LMFDB - Embedded newform 3042.2.b.n.1351.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1351,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3042.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (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:	$24.2904922949$
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			1351.5
Root			$0.445042i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.n.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +0.890084i q^{5} -4.49396i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +0.890084i q^{5} -4.49396i q^{7} -1.00000i q^{8} -0.890084 q^{10} -2.69202i q^{11} +4.49396 q^{14} +1.00000 q^{16} +3.58211 q^{17} +2.93900i q^{19} -0.890084i q^{20} +2.69202 q^{22} -6.09783 q^{23} +4.20775 q^{25} +4.49396i q^{28} -2.98792 q^{29} +2.39612i q^{31} +1.00000i q^{32} +3.58211i q^{34} +4.00000 q^{35} -1.50604i q^{37} -2.93900 q^{38} +0.890084 q^{40} -3.65279i q^{41} -0.170915 q^{43} +2.69202i q^{44} -6.09783i q^{46} -5.20775i q^{47} -13.1957 q^{49} +4.20775i q^{50} -6.09783 q^{53} +2.39612 q^{55} -4.49396 q^{56} -2.98792i q^{58} -3.07069i q^{59} -13.9758 q^{61} -2.39612 q^{62} -1.00000 q^{64} -11.0707i q^{67} -3.58211 q^{68} +4.00000i q^{70} -10.0978i q^{71} +10.9487i q^{73} +1.50604 q^{74} -2.93900i q^{76} -12.0978 q^{77} +2.81163 q^{79} +0.890084i q^{80} +3.65279 q^{82} +2.93900i q^{83} +3.18837i q^{85} -0.170915i q^{86} -2.69202 q^{88} -12.1806i q^{89} +6.09783 q^{92} +5.20775 q^{94} -2.61596 q^{95} -12.9661i q^{97} -13.1957i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4}+O(q^{10}) $ 6 * q - 6 * q^4	$ 6 q - 6 q^{4} - 4 q^{10} + 8 q^{14} + 6 q^{16} + 10 q^{17} + 6 q^{22} - 10 q^{25} + 20 q^{29} + 24 q^{35} + 2 q^{38} + 4 q^{40} - 22 q^{43} - 6 q^{49} + 32 q^{55} - 8 q^{56} - 8 q^{61} - 32 q^{62} - 6 q^{64} - 10 q^{68} + 28 q^{74} - 36 q^{77} - 36 q^{79} - 14 q^{82} - 6 q^{88} - 4 q^{94} - 36 q^{95}+O(q^{100}) $ 6 * q - 6 * q^4 - 4 * q^10 + 8 * q^14 + 6 * q^16 + 10 * q^17 + 6 * q^22 - 10 * q^25 + 20 * q^29 + 24 * q^35 + 2 * q^38 + 4 * q^40 - 22 * q^43 - 6 * q^49 + 32 * q^55 - 8 * q^56 - 8 * q^61 - 32 * q^62 - 6 * q^64 - 10 * q^68 + 28 * q^74 - 36 * q^77 - 36 * q^79 - 14 * q^82 - 6 * q^88 - 4 * q^94 - 36 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	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	$-0.322963\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	−0.890084	−0.281469
$11$	− 2.69202i	− 0.811675i	−0.913945	−	0.405838i	$-0.866980\pi$
$11$	− 2.69202i	− 0.811675i	0.913945	−	0.405838i	$-0.133020\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	4.49396	1.20106
$15$	0	0
$16$	1.00000	0.250000
$17$	3.58211	0.868788	0.434394	−	0.900723i	$-0.356963\pi$
$17$	3.58211	0.868788	0.434394	+	0.900723i	$0.356963\pi$
$18$	0	0
$19$	2.93900i	0.674253i	0.941459	+	0.337127i	$0.109455\pi$
$19$	2.93900i	0.674253i	−0.941459	+	0.337127i	$0.890545\pi$
$20$	− 0.890084i	− 0.199029i
$21$	0	0
$22$	2.69202	0.573941
$23$	−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
$27$	0	0
$28$	4.49396i	0.849278i
$29$	−2.98792	−0.554843	−0.277421	−	0.960748i	$-0.589480\pi$
$29$	−2.98792	−0.554843	−0.277421	+	0.960748i	$0.589480\pi$
$30$	0	0
$31$	2.39612i	0.430357i	0.976575	+	0.215178i	$0.0690333\pi$
$31$	2.39612i	0.430357i	−0.976575	+	0.215178i	$0.930967\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	3.58211i	0.614326i
$35$	4.00000	0.676123
$36$	0	0
$37$	− 1.50604i	− 0.247592i	−0.992308	−	0.123796i	$-0.960493\pi$
$37$	− 1.50604i	− 0.247592i	0.992308	−	0.123796i	$-0.0395068\pi$
$38$	−2.93900	−0.476769
$39$	0	0
$40$	0.890084	0.140735
$41$	− 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	$-0.504148\pi$
$43$	−0.170915	−0.0260643	−0.0130322	+	0.999915i	$0.504148\pi$
$44$	2.69202i	0.405838i
$45$	0	0
$46$	− 6.09783i	− 0.899077i
$47$	− 5.20775i	− 0.759629i	−0.925063	−	0.379814i	$-0.875988\pi$
$47$	− 5.20775i	− 0.759629i	0.925063	−	0.379814i	$-0.124012\pi$
$48$	0	0
$49$	−13.1957	−1.88510
$50$	4.20775i	0.595066i
$51$	0	0
$52$	0	0
$53$	−6.09783	−0.837602	−0.418801	−	0.908078i	$-0.637550\pi$
$53$	−6.09783	−0.837602	−0.418801	+	0.908078i	$0.637550\pi$
$54$	0	0
$55$	2.39612	0.323093
$56$	−4.49396	−0.600531
$57$	0	0
$58$	− 2.98792i	− 0.392333i
$59$	− 3.07069i	− 0.399769i	−0.979819	−	0.199885i	$-0.935943\pi$
$59$	− 3.07069i	− 0.399769i	0.979819	−	0.199885i	$-0.0640568\pi$
