
$654$	0	0
$655$	4.59179i	0.179416i
$656$	0	0
$657$	− 27.9734i	− 1.09135i
$658$	0	0
$659$	32.7724	1.27663	0.638316	−	0.769775i	$-0.279631\pi$
$659$	32.7724	1.27663	0.638316	+	0.769775i	$0.279631\pi$
$660$	0	0
$661$	− 20.1957i	− 0.785520i	−0.919641	−	0.392760i	$-0.871520\pi$
$661$	− 20.1957i	− 0.785520i	0.919641	−	0.392760i	$-0.128480\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.7560i	0.455878i
$666$	0	0
$667$	−18.2198	−0.705475
$668$	0	0
$669$	− 1.86725i	− 0.0721920i
$670$	0	0
$671$	− 37.6233i	− 1.45243i
$672$	0	0
$673$	30.3435	1.16966	0.584828	−	0.811158i	$-0.301162\pi$
$673$	30.3435	1.16966	0.584828	+	0.811158i	$0.301162\pi$
$674$	0	0
$675$	4.41358	0.169879
$676$	0	0
$677$	−21.1642	−0.813407	−0.406703	−	0.913560i	$-0.633322\pi$
$677$	−21.1642	−0.813407	−0.406703	+	0.913560i	$0.633322\pi$
$678$	0	0
$679$	58.2693	2.23617
$680$	0	0
$681$	− 51.2868i	− 1.96531i
$682$	0	0
$683$	35.4873i	1.35788i	0.734193	+	0.678941i	$0.237561\pi$
$683$	35.4873i	1.35788i	−0.734193	+	0.678941i	$0.762439\pi$
$684$	0	0
$685$	−2.93841	−0.112271
$686$	0	0
$687$	− 4.65684i	− 0.177670i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	36.7472i	1.39793i	0.715157	+	0.698964i	$0.246355\pi$
$691$	36.7472i	1.39793i	−0.715157	+	0.698964i	$0.753645\pi$
$692$	0	0
$693$	30.9095	1.17415
$694$	0	0
$695$	5.77804i	0.219173i
$696$	0	0
$697$	13.0847i	0.495618i
$698$	0	0
$699$	42.9778	1.62557
$700$	0	0
$701$	−19.7668	−0.746580	−0.373290	−	0.927715i	$-0.621770\pi$
$701$	−19.7668	−0.746580	−0.373290	+	0.927715i	$0.621770\pi$
$702$	0	0
$703$	−4.42626	−0.166939
$704$	0	0
$705$	10.9250	0.411459
$706$	0	0
$707$	− 45.8189i	− 1.72320i
$708$	0	0
$709$	11.6280i	0.436700i	0.975871	+	0.218350i	$0.0700675\pi$
$709$	11.6280i	0.436700i	−0.975871	+	0.218350i	$0.929933\pi$
$710$	0	0
$711$	−7.18359	−0.269406
$712$	0	0
$713$	− 14.6112i	− 0.547193i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 52.0253i	− 1.94292i
$718$	0	0
$719$	−1.06638	−0.0397691	−0.0198846	−	0.999802i	$-0.506330\pi$
$719$	−1.06638	−0.0397691	−0.0198846	+	0.999802i	$0.506330\pi$
$720$	0	0
$721$	− 47.9517i	− 1.78581i
$722$	0	0
$723$	16.4698i	0.612518i
$724$	0	0
$725$	12.5724	0.466928
$726$	0	0
$727$	−1.26205	−0.0468067	−0.0234033	−	0.999726i	$-0.507450\pi$
$727$	−1.26205	−0.0468067	−0.0234033	+	0.999726i	$0.507450\pi$
$728$	0	0
$729$	−18.4832	−0.684563
$730$	0	0
$731$	−0.612236	−0.0226444
$732$	0	0
$733$	20.6789i	0.763792i	0.924205	+	0.381896i	$0.124729\pi$
$733$	20.6789i	0.763792i	−0.924205	+	0.381896i	$0.875271\pi$
$734$	0	0
$735$	27.6823i	1.02108i
$736$	0	0
$737$	29.8025	1.09779
$738$	0	0
$739$	− 5.17331i	− 0.190303i	−0.995463	−	0.0951516i	$-0.969666\pi$
$739$	− 5.17331i	− 0.190303i	0.995463	−	0.0951516i	$-0.0303336\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 10.4397i	− 0.382994i	−0.981493	−	0.191497i	$-0.938666\pi$
