LMFDB - Embedded newform 7623.2.a.cy.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 $ x^10 - 19x^8 - x^7 + 124x^6 + 6x^5 - 316x^4 + 17x^3 + 253x^2 - 70*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.80545$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.80545 q^{2} +1.25966 q^{4} -2.77715 q^{5} +1.00000 q^{7} -1.33666 q^{8} +O(q^{10})$	$q+1.80545 q^{2} +1.25966 q^{4} -2.77715 q^{5} +1.00000 q^{7} -1.33666 q^{8} -5.01400 q^{10} -2.31311 q^{13} +1.80545 q^{14} -4.93258 q^{16} +2.66163 q^{17} -8.08270 q^{19} -3.49825 q^{20} +2.43007 q^{23} +2.71254 q^{25} -4.17621 q^{26} +1.25966 q^{28} -7.55762 q^{29} +9.24567 q^{31} -6.23222 q^{32} +4.80544 q^{34} -2.77715 q^{35} +11.1976 q^{37} -14.5929 q^{38} +3.71209 q^{40} -0.299894 q^{41} +7.29328 q^{43} +4.38737 q^{46} -0.457855 q^{47} +1.00000 q^{49} +4.89737 q^{50} -2.91372 q^{52} +5.19340 q^{53} -1.33666 q^{56} -13.6449 q^{58} -12.2826 q^{59} +1.81788 q^{61} +16.6926 q^{62} -1.38682 q^{64} +6.42384 q^{65} -1.42267 q^{67} +3.35273 q^{68} -5.01400 q^{70} -0.558099 q^{71} +9.66283 q^{73} +20.2168 q^{74} -10.1814 q^{76} +10.3200 q^{79} +13.6985 q^{80} -0.541445 q^{82} -8.51075 q^{83} -7.39173 q^{85} +13.1677 q^{86} +1.56616 q^{89} -2.31311 q^{91} +3.06105 q^{92} -0.826634 q^{94} +22.4468 q^{95} -0.533862 q^{97} +1.80545 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8}+O(q^{10}) $ 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8	$ 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8} - 6 q^{10} + 6 q^{13} + 38 q^{16} - 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} + 18 q^{28} + 14 q^{29} + 26 q^{31} + 41 q^{32} + 21 q^{34} - 5 q^{35} + 24 q^{37} - 8 q^{38} - 5 q^{40} - 19 q^{41} - 6 q^{43} - q^{46} - 15 q^{47} + 10 q^{49} + q^{50} - 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} - 11 q^{62} + 53 q^{64} + 29 q^{65} + 38 q^{67} - 87 q^{68} - 6 q^{70} - 26 q^{71} - q^{73} + 39 q^{74} - 2 q^{76} + 5 q^{79} - 6 q^{80} + 5 q^{82} - 6 q^{83} - q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} + 42 q^{95} + 24 q^{97}+O(q^{100}) $ 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8 - 6 * q^10 + 6 * q^13 + 38 * q^16 - 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 + 18 * q^28 + 14 * q^29 + 26 * q^31 + 41 * q^32 + 21 * q^34 - 5 * q^35 + 24 * q^37 - 8 * q^38 - 5 * q^40 - 19 * q^41 - 6 * q^43 - q^46 - 15 * q^47 + 10 * q^49 + q^50 - 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 - 11 * q^62 + 53 * q^64 + 29 * q^65 + 38 * q^67 - 87 * q^68 - 6 * q^70 - 26 * q^71 - q^73 + 39 * q^74 - 2 * q^76 + 5 * q^79 - 6 * q^80 + 5 * q^82 - 6 * q^83 - q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 + 42 * q^95 + 24 * 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 $

$2$	1.80545	1.27665	0.638324	−	0.769768i	$-0.279628\pi$
$2$	1.80545	1.27665	0.638324	+	0.769768i	$0.279628\pi$
$3$	0	0
$3$	0	0
$4$	1.25966	0.629828
$5$	−2.77715	−1.24198	−0.620989	−	0.783819i	$-0.713269\pi$
$5$	−2.77715	−1.24198	−0.620989	+	0.783819i	$0.713269\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.33666	−0.472579
$9$	0	0
$10$	−5.01400	−1.58557
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.31311	−0.641541	−0.320771	−	0.947157i	$-0.603942\pi$