$60$	0	0
$61$	−13.9758	−1.78942	−0.894711	−	0.446645i	$-0.852619\pi$
$61$	−13.9758	−1.78942	−0.894711	+	0.446645i	$0.852619\pi$
$62$	−2.39612	−0.304308
$63$	0	0
$64$	−1.00000	−0.125000
$65$	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	$-0.236396\pi$
$68$	−3.58211	−0.434394
$69$	0	0
$70$	4.00000i	0.478091i
$71$	− 10.0978i	− 1.19839i	−0.800602	−	0.599196i	$-0.795487\pi$
$71$	− 10.0978i	− 1.19839i	0.800602	−	0.599196i	$-0.204513\pi$
$72$	0	0
$73$	10.9487i	1.28145i	0.767772	+	0.640724i	$0.221366\pi$
$73$	10.9487i	1.28145i	−0.767772	+	0.640724i	$0.778634\pi$
$74$	1.50604	0.175074
$75$	0	0
$76$	− 2.93900i	− 0.337127i
$77$	−12.0978	−1.37868
$78$	0	0
$79$	2.81163	0.316333	0.158166	−	0.987412i	$-0.449442\pi$
$79$	2.81163	0.316333	0.158166	+	0.987412i	$0.449442\pi$
$80$	0.890084i	0.0995144i
$81$	0	0
$82$	3.65279	0.403383
$83$	2.93900i	0.322597i	0.986906	+	0.161299i	$0.0515682\pi$
$83$	2.93900i	0.322597i	−0.986906	+	0.161299i	$0.948432\pi$
$84$	0	0
$85$	3.18837i	0.345828i
$86$	− 0.170915i	− 0.0184303i
$87$	0	0
$88$	−2.69202	−0.286970
$89$	− 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
$92$	6.09783	0.635743
$93$	0	0
$94$	5.20775	0.537138
$95$	−2.61596	−0.268392
$96$	0	0
$97$	− 12.9661i	− 1.31651i	−0.752794	−	0.658256i	$-0.771294\pi$
$97$	− 12.9661i	− 1.31651i	0.752794	−	0.658256i	$-0.228706\pi$
$98$	− 13.1957i	− 1.33296i
$99$	0	0
$100$	−4.20775	−0.420775
$101$	−10.1957	−1.01451	−0.507254	−	0.861797i	$-0.669339\pi$
$101$	−10.1957	−1.01451	−0.507254	+	0.861797i	$0.669339\pi$
$102$	0	0
$103$	−10.6703	−1.05137	−0.525686	−	0.850679i	$-0.676191\pi$
$103$	−10.6703	−1.05137	−0.525686	+	0.850679i	$0.676191\pi$
$104$	0	0
$105$	0	0
$106$	− 6.09783i	− 0.592274i
$107$	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	$-0.958513\pi$
$109$	− 2.71379i	− 0.259934i	0.991518	−	0.129967i	$-0.0414872\pi$
$110$	2.39612i	0.228462i
$111$	0	0
$112$	− 4.49396i	− 0.424639i
$113$	−20.6504	−1.94263	−0.971313	−	0.237804i	$-0.923572\pi$
$113$	−20.6504	−1.94263	−0.971313	+	0.237804i	$0.923572\pi$
$114$	0	0
$115$	− 5.42758i	− 0.506125i
$116$	2.98792	0.277421
$117$	0	0
$118$	3.07069	0.282680
$119$	− 16.0978i	− 1.47569i
$120$	0	0
$121$	3.75302	0.341184
$122$	− 13.9758i	− 1.26531i
$123$	0	0
$124$	− 2.39612i	− 0.215178i
$125$	8.19567i	0.733043i
$126$	0	0
$127$	−12.7681	−1.13298	−0.566492	−	0.824067i	$-0.691700\pi$
$127$	−12.7681	−1.13298	−0.566492	+	0.824067i	$0.691700\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	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
$134$	11.0707	0.956362
$135$	0	0
$136$	− 3.58211i	− 0.307163i
$137$	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	$-0.588778\pi$
$139$	−6.49157	−0.550607	−0.275304	+	0.961357i	$0.588778\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	10.0978	0.847391
$143$	0	0
$144$	0	0
$145$	− 2.65950i	− 0.220859i
$146$	−10.9487	−0.906120
$147$	0	0
$148$	1.50604i	0.123796i
$149$	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	$-0.0223717\pi$
$152$	2.93900	0.238384
$153$	0	0
$154$	− 12.0978i	− 0.974871i
$155$	−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	$0.287636\pi$
$158$	2.81163i	0.223681i
$159$	0	0
$160$	−0.890084	−0.0703673
$161$	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	$-0.280959\pi$
$164$	3.65279i	0.285235i
$165$	0	0
$166$	−2.93900	−0.228111
$167$	14.0000i	1.08335i	0.840587	+	0.541676i	$0.182210\pi$
$167$	14.0000i	1.08335i	−0.840587	+	0.541676i	$0.817790\pi$
$168$	0	0
$169$	0	0
$170$	−3.18837	−0.244537
$171$	0	0
$172$	0.170915	0.0130322
$173$	1.20775	0.0918236	0.0459118	−	0.998945i	$-0.485381\pi$
$173$	1.20775	0.0918236	0.0459118	+	0.998945i	$0.485381\pi$
$174$	0	0
$175$	− 18.9095i	− 1.42942i
$176$	− 2.69202i	− 0.202919i
$177$	0	0
$178$	12.1806	0.912975
$179$	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
$183$	0	0
$184$	6.09783i	0.449538i
$185$	1.34050	0.0985557
$186$	0	0
$187$	− 9.64310i	− 0.705174i
$188$	5.20775i	0.379814i
$189$	0	0
$190$	− 2.61596i	− 0.189781i
$191$	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	$-0.953953\pi$
$193$	− 4.00538i	− 0.288313i	0.989555	−	0.144157i	$-0.0460469\pi$
$194$	12.9661	0.930915
$195$	0	0
$196$	13.1957	0.942548
$197$	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	$-0.699359\pi$
$199$	−16.5375	−1.17231	−0.586156	+	0.810198i	$0.699359\pi$