$743$	− 10.4397i	− 0.382994i	0.981493	−	0.191497i	$-0.0613343\pi$
$744$	0	0
$745$	−4.47842	−0.164077
$746$	0	0
$747$	− 7.50902i	− 0.274741i
$748$	0	0
$749$	− 92.4059i	− 3.37644i
$750$	0	0
$751$	−14.0435	−0.512456	−0.256228	−	0.966616i	$-0.582480\pi$
$751$	−14.0435	−0.512456	−0.256228	+	0.966616i	$0.582480\pi$
$752$	0	0
$753$	29.6009	1.07872
$754$	0	0
$755$	1.53617	0.0559070
$756$	0	0
$757$	−9.30559	−0.338217	−0.169109	−	0.985597i	$-0.554089\pi$
$757$	−9.30559	−0.338217	−0.169109	+	0.985597i	$0.554089\pi$
$758$	0	0
$759$	38.6896i	1.40434i
$760$	0	0
$761$	− 9.56273i	− 0.346649i	−0.984865	−	0.173324i	$-0.944549\pi$
$761$	− 9.56273i	− 0.346649i	0.984865	−	0.173324i	$-0.0554509\pi$
$762$	0	0
$763$	12.1957	0.441513
$764$	0	0
$765$	− 8.14616i	− 0.294525i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	31.9299i	1.15142i	0.817653	+	0.575711i	$0.195275\pi$
$769$	31.9299i	1.15142i	−0.817653	+	0.575711i	$0.804725\pi$
$770$	0	0
$771$	−40.0514	−1.44242
$772$	0	0
$773$	34.1172i	1.22711i	0.789652	+	0.613555i	$0.210261\pi$
$773$	34.1172i	1.22711i	−0.789652	+	0.613555i	$0.789739\pi$
$774$	0	0
$775$	10.0823i	0.362167i
$776$	0	0
$777$	−15.9517	−0.572263
$778$	0	0
$779$	10.7356	0.384641
$780$	0	0
$781$	−27.1836	−0.972705
$782$	0	0
$783$	3.13408	0.112003
$784$	0	0
$785$	13.8017i	0.492603i
$786$	0	0
$787$	37.3467i	1.33127i	0.746279	+	0.665634i	$0.231839\pi$
$787$	37.3467i	1.33127i	−0.746279	+	0.665634i	$0.768161\pi$
$788$	0	0
$789$	−10.3612	−0.368869
$790$	0	0
$791$	− 92.8021i	− 3.29966i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 12.7922i	− 0.453694i
$796$	0	0
$797$	14.9831	0.530730	0.265365	−	0.964148i	$-0.414508\pi$
$797$	14.9831	0.530730	0.265365	+	0.964148i	$0.414508\pi$
$798$	0	0
$799$	− 18.6547i	− 0.659956i
$800$	0	0
$801$	− 31.1209i	− 1.09960i
$802$	0	0
$803$	29.4741	1.04012
$804$	0	0
$805$	−24.3913	−0.859682
$806$	0	0
$807$	−36.6741	−1.29099
$808$	0	0
$809$	25.6770	0.902754	0.451377	−	0.892333i	$-0.350933\pi$
$809$	25.6770	0.902754	0.451377	+	0.892333i	$0.350933\pi$
$810$	0	0
$811$	8.66786i	0.304370i	0.988352	+	0.152185i	$0.0486309\pi$
$811$	8.66786i	0.304370i	−0.988352	+	0.152185i	$0.951369\pi$
$812$	0	0
$813$	51.8404i	1.81812i
$814$	0	0
$815$	17.5555	0.614944
$816$	0	0
$817$	0.502320i	0.0175739i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	54.6547i	1.90746i	0.300660	+	0.953731i	$0.402793\pi$
$821$	54.6547i	1.90746i	−0.300660	+	0.953731i	$0.597207\pi$
$822$	0	0
$823$	−6.33704	−0.220895	−0.110448	−	0.993882i	$-0.535228\pi$
$823$	−6.33704	−0.220895	−0.110448	+	0.993882i	$0.535228\pi$
$824$	0	0
$825$	− 26.6974i	− 0.929484i
$826$	0	0
$827$	− 35.5405i	− 1.23586i	−0.786232	−	0.617932i	$-0.787971\pi$
$827$	− 35.5405i	− 1.23586i	0.786232	−	0.617932i	$-0.212029\pi$
$828$	0	0
$829$	41.5555	1.44328	0.721642	−	0.692267i	$-0.243388\pi$
$829$	41.5555	1.44328	0.721642	+	0.692267i	$0.243388\pi$
$830$	0	0
$831$	−4.42626	−0.153545
$832$	0	0
$833$	47.2683	1.63775
$834$	0	0
$835$	12.4612	0.431237
$836$	0	0
$837$	2.51334i	0.0868736i
$838$	0	0
$839$	− 17.5496i	− 0.605879i	−0.953010	−	0.302939i	$-0.902032\pi$
$839$	− 17.5496i	− 0.605879i	0.953010	−	0.302939i	$-0.0979680\pi$
$840$	0	0
$841$	−20.0723	−0.692150
$842$	0	0
$843$	21.6961i	0.747252i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 16.8659i	− 0.579520i
$848$	0	0
$849$	−11.0972	−0.380856
$850$	0	0
$851$	− 9.18359i	− 0.314809i
$852$	0	0
$853$	20.1414i	0.689628i	0.938671	+	0.344814i	$0.112058\pi$
$853$	20.1414i	0.689628i	−0.938671	+	0.344814i	$0.887942\pi$
$854$	0	0
$855$	−6.68366	−0.228576
$856$	0	0
$857$	8.85756	0.302568	0.151284	−	0.988490i	$-0.451659\pi$
$857$	8.85756	0.302568	0.151284	+	0.988490i	$0.451659\pi$
$858$	0	0
$859$	−28.8810	−0.985407	−0.492703	−	0.870197i	$-0.663991\pi$
$859$	−28.8810	−0.985407	−0.492703	+	0.870197i	$0.663991\pi$
$860$	0	0
$861$	38.6896	1.31854
$862$	0	0
$863$	3.90813i	0.133034i	0.997785	+	0.0665172i	$0.0211887\pi$
$863$	3.90813i	0.133034i	−0.997785	+	0.0665172i	$0.978811\pi$
$864$	0	0
$865$	− 1.07500i	− 0.0365511i
$866$	0	0
$867$	9.82477	0.333667
$868$	0	0
$869$	− 7.56896i	− 0.256759i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	33.1280i	1.12121i
$874$	0	0
$875$	36.8310	1.24512
$876$	0	0
$877$	44.9879i	1.51913i	0.650429	+	0.759567i	$0.274589\pi$
$877$	44.9879i	1.51913i	−0.650429	+	0.759567i	$0.725411\pi$
$878$	0	0
$879$	− 32.2935i	− 1.08923i
$880$	0	0
$881$	14.1933	0.478184	0.239092	−	0.970997i	$-0.423150\pi$
$881$	14.1933	0.478184	0.239092	+	0.970997i	$0.423150\pi$
$882$	0	0
$883$	48.4626	1.63090	0.815448	−	0.578830i	$-0.196490\pi$
$883$	48.4626	1.63090	0.815448	+	0.578830i	$0.196490\pi$
$884$	0	0
$885$	6.44179	0.216539
$886$	0	0
$887$	−30.8611	−1.03622	−0.518108	−	0.855315i	$-0.673363\pi$
$887$	−30.8611	−1.03622	−0.518108	+	0.855315i	$0.673363\pi$
$888$	0	0
$889$	− 57.3793i	− 1.92444i
$890$	0	0
$891$	− 27.2892i	− 0.914222i
$892$	0	0
$893$	−15.3056	−0.512182
$894$	0	0
$895$	14.7004i	0.491380i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.15942i	0.238780i
$900$	0	0
$901$	−21.8431	−0.727699
$902$	0	0
$903$	1.81030i	0.0602430i
$904$	0	0
$905$	− 5.67158i	− 0.188530i
$906$	0	0
$907$	3.94139	0.130872	0.0654359	−	0.997857i	$-0.479156\pi$
$907$	3.94139	0.130872	0.0654359	+	0.997857i	$0.479156\pi$
$908$	0	0
$909$	26.0495	0.864008
$910$	0	0
$911$	37.1943	1.23230	0.616152	−	0.787627i	$-0.288691\pi$
$911$	37.1943	1.23230	0.616152	+	0.787627i	$0.288691\pi$
$912$	0	0
$913$	7.91185	0.261844
$914$	0	0
$915$	29.3190i	0.969256i
$916$	0	0
$917$	− 23.1836i	− 0.765590i
$918$	0	0
$919$	−0.681005	−0.0224643	−0.0112321	−	0.999937i	$-0.503575\pi$