$200$	− 4.20775i	− 0.297533i
$201$	0	0
$202$	− 10.1957i	− 0.717365i
$203$	13.4276i	0.942432i
$204$	0	0
$205$	3.25129	0.227080
$206$	− 10.6703i	− 0.743432i
$207$	0	0
$208$	0	0
$209$	7.91185	0.547274
$210$	0	0
$211$	−1.66056	−0.114318	−0.0571589	−	0.998365i	$-0.518204\pi$
$211$	−1.66056	−0.114318	−0.0571589	+	0.998365i	$0.518204\pi$
$212$	6.09783	0.418801
$213$	0	0
$214$	20.5623i	1.40561i
$215$	− 0.152129i	− 0.0103751i
$216$	0	0
$217$	10.7681	0.730985
$218$	2.71379	0.183801
$219$	0	0
$220$	−2.39612	−0.161547
$221$	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	$0.991555\pi$
$224$	4.49396	0.300265
$225$	0	0
$226$	− 20.6504i	− 1.37364i
$227$	− 21.7603i	− 1.44428i	−0.691745	−	0.722141i	$-0.743158\pi$
$227$	− 21.7603i	− 1.44428i	0.691745	−	0.722141i	$-0.256842\pi$
$228$	0	0
$229$	− 1.97584i	− 0.130567i	−0.997867	−	0.0652835i	$-0.979205\pi$
$229$	− 1.97584i	− 0.130567i	0.997867	−	0.0652835i	$-0.0207952\pi$
$230$	5.42758	0.357884
$231$	0	0
$232$	2.98792i	0.196166i
$233$	18.2349	1.19461	0.597304	−	0.802015i	$-0.296239\pi$
$233$	18.2349	1.19461	0.597304	+	0.802015i	$0.296239\pi$
$234$	0	0
$235$	4.63533	0.302376
$236$	3.07069i	0.199885i
$237$	0	0
$238$	16.0978	1.04347
$239$	− 22.0737i	− 1.42783i	−0.700234	−	0.713914i	$-0.746921\pi$
$239$	− 22.0737i	− 1.42783i	0.700234	−	0.713914i	$-0.253079\pi$
$240$	0	0
$241$	6.98792i	0.450131i	0.974344	+	0.225066i	$0.0722597\pi$
$241$	6.98792i	0.450131i	−0.974344	+	0.225066i	$0.927740\pi$
$242$	3.75302i	0.241253i
$243$	0	0
$244$	13.9758	0.894711
$245$	− 11.7453i	− 0.750377i
$246$	0	0
$247$	0	0
$248$	2.39612	0.152154
$249$	0	0
$250$	−8.19567	−0.518340
$251$	−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
$254$	− 12.7681i	− 0.801141i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.9933	−1.06001	−0.530006	−	0.847994i	$-0.677810\pi$
$257$	−16.9933	−1.06001	−0.530006	+	0.847994i	$0.677810\pi$
$258$	0	0
$259$	−6.76809	−0.420548
$260$	0	0
$261$	0	0
$262$	5.15883i	0.318714i
$263$	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
$266$	13.2078i	0.809819i
$267$	0	0
$268$	11.0707i	0.676250i
$269$	−15.5603	−0.948730	−0.474365	−	0.880328i	$-0.657322\pi$
$269$	−15.5603	−0.948730	−0.474365	+	0.880328i	$0.657322\pi$
$270$	0	0
$271$	− 21.9952i	− 1.33611i	−0.744110	−	0.668057i	$-0.767126\pi$
$271$	− 21.9952i	− 1.33611i	0.744110	−	0.668057i	$-0.232874\pi$
$272$	3.58211	0.217197
$273$	0	0
$274$	−3.30127	−0.199437
$275$	− 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	$0.482032\pi$
$278$	− 6.49157i	− 0.389338i
$279$	0	0
$280$	− 4.00000i	− 0.239046i
$281$	− 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	$-0.544692\pi$
$283$	−4.70841	−0.279886	−0.139943	+	0.990160i	$0.544692\pi$
$284$	10.0978i	0.599196i
$285$	0	0
$286$	0	0
$287$	−16.4155	−0.968976
$288$	0	0
$289$	−4.16852	−0.245207
$290$	2.65950	0.156171
$291$	0	0
$292$	− 10.9487i	− 0.640724i
$293$	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
$296$	−1.50604	−0.0875368
$297$	0	0
$298$	−5.03146	−0.291465
$299$	0	0
$300$	0	0
$301$	0.768086i	0.0442717i
$302$	1.72587	0.0993128
$303$	0	0
$304$	2.93900i	0.168563i
$305$	− 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	$0.926382\pi$
$308$	12.0978	0.689338
$309$	0	0
$310$	− 2.13275i	− 0.121132i
$311$	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	$0.460548\pi$
$314$	15.5060i	0.875057i
$315$	0	0
$316$	−2.81163	−0.158166
$317$	− 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
$320$	− 0.890084i	− 0.0497572i
$321$	0	0
$322$	−27.4034	−1.52713
$323$	10.5278i	0.585783i
$324$	0	0
$325$	0	0
$326$	19.7235	1.09238
$327$	0	0
$328$	−3.65279	−0.201692
$329$	−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	$-0.0539063\pi$
$332$	− 2.93900i	− 0.161299i
$333$	0	0
$334$	−14.0000	−0.766046
$335$	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
$339$	0	0
$340$	− 3.18837i	− 0.172914i
$341$	6.45042	0.349310
$342$	0	0
$343$	27.8431i	1.50339i
$344$	0.170915i	0.00921513i
$345$	0	0
$346$	1.20775i	0.0649291i
$347$	−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	$-0.820994\pi$
$349$	− 19.9215i	− 1.06638i	0.845997	−	0.533188i	$-0.179006\pi$
$350$	18.9095	1.01075
$351$	0	0
$352$	2.69202	0.143485
$353$	− 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
$356$	12.1806i	0.645571i
$357$	0	0
$358$	16.5157i	0.872883i