$919$	−0.681005	−0.0224643	−0.0112321	+	0.999937i	$0.503575\pi$
$920$	0	0
$921$	− 18.9318i	− 0.623825i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	6.33704i	0.208361i
$926$	0	0
$927$	27.2620	0.895403
$928$	0	0
$929$	− 32.4355i	− 1.06417i	−0.846690	−	0.532087i	$-0.821408\pi$
$929$	− 32.4355i	− 1.06417i	0.846690	−	0.532087i	$-0.178592\pi$
$930$	0	0
$931$	− 38.7821i	− 1.27103i
$932$	0	0
$933$	−9.65817	−0.316194
$934$	0	0
$935$	8.58317	0.280700
$936$	0	0
$937$	−14.6165	−0.477502	−0.238751	−	0.971081i	$-0.576738\pi$
$937$	−14.6165	−0.477502	−0.238751	+	0.971081i	$0.576738\pi$
$938$	0	0
$939$	−10.3099	−0.336451
$940$	0	0
$941$	30.1763i	0.983719i	0.870675	+	0.491860i	$0.163683\pi$
$941$	30.1763i	0.983719i	−0.870675	+	0.491860i	$0.836317\pi$
$942$	0	0
$943$	22.2741i	0.725345i
$944$	0	0
$945$	4.19567	0.136485
$946$	0	0
$947$	20.6708i	0.671712i	0.941913	+	0.335856i	$0.109026\pi$
$947$	20.6708i	0.671712i	−0.941913	+	0.335856i	$0.890974\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	47.9892i	1.55616i
$952$	0	0
$953$	47.2411	1.53029	0.765145	−	0.643858i	$-0.222667\pi$
$953$	47.2411	1.53029	0.765145	+	0.643858i	$0.222667\pi$
$954$	0	0
$955$	2.21983i	0.0718321i
$956$	0	0
$957$	− 18.9578i	− 0.612818i
$958$	0	0
$959$	14.8358	0.479073
$960$	0	0
$961$	25.2586	0.814793
$962$	0	0
$963$	52.5357	1.69294
$964$	0	0
$965$	−3.56512	−0.114765
$966$	0	0
$967$	− 53.9517i	− 1.73497i	−0.497464	−	0.867484i	$-0.665735\pi$
$967$	− 53.9517i	− 1.73497i	0.497464	−	0.867484i	$-0.334265\pi$
$968$	0	0
$969$	24.8130i	0.797107i
$970$	0	0
$971$	−49.8920	−1.60111	−0.800555	−	0.599259i	$-0.795462\pi$
$971$	−49.8920	−1.60111	−0.800555	+	0.599259i	$0.795462\pi$
$972$	0	0
$973$	− 29.1728i	− 0.935238i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.1299i	0.931948i	0.884799	+	0.465974i	$0.154296\pi$
$977$	29.1299i	0.931948i	−0.884799	+	0.465974i	$0.845704\pi$
$978$	0	0
$979$	32.7904	1.04799
$980$	0	0
$981$	6.93362i	0.221374i
$982$	0	0
$983$	− 41.3309i	− 1.31825i	−0.752033	−	0.659126i	$-0.770926\pi$
$983$	− 41.3309i	− 1.31825i	0.752033	−	0.659126i	$-0.229074\pi$
$984$	0	0
$985$	−21.0362	−0.670270
$986$	0	0
$987$	−55.1594	−1.75574
$988$	0	0
$989$	−1.04221	−0.0331404
$990$	0	0
$991$	−19.2185	−0.610496	−0.305248	−	0.952273i	$-0.598739\pi$
$991$	−19.2185	−0.610496	−0.305248	+	0.952273i	$0.598739\pi$
$992$	0	0
$993$	14.4543i	0.458692i
$994$	0	0
$995$	14.7198i	0.466648i
$996$	0	0
$997$	22.4940	0.712391	0.356195	−	0.934411i	$-0.384074\pi$
$997$	22.4940	0.712391	0.356195	+	0.934411i	$0.384074\pi$
$998$	0	0
$999$	1.57971i	0.0499799i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.f.m.337.4		6
4.3	odd	2		338.2.b.d.337.5		6
12.11	even	2		3042.2.b.n.1351.2		6
13.5	odd	4		2704.2.a.w.1.2		3
13.8	odd	4		2704.2.a.v.1.2		3
13.12	even	2	inner	2704.2.f.m.337.3		6
52.3	odd	6		338.2.e.e.147.2		12
52.7	even	12		338.2.c.i.315.2		6
52.11	even	12		338.2.c.i.191.2		6
52.15	even	12		338.2.c.h.191.2		6
52.19	even	12		338.2.c.h.315.2		6
52.23	odd	6		338.2.e.e.147.5		12
52.31	even	4		338.2.a.h.1.2	yes	3
52.35	odd	6		338.2.e.e.23.5		12
52.43	odd	6		338.2.e.e.23.2		12
52.47	even	4		338.2.a.g.1.2	✓	3
52.51	odd	2		338.2.b.d.337.2		6
156.47	odd	4		3042.2.a.bi.1.2		3
156.83	odd	4		3042.2.a.z.1.2		3
156.155	even	2		3042.2.b.n.1351.5		6
260.99	even	4		8450.2.a.bx.1.2		3
260.239	even	4		8450.2.a.bn.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.2.a.g.1.2	✓	3	52.47	even	4
338.2.a.h.1.2	yes	3	52.31	even	4
338.2.b.d.337.2		6	52.51	odd	2
338.2.b.d.337.5		6	4.3	odd	2
338.2.c.h.191.2		6	52.15	even	12
338.2.c.h.315.2		6	52.19	even	12
338.2.c.i.191.2		6	52.11	even	12
338.2.c.i.315.2		6	52.7	even	12
338.2.e.e.23.2		12	52.43	odd	6
338.2.e.e.23.5		12	52.35	odd	6
338.2.e.e.147.2		12	52.3	odd	6
338.2.e.e.147.5		12	52.23	odd	6
2704.2.a.v.1.2		3	13.8	odd	4
2704.2.a.w.1.2		3	13.5	odd	4
2704.2.f.m.337.3		6	13.12	even	2	inner
2704.2.f.m.337.4		6	1.1	even	1	trivial
3042.2.a.z.1.2		3	156.83	odd	4
3042.2.a.bi.1.2		3	156.47	odd	4
3042.2.b.n.1351.2		6	12.11	even	2
3042.2.b.n.1351.5		6	156.155	even	2
8450.2.a.bn.1.2		3	260.239	even	4
8450.2.a.bx.1.2		3	260.99	even	4

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:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.35690 q^{3} +0.890084i q^{5} -4.49396i q^{7} +2.55496 q^{9} +2.69202i q^{11} -2.09783i q^{15} -3.58211 q^{17} +2.93900i q^{19} +10.5918i q^{21} -6.09783 q^{23} +4.20775 q^{25} +1.04892 q^{27} +2.98792 q^{29} +2.39612i q^{31} -6.34481i q^{33} +4.00000 q^{35} +1.50604i q^{37} -3.65279i q^{41} +0.170915 q^{43} +2.27413i q^{45} +5.20775i q^{47} -13.1957 q^{49} +8.44265 q^{51} +6.09783 q^{53} -2.39612 q^{55} -6.92692i q^{57} +3.07069i q^{59} -13.9758 q^{61} -11.4819i q^{63} -11.0707i q^{67} +14.3720 q^{69} +10.0978i q^{71} -10.9487i q^{73} -9.91723 q^{75} +12.0978 q^{77} -2.81163 q^{79} -10.1371 q^{81} -2.93900i q^{83} -3.18837i q^{85} -7.04221 q^{87} -12.1806i q^{89} -5.64742i q^{93} -2.61596 q^{95} +12.9661i q^{97} +6.87800i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 16 q^{9} - 10 q^{17} - 10 q^{25} - 12 q^{27} - 20 q^{29} + 24 q^{35} + 22 q^{43} - 6 q^{49} - 32 q^{51} - 32 q^{55} - 8 q^{61} + 28 q^{69} - 46 q^{75} + 36 q^{77} + 36 q^{79} - 50 q^{81}+ \cdots - 36 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 + 16 * q^9 - 10 * q^17 - 10 * q^25 - 12 * q^27 - 20 * q^29 + 24 * q^35 + 22 * q^43 - 6 * q^49 - 32 * q^51 - 32 * q^55 - 8 * q^61 + 28 * q^69 - 46 * q^75 + 36 * q^77 + 36 * q^79 - 50 * q^81 + 20 * q^87 - 36 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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