$359$	21.6039i	1.14021i	0.821572	+	0.570104i	$0.193097\pi$
$359$	21.6039i	1.14021i	−0.821572	+	0.570104i	$0.806903\pi$
$360$	0	0
$361$	10.3623	0.545383
$362$	− 6.37196i	− 0.334903i
$363$	0	0
$364$	0	0
$365$	−9.74525	−0.510090
$366$	0	0
$367$	18.7681	0.979686	0.489843	−	0.871811i	$-0.337054\pi$
$367$	18.7681	0.979686	0.489843	+	0.871811i	$0.337054\pi$
$368$	−6.09783	−0.317872
$369$	0	0
$370$	1.34050i	0.0696894i
$371$	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	$0.578483\pi$
$374$	9.64310	0.498633
$375$	0	0
$376$	−5.20775	−0.268569
$377$	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	$-0.307214\pi$
$380$	2.61596	0.134196
$381$	0	0
$382$	2.49396i	0.127602i
$383$	15.9517i	0.815092i	0.913185	+	0.407546i	$0.133615\pi$
$383$	15.9517i	0.815092i	−0.913185	+	0.407546i	$0.866385\pi$
$384$	0	0
$385$	− 10.7681i	− 0.548792i
$386$	4.00538	0.203868
$387$	0	0
$388$	12.9661i	0.658256i
$389$	−18.8659	−0.956540	−0.478270	−	0.878213i	$-0.658736\pi$
$389$	−18.8659	−0.956540	−0.478270	+	0.878213i	$0.658736\pi$
$390$	0	0
$391$	−21.8431	−1.10465
$392$	13.1957i	0.666482i
$393$	0	0
$394$	−23.6340	−1.19066
$395$	2.50258i	0.125919i
$396$	0	0
$397$	37.6969i	1.89195i	0.324233	+	0.945977i	$0.394894\pi$
$397$	37.6969i	1.89195i	−0.324233	+	0.945977i	$0.605106\pi$
$398$	− 16.5375i	− 0.828950i
$399$	0	0
$400$	4.20775	0.210388
$401$	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
$404$	10.1957	0.507254
$405$	0	0
$406$	−13.4276	−0.666400
$407$	−4.05429	−0.200964
$408$	0	0
$409$	3.54527i	0.175302i	0.996151	+	0.0876511i	$0.0279361\pi$
$409$	3.54527i	0.175302i	−0.996151	+	0.0876511i	$0.972064\pi$
$410$	3.25129i	0.160570i
$411$	0	0
$412$	10.6703	0.525686
$413$	−13.7995	−0.679031
$414$	0	0
$415$	−2.61596	−0.128412
$416$	0	0
$417$	0	0
$418$	7.91185i	0.386981i
$419$	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	$-0.920861\pi$
$421$	− 10.0978i	− 0.492138i	0.969252	−	0.246069i	$-0.0791390\pi$
$422$	− 1.66056i	− 0.0808349i
$423$	0	0
$424$	6.09783i	0.296137i
$425$	15.0726	0.731129
$426$	0	0
$427$	62.8068i	3.03944i
$428$	−20.5623	−0.993914
$429$	0	0
$430$	0.152129	0.00733630
$431$	− 20.2198i	− 0.973955i	−0.873414	−	0.486978i	$-0.838099\pi$
$431$	− 20.2198i	− 0.973955i	0.873414	−	0.486978i	$-0.161901\pi$
$432$	0	0
$433$	−36.4849	−1.75335	−0.876675	−	0.481083i	$-0.840244\pi$
$433$	−36.4849	−1.75335	−0.876675	+	0.481083i	$0.840244\pi$
$434$	10.7681i	0.516885i
$435$	0	0
$436$	2.71379i	0.129967i
$437$	− 17.9215i	− 0.857304i
$438$	0	0
$439$	28.3612	1.35361	0.676803	−	0.736164i	$-0.263365\pi$
$439$	28.3612	1.35361	0.676803	+	0.736164i	$0.263365\pi$
$440$	− 2.39612i	− 0.114231i
$441$	0	0
$442$	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
$446$	−0.792249	−0.0375141
$447$	0	0
$448$	4.49396i	0.212320i
$449$	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
$452$	20.6504	0.971313
$453$	0	0
$454$	21.7603	1.02126
$455$	0	0
$456$	0	0
$457$	13.6474i	0.638399i	0.947688	+	0.319200i	$0.103414\pi$
$457$	13.6474i	0.638399i	−0.947688	+	0.319200i	$0.896586\pi$
$458$	1.97584	0.0923248
$459$	0	0
$460$	5.42758i	0.253062i
$461$	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	$-0.0314864\pi$
$464$	−2.98792	−0.138711
$465$	0	0
$466$	18.2349i	0.844715i
$467$	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
$470$	4.63533i	0.213812i
$471$	0	0
$472$	−3.07069	−0.141340
$473$	0.460107i	0.0211558i
$474$	0	0
$475$	12.3666i	0.567418i
$476$	16.0978i	0.737843i
$477$	0	0
$478$	22.0737	1.00963
$479$	− 31.0267i	− 1.41764i	−0.705387	−	0.708822i	$-0.749227\pi$
$479$	− 31.0267i	− 1.41764i	0.705387	−	0.708822i	$-0.250773\pi$
$480$	0	0
$481$	0	0
$482$	−6.98792	−0.318291
$483$	0	0
$484$	−3.75302	−0.170592
$485$	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	$0.919833\pi$
$488$	13.9758i	0.632656i
$489$	0	0
$490$	11.7453	0.530596
$491$	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
$495$	0	0
$496$	2.39612i	0.107589i
$497$	−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	$0.876798\pi$
$500$	− 8.19567i	− 0.366521i
$501$	0	0
$502$	− 12.5593i	− 0.560548i
$503$	−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
$506$	−16.4155	−0.729758
$507$	0	0
$508$	12.7681	0.566492
$509$	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
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 16.9933i	− 0.749542i
$515$	− 9.49742i	− 0.418506i
$516$	0	0
$517$	−14.0194	−0.616572
$518$	− 6.76809i	− 0.297373i
$519$	0	0
$520$	0	0
$521$	−26.1564	−1.14593	−0.572967	−	0.819578i	$-0.694208\pi$
$521$	−26.1564	−1.14593	−0.572967	+	0.819578i	$0.694208\pi$
$522$	0	0
$523$	4.91185	0.214780	0.107390	−	0.994217i	$-0.465751\pi$
$523$	4.91185	0.214780	0.107390	+	0.994217i	$0.465751\pi$
$524$	−5.15883	−0.225365
$525$	0	0
$526$	4.39612i	0.191680i
$527$	8.58317i	0.373889i
$528$	0	0
$529$	14.1836	0.616678
$530$	5.42758	0.235759
$531$	0	0
$532$	−13.2078	−0.572629
$533$	0	0
$534$	0	0
$535$	18.3021i	0.791270i
$536$	−11.0707	−0.478181
$537$	0	0
$538$	− 15.5603i	− 0.670854i
$539$	35.5230i	1.53009i
$540$	0	0
$541$	0.459042i	0.0197358i	0.999951	+	0.00986789i	$0.00314110\pi$
$541$	0.459042i	0.0197358i	−0.999951	+	0.00986789i	$0.996859\pi$
$542$	21.9952	0.944775
$543$	0	0
$544$	3.58211i	0.153581i
$545$	2.41550	0.103469
$546$	0	0
$547$	−20.3327	−0.869365	−0.434682	−	0.900584i	$-0.643139\pi$
$547$	−20.3327	−0.869365	−0.434682	+	0.900584i	$0.643139\pi$
$548$	− 3.30127i	− 0.141023i
$549$	0	0
$550$	11.3274	0.483000
$551$	− 8.78150i	− 0.374104i
$552$	0	0
$553$	− 12.6353i	− 0.537309i
$554$	1.87800i	0.0797887i
$555$	0	0
$556$	6.49157	0.275304
$557$	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
$560$	4.00000	0.169031
$561$	0	0
$562$	9.20536	0.388305
$563$	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
$566$	− 4.70841i	− 0.197909i
$567$	0	0
$568$	−10.0978	−0.423696
$569$	8.28919	0.347501	0.173751	−	0.984790i	$-0.444411\pi$
$569$	8.28919	0.347501	0.173751	+	0.984790i	$0.444411\pi$
$570$	0	0
$571$	22.9836	0.961834	0.480917	−	0.876766i	$-0.340304\pi$
$571$	22.9836	0.961834	0.480917	+	0.876766i	$0.340304\pi$
$572$	0	0
$573$	0	0
$574$	− 16.4155i	− 0.685170i
$575$	−25.6582	−1.07002
$576$	0	0
$577$	− 0.553630i	− 0.0230479i	−0.999934	−	0.0115240i	$-0.996332\pi$
$577$	− 0.553630i	− 0.0230479i	0.999934	−	0.0115240i	$-0.00366827\pi$
$578$	− 4.16852i	− 0.173388i
$579$	0	0
$580$	2.65950i	0.110430i
$581$	13.2078	0.547950
$582$	0	0
$583$	16.4155i	0.679861i
$584$	10.9487	0.453060
$585$	0	0
$586$	−13.7017	−0.566012
$587$	− 16.0355i	− 0.661856i	−0.943656	−	0.330928i	$-0.892638\pi$
$587$	− 16.0355i	− 0.661856i	0.943656	−	0.330928i	$-0.107362\pi$
$588$	0	0
$589$	−7.04221	−0.290169
$590$	2.73317i	0.112523i
$591$	0	0
$592$	− 1.50604i	− 0.0618979i
$593$	− 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
$596$	− 5.03146i	− 0.206097i
$597$	0	0
$598$	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	$0.705659\pi$
$602$	−0.768086	−0.0313048
$603$	0	0
$604$	1.72587i	0.0702248i
$605$	3.34050i	0.135811i
$606$	0	0
$607$	37.4228	1.51894	0.759472	−	0.650540i	$-0.225457\pi$
$607$	37.4228	1.51894	0.759472	+	0.650540i	$0.225457\pi$
$608$	−2.93900	−0.119192
$609$	0	0
$610$	12.4397	0.503667
$611$	0	0
$612$	0	0
$613$	38.0737i	1.53778i	0.639380	+	0.768891i	$0.279191\pi$
$613$	38.0737i	1.53778i	−0.639380	+	0.768891i	$0.720809\pi$
$614$	−8.03252	−0.324166
$615$	0	0
$616$	12.0978i	0.487436i
$617$	− 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	$0.989305\pi$
$620$	2.13275	0.0856534
$621$	0	0
$622$	4.09783i	0.164308i
$623$	−54.7391	−2.19308
$624$	0	0
$625$	13.7439	0.549757
$626$	4.37435i	0.174834i
$627$	0	0
$628$	−15.5060	−0.618758
$629$	− 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	$-0.0554208\pi$
$632$	− 2.81163i	− 0.111840i
$633$	0	0
$634$	20.3612	0.808647
$635$	− 11.3647i	− 0.450993i
$636$	0	0
$637$	0	0
$638$	−8.04354	−0.318447
$639$	0	0
$640$	0.890084	0.0351836
$641$	−19.9075	−0.786301	−0.393150	−	0.919474i	$-0.628615\pi$
$641$	−19.9075	−0.786301	−0.393150	+	0.919474i	$0.628615\pi$
$642$	0	0
$643$	27.0756i	1.06776i	0.845561	+	0.533879i	$0.179266\pi$
$643$	27.0756i	1.06776i	−0.845561	+	0.533879i	$0.820734\pi$
$644$	− 27.4034i	− 1.07985i
$645$	0	0
$646$	−10.5278	−0.414211
$647$	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
$651$	0	0
$652$	19.7235i	0.772431i
$653$	37.6292	1.47255	0.736273	−	0.676685i	$-0.236584\pi$
$653$	37.6292	1.47255	0.736273	+	0.676685i	$0.236584\pi$
$654$	0	0
$655$	4.59179i	0.179416i
$656$	− 3.65279i	− 0.142618i
$657$	0	0
$658$	− 23.4034i	− 0.912360i
$659$	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	$0.128480\pi$
$661$	20.1957i	0.785520i	−0.919641	+	0.392760i	$0.871520\pi$
$662$	6.13275	0.238356
$663$	0	0
$664$	2.93900	0.114055
$665$	11.7560i	0.455878i
$666$	0	0
$667$	18.2198	0.705475
$668$	− 14.0000i	− 0.541676i
$669$	0	0
$670$	9.85384i	0.380687i
$671$	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	$0.301162\pi$
$674$	27.8485i	1.07268i
$675$	0	0
$676$	0	0
$677$	21.1642	0.813407	0.406703	−	0.913560i	$-0.366678\pi$
$677$	21.1642	0.813407	0.406703	+	0.913560i	$0.366678\pi$
$678$	0	0
$679$	−58.2693	−2.23617
$680$	3.18837	0.122269
$681$	0	0
$682$	6.45042i	0.246999i
$683$	− 35.4873i	− 1.35788i	−0.734193	−	0.678941i	$-0.762439\pi$
$683$	− 35.4873i	− 1.35788i	0.734193	−	0.678941i	$-0.237561\pi$
$684$	0	0
$685$	−2.93841	−0.112271
$686$	−27.8431	−1.06305
$687$	0	0
$688$	−0.170915	−0.00651608
$689$	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	$0.753645\pi$
$692$	−1.20775	−0.0459118
$693$	0	0
$694$	− 4.43967i	− 0.168527i
$695$	− 5.77804i	− 0.219173i
$696$	0	0
$697$	− 13.0847i	− 0.495618i
$698$	19.9215	0.754042
$699$	0	0
$700$	18.9095i	0.714710i
$701$	19.7668	0.746580	0.373290	−	0.927715i	$-0.378230\pi$
$701$	19.7668	0.746580	0.373290	+	0.927715i	$0.378230\pi$
$702$	0	0
$703$	4.42626	0.166939
$704$	2.69202i	0.101459i
$705$	0	0
$706$	30.5894	1.15125
$707$	45.8189i	1.72320i
$708$	0	0
$709$	− 11.6280i	− 0.436700i	−0.975871	−	0.218350i	$-0.929933\pi$
$709$	− 11.6280i	− 0.436700i	0.975871	−	0.218350i	$-0.0700675\pi$
$710$	8.98792i	0.337311i
$711$	0	0
$712$	−12.1806	−0.456487
$713$	− 14.6112i	− 0.547193i
$714$	0	0
$715$	0	0
$716$	−16.5157	−0.617222
$717$	0	0
$718$	−21.6039	−0.806249
$719$	−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
$722$	10.3623i	0.385644i
$723$	0	0
$724$	6.37196	0.236812
$725$	−12.5724	−0.466928
$726$	0	0
$727$	1.26205	0.0468067	0.0234033	−	0.999726i	$-0.492550\pi$
$727$	1.26205	0.0468067	0.0234033	+	0.999726i	$0.492550\pi$
$728$	0	0
$729$	0	0
$730$	− 9.74525i	− 0.360688i
$731$	−0.612236	−0.0226444
$732$	0	0
$733$	− 20.6789i	− 0.763792i	−0.924205	−	0.381896i	$-0.875271\pi$
$733$	− 20.6789i	− 0.763792i	0.924205	−	0.381896i	$-0.124729\pi$
$734$	18.7681i	0.692743i
$735$	0	0
$736$	− 6.09783i	− 0.224769i
$737$	−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	$-0.0303336\pi$
$740$	−1.34050	−0.0492778
$741$	0	0
$742$	−27.4034	−1.00601
$743$	10.4397i	0.382994i	0.981493	+	0.191497i	$0.0613343\pi$
$743$	10.4397i	0.382994i	−0.981493	+	0.191497i	$0.938666\pi$
$744$	0	0
$745$	−4.47842	−0.164077
$746$	− 9.42758i	− 0.345168i
$747$	0	0
$748$	9.64310i	0.352587i
$749$	− 92.4059i	− 3.37644i
$750$	0	0
$751$	14.0435	0.512456	0.256228	−	0.966616i	$-0.417520\pi$
$751$	14.0435	0.512456	0.256228	+	0.966616i	$0.417520\pi$
$752$	− 5.20775i	− 0.189907i
$753$	0	0
$754$	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	$0.554089\pi$
$758$	32.0103	1.16267
$759$	0	0
$760$	2.61596i	0.0948907i
$761$	− 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
$764$	−2.49396	−0.0902283
$765$	0	0
$766$	−15.9517	−0.576357
$767$	0	0
$768$	0	0
$769$	− 31.9299i	− 1.15142i	−0.817653	−	0.575711i	$-0.804725\pi$
$769$	− 31.9299i	− 1.15142i	0.817653	−	0.575711i	$-0.195275\pi$
$770$	10.7681	0.388055
$771$	0	0
$772$	4.00538i	0.144157i
$773$	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
$776$	−12.9661	−0.465458
$777$	0	0
$778$	− 18.8659i	− 0.676376i
$779$	10.7356	0.384641
$780$	0	0
$781$	−27.1836	−0.972705
$782$	− 21.8431i	− 0.781107i
$783$	0	0
$784$	−13.1957	−0.471274
$785$	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	$0.768161\pi$
$788$	− 23.6340i	− 0.841927i
$789$	0	0
$790$	−2.50258	−0.0890379
$791$	92.8021i	3.29966i
$792$	0	0
$793$	0	0
$794$	−37.6969	−1.33781
$795$	0	0
$796$	16.5375	0.586156
$797$	−14.9831	−0.530730	−0.265365	−	0.964148i	$-0.585492\pi$
$797$	−14.9831	−0.530730	−0.265365	+	0.964148i	$0.585492\pi$
$798$	0	0
$799$	− 18.6547i	− 0.659956i
$800$	4.20775i	0.148766i
$801$	0	0
$802$	−3.22952	−0.114038
$803$	29.4741	1.04012
$804$	0	0
$805$	−24.3913	−0.859682
$806$	0	0
$807$	0	0
$808$	10.1957i	0.358682i
$809$	−25.6770	−0.902754	−0.451377	−	0.892333i	$-0.649067\pi$
$809$	−25.6770	−0.902754	−0.451377	+	0.892333i	$0.649067\pi$
$810$	0	0
$811$	8.66786i	0.304370i	0.988352	+	0.152185i	$0.0486309\pi$
$811$	8.66786i	0.304370i	−0.988352	+	0.152185i	$0.951369\pi$
$812$	− 13.4276i	− 0.471216i
$813$	0	0
$814$	− 4.05429i	− 0.142103i
$815$	17.5555	0.614944
$816$	0	0
$817$	− 0.502320i	− 0.0175739i
$818$	−3.54527	−0.123957
$819$	0	0
$820$	−3.25129	−0.113540
$821$	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	$-0.464772\pi$
$823$	6.33704	0.220895	0.110448	+	0.993882i	$0.464772\pi$
$824$	10.6703i	0.371716i
$825$	0	0
$826$	− 13.7995i	− 0.480148i
$827$	35.5405i	1.23586i	0.786232	+	0.617932i	$0.212029\pi$
$827$	35.5405i	1.23586i	−0.786232	+	0.617932i	$0.787971\pi$
$828$	0	0
$829$	41.5555	1.44328	0.721642	−	0.692267i	$-0.243388\pi$
$829$	41.5555	1.44328	0.721642	+	0.692267i	$0.243388\pi$
$830$	− 2.61596i	− 0.0908012i
$831$	0	0
$832$	0	0
$833$	−47.2683	−1.63775
$834$	0	0
$835$	−12.4612	−0.431237
$836$	−7.91185	−0.273637
$837$	0	0
$838$	14.4155i	0.497975i
$839$	17.5496i	0.605879i	0.953010	+	0.302939i	$0.0979680\pi$
$839$	17.5496i	0.605879i	−0.953010	+	0.302939i	$0.902032\pi$
$840$	0	0
$841$	−20.0723	−0.692150
$842$	10.0978	0.347994
$843$	0	0
$844$	1.66056	0.0571589
$845$	0	0
$846$	0	0
$847$	− 16.8659i	− 0.579520i
$848$	−6.09783	−0.209401
$849$	0	0
$850$	15.0726i	0.516986i
$851$	9.18359i	0.314809i
$852$	0	0
$853$	− 20.1414i	− 0.689628i	−0.938671	−	0.344814i	$-0.887942\pi$
$853$	− 20.1414i	− 0.689628i	0.938671	−	0.344814i	$-0.112058\pi$
$854$	−62.8068	−2.14921
$855$	0	0
$856$	− 20.5623i	− 0.702803i
$857$	−8.85756	−0.302568	−0.151284	−	0.988490i	$-0.548341\pi$
$857$	−8.85756	−0.302568	−0.151284	+	0.988490i	$0.548341\pi$
$858$	0	0
$859$	28.8810	0.985407	0.492703	−	0.870197i	$-0.336009\pi$
$859$	28.8810	0.985407	0.492703	+	0.870197i	$0.336009\pi$
$860$	0.152129i	0.00518755i
$861$	0	0
$862$	20.2198	0.688690
$863$	− 3.90813i	− 0.133034i	−0.997785	−	0.0665172i	$-0.978811\pi$
$863$	− 3.90813i	− 0.133034i	0.997785	−	0.0665172i	$-0.0211887\pi$
$864$	0	0
$865$	1.07500i	0.0365511i
$866$	− 36.4849i	− 1.23981i
$867$	0	0
$868$	−10.7681	−0.365493
$869$	− 7.56896i	− 0.256759i
$870$	0	0
$871$	0	0
$872$	−2.71379	−0.0919006
$873$	0	0
$874$	17.9215	0.606205
$875$	36.8310	1.24512
$876$	0	0
$877$	− 44.9879i	− 1.51913i	−0.650429	−	0.759567i	$-0.725411\pi$
$877$	− 44.9879i	− 1.51913i	0.650429	−	0.759567i	$-0.274589\pi$
$878$	28.3612i	0.957144i
$879$	0	0
$880$	2.39612	0.0807733
$881$	−14.1933	−0.478184	−0.239092	−	0.970997i	$-0.576850\pi$
$881$	−14.1933	−0.478184	−0.239092	+	0.970997i	$0.576850\pi$
$882$	0	0
$883$	−48.4626	−1.63090	−0.815448	−	0.578830i	$-0.803510\pi$
$883$	−48.4626	−1.63090	−0.815448	+	0.578830i	$0.803510\pi$
$884$	0	0
$885$	0	0
$886$	− 2.80061i	− 0.0940883i
$887$	−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
$890$	10.8418i	0.363417i
$891$	0	0
$892$	− 0.792249i	− 0.0265265i
$893$	15.3056	0.512182
$894$	0	0
$895$	14.7004i	0.491380i
$896$	−4.49396	−0.150133
$897$	0	0
$898$	−13.2760	−0.443027
$899$	− 7.15942i	− 0.238780i
$900$	0	0
$901$	−21.8431	−0.727699
$902$	− 9.83340i	− 0.327416i
$903$	0	0
$904$	20.6504i	0.686822i
$905$	− 5.67158i	− 0.188530i
$906$	0	0
$907$	−3.94139	−0.130872	−0.0654359	−	0.997857i	$-0.520844\pi$
$907$	−3.94139	−0.130872	−0.0654359	+	0.997857i	$0.520844\pi$
$908$	21.7603i	0.722141i
$909$	0	0
$910$	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
$914$	−13.6474	−0.451416
$915$	0	0
$916$	1.97584i	0.0652835i
$917$	− 23.1836i	− 0.765590i
$918$	0	0
$919$	0.681005	0.0224643	0.0112321	−	0.999937i	$-0.496425\pi$
$919$	0.681005	0.0224643	0.0112321	+	0.999937i	$0.496425\pi$
$920$	−5.42758	−0.178942
$921$	0	0
$922$	−12.1655	−0.400651
$923$	0	0
$924$	0	0
$925$	− 6.33704i	− 0.208361i
$926$	4.24996	0.139662
$927$	0	0
$928$	− 2.98792i	− 0.0980832i
$929$	− 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
$932$	−18.2349	−0.597304
$933$	0	0
$934$	8.21552i	0.268820i
$935$	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	$0.576738\pi$
$938$	− 49.7512i	− 1.62443i
$939$	0	0
$940$	−4.63533	−0.151188
$941$	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
$944$	− 3.07069i	− 0.0999424i
$945$	0	0
$946$	−0.460107	−0.0149594
$947$	− 20.6708i	− 0.671712i	−0.941913	−	0.335856i	$-0.890974\pi$
$947$	− 20.6708i	− 0.671712i	0.941913	−	0.335856i	$-0.109026\pi$
$948$	0	0
$949$	0	0
$950$	−12.3666	−0.401225
$951$	0	0
$952$	−16.0978	−0.521734
$953$	−47.2411	−1.53029	−0.765145	−	0.643858i	$-0.777333\pi$
$953$	−47.2411	−1.53029	−0.765145	+	0.643858i	$0.777333\pi$
$954$	0	0
$955$	2.21983i	0.0718321i
$956$	22.0737i	0.713914i
$957$	0	0
$958$	31.0267	1.00243
$959$	14.8358	0.479073
$960$	0	0
$961$	25.2586	0.814793
$962$	0	0
$963$	0	0
$964$	− 6.98792i	− 0.225066i
$965$	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	$-0.334265\pi$
$968$	− 3.75302i	− 0.120627i
$969$	0	0
$970$	11.5410i	0.370558i
$971$	−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
$974$	−10.9987	−0.352420
$975$	0	0
$976$	−13.9758	−0.447356
$977$	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
$980$	11.7453i	0.375188i
$981$	0	0
$982$	21.2336i	0.677590i
$983$	41.3309i	1.31825i	0.752033	+	0.659126i	$0.229074\pi$
$983$	41.3309i	1.31825i	−0.752033	+	0.659126i	$0.770926\pi$
$984$	0	0
$985$	−21.0362	−0.670270
$986$	− 10.7030i	− 0.340854i
$987$	0	0
$988$	0	0
$989$	1.04221	0.0331404
$990$	0	0
$991$	19.2185	0.610496	0.305248	−	0.952273i	$-0.401261\pi$
$991$	19.2185	0.610496	0.305248	+	0.952273i	$0.401261\pi$
$992$	−2.39612	−0.0760770
$993$	0	0
$994$	− 45.3793i	− 1.43934i
$995$	− 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	$0.384074\pi$
$998$	−16.8635	−0.533806
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.n.1351.5		6
3.2	odd	2		338.2.b.d.337.2		6
12.11	even	2		2704.2.f.m.337.3		6
13.5	odd	4		3042.2.a.bi.1.2		3
13.8	odd	4		3042.2.a.z.1.2		3
13.12	even	2	inner	3042.2.b.n.1351.2		6
39.2	even	12		338.2.c.i.191.2		6
39.5	even	4		338.2.a.g.1.2	✓	3
39.8	even	4		338.2.a.h.1.2	yes	3
39.11	even	12		338.2.c.h.191.2		6
39.17	odd	6		338.2.e.e.23.5		12
39.20	even	12		338.2.c.h.315.2		6
39.23	odd	6		338.2.e.e.147.2		12
39.29	odd	6		338.2.e.e.147.5		12
39.32	even	12		338.2.c.i.315.2		6
39.35	odd	6		338.2.e.e.23.2		12
39.38	odd	2		338.2.b.d.337.5		6
156.47	odd	4		2704.2.a.w.1.2		3
156.83	odd	4		2704.2.a.v.1.2		3
156.155	even	2		2704.2.f.m.337.4		6
195.44	even	4		8450.2.a.bx.1.2		3
195.164	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	39.5	even	4
338.2.a.h.1.2	yes	3	39.8	even	4
338.2.b.d.337.2		6	3.2	odd	2
338.2.b.d.337.5		6	39.38	odd	2
338.2.c.h.191.2		6	39.11	even	12
338.2.c.h.315.2		6	39.20	even	12
338.2.c.i.191.2		6	39.2	even	12
338.2.c.i.315.2		6	39.32	even	12
338.2.e.e.23.2		12	39.35	odd	6
338.2.e.e.23.5		12	39.17	odd	6
338.2.e.e.147.2		12	39.23	odd	6
338.2.e.e.147.5		12	39.29	odd	6
2704.2.a.v.1.2		3	156.83	odd	4
2704.2.a.w.1.2		3	156.47	odd	4
2704.2.f.m.337.3		6	12.11	even	2
2704.2.f.m.337.4		6	156.155	even	2
3042.2.a.z.1.2		3	13.8	odd	4
3042.2.a.bi.1.2		3	13.5	odd	4
3042.2.b.n.1351.2		6	13.12	even	2	inner
3042.2.b.n.1351.5		6	1.1	even	1	trivial
8450.2.a.bn.1.2		3	195.164	even	4
8450.2.a.bx.1.2		3	195.44	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.890084i q^{5} -4.49396i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.890084i q^{5} -4.49396i q^{7} -1.00000i q^{8} -0.890084 q^{10} -2.69202i q^{11} +4.49396 q^{14} +1.00000 q^{16} +3.58211 q^{17} +2.93900i q^{19} -0.890084i q^{20} +2.69202 q^{22} -6.09783 q^{23} +4.20775 q^{25} +4.49396i q^{28} -2.98792 q^{29} +2.39612i q^{31} +1.00000i q^{32} +3.58211i q^{34} +4.00000 q^{35} -1.50604i q^{37} -2.93900 q^{38} +0.890084 q^{40} -3.65279i q^{41} -0.170915 q^{43} +2.69202i q^{44} -6.09783i q^{46} -5.20775i q^{47} -13.1957 q^{49} +4.20775i q^{50} -6.09783 q^{53} +2.39612 q^{55} -4.49396 q^{56} -2.98792i q^{58} -3.07069i q^{59} -13.9758 q^{61} -2.39612 q^{62} -1.00000 q^{64} -11.0707i q^{67} -3.58211 q^{68} +4.00000i q^{70} -10.0978i q^{71} +10.9487i q^{73} +1.50604 q^{74} -2.93900i q^{76} -12.0978 q^{77} +2.81163 q^{79} +0.890084i q^{80} +3.65279 q^{82} +2.93900i q^{83} +3.18837i q^{85} -0.170915i q^{86} -2.69202 q^{88} -12.1806i q^{89} +6.09783 q^{92} +5.20775 q^{94} -2.61596 q^{95} -12.9661i q^{97} -13.1957i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4}+O(q^{10}) \) 6 * q - 6 * q^4	\( 6 q - 6 q^{4} - 4 q^{10} + 8 q^{14} + 6 q^{16} + 10 q^{17} + 6 q^{22} - 10 q^{25} + 20 q^{29} + 24 q^{35} + 2 q^{38} + 4 q^{40} - 22 q^{43} - 6 q^{49} + 32 q^{55} - 8 q^{56} - 8 q^{61} - 32 q^{62} - 6 q^{64} - 10 q^{68} + 28 q^{74} - 36 q^{77} - 36 q^{79} - 14 q^{82} - 6 q^{88} - 4 q^{94} - 36 q^{95}+O(q^{100}) \) 6 * q - 6 * q^4 - 4 * q^10 + 8 * q^14 + 6 * q^16 + 10 * q^17 + 6 * q^22 - 10 * q^25 + 20 * q^29 + 24 * q^35 + 2 * q^38 + 4 * q^40 - 22 * q^43 - 6 * q^49 + 32 * q^55 - 8 * q^56 - 8 * q^61 - 32 * q^62 - 6 * q^64 - 10 * q^68 + 28 * q^74 - 36 * q^77 - 36 * q^79 - 14 * q^82 - 6 * q^88 - 4 * q^94 - 36 * q^95

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