$13$	−2.31311	−0.641541	−0.320771	+	0.947157i	$0.603942\pi$
$14$	1.80545	0.482527
$15$	0	0
$16$	−4.93258	−1.23314
$17$	2.66163	0.645539	0.322770	−	0.946478i	$-0.395386\pi$
$17$	2.66163	0.645539	0.322770	+	0.946478i	$0.395386\pi$
$18$	0	0
$19$	−8.08270	−1.85430	−0.927149	−	0.374693i	$-0.877748\pi$
$19$	−8.08270	−1.85430	−0.927149	+	0.374693i	$0.877748\pi$
$20$	−3.49825	−0.782232
$21$	0	0
$22$	0	0
$23$	2.43007	0.506704	0.253352	−	0.967374i	$-0.418467\pi$
$23$	2.43007	0.506704	0.253352	+	0.967374i	$0.418467\pi$
$24$	0	0
$25$	2.71254	0.542509
$26$	−4.17621	−0.819022
$27$	0	0
$28$	1.25966	0.238053
$29$	−7.55762	−1.40342	−0.701708	−	0.712465i	$-0.747579\pi$
$29$	−7.55762	−1.40342	−0.701708	+	0.712465i	$0.747579\pi$
$30$	0	0
$31$	9.24567	1.66057	0.830285	−	0.557339i	$-0.188177\pi$
$31$	9.24567	1.66057	0.830285	+	0.557339i	$0.188177\pi$
$32$	−6.23222	−1.10171
$33$	0	0
$34$	4.80544	0.824126
$35$	−2.77715	−0.469423
$36$	0	0
$37$	11.1976	1.84088	0.920438	−	0.390888i	$-0.127832\pi$
$37$	11.1976	1.84088	0.920438	+	0.390888i	$0.127832\pi$
$38$	−14.5929	−2.36728
$39$	0	0
$40$	3.71209	0.586933
$41$	−0.299894	−0.0468356	−0.0234178	−	0.999726i	$-0.507455\pi$
$41$	−0.299894	−0.0468356	−0.0234178	+	0.999726i	$0.507455\pi$
$42$	0	0
$43$	7.29328	1.11221	0.556107	−	0.831110i	$-0.312294\pi$
$43$	7.29328	1.11221	0.556107	+	0.831110i	$0.312294\pi$
$44$	0	0
$45$	0	0
$46$	4.38737	0.646882
$47$	−0.457855	−0.0667849	−0.0333925	−	0.999442i	$-0.510631\pi$
$47$	−0.457855	−0.0667849	−0.0333925	+	0.999442i	$0.510631\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.89737	0.692592
$51$	0	0
$52$	−2.91372	−0.404060
$53$	5.19340	0.713369	0.356684	−	0.934225i	$-0.383907\pi$
$53$	5.19340	0.713369	0.356684	+	0.934225i	$0.383907\pi$
$54$	0	0
$55$	0	0
$56$	−1.33666	−0.178618
$57$	0	0
$58$	−13.6449	−1.79167
$59$	−12.2826	−1.59906	−0.799532	−	0.600623i	$-0.794919\pi$
$59$	−12.2826	−1.59906	−0.799532	+	0.600623i	$0.794919\pi$
$60$	0	0
$61$	1.81788	0.232755	0.116378	−	0.993205i	$-0.462872\pi$
$61$	1.81788	0.232755	0.116378	+	0.993205i	$0.462872\pi$
$62$	16.6926	2.11996
$63$	0	0
$64$	−1.38682	−0.173352
$65$	6.42384	0.796780
$66$	0	0
$67$	−1.42267	−0.173807	−0.0869036	−	0.996217i	$-0.527697\pi$
$67$	−1.42267	−0.173807	−0.0869036	+	0.996217i	$0.527697\pi$
$68$	3.35273	0.406579
$69$	0	0
$70$	−5.01400	−0.599288
$71$	−0.558099	−0.0662342	−0.0331171	−	0.999451i	$-0.510543\pi$
$71$	−0.558099	−0.0662342	−0.0331171	+	0.999451i	$0.510543\pi$
$72$	0	0
$73$	9.66283	1.13095	0.565474	−	0.824766i	$-0.308693\pi$
$73$	9.66283	1.13095	0.565474	+	0.824766i	$0.308693\pi$
$74$	20.2168	2.35015
$75$	0	0
$76$	−10.1814	−1.16789
$77$	0	0
$78$	0	0
$79$	10.3200	1.16109	0.580544	−	0.814229i	$-0.302840\pi$
$79$	10.3200	1.16109	0.580544	+	0.814229i	$0.302840\pi$
$80$	13.6985	1.53154
$81$	0	0
$82$	−0.541445	−0.0597926
$83$	−8.51075	−0.934177	−0.467088	−	0.884211i	$-0.654697\pi$
$83$	−8.51075	−0.934177	−0.467088	+	0.884211i	$0.654697\pi$
$84$	0	0
$85$	−7.39173	−0.801745
$86$	13.1677	1.41991
$87$	0	0
$88$	0	0
$89$	1.56616	0.166012	0.0830062	−	0.996549i	$-0.473548\pi$
$89$	1.56616	0.166012	0.0830062	+	0.996549i	$0.473548\pi$
$90$	0	0
$91$	−2.31311	−0.242480
$92$	3.06105	0.319136
$93$	0	0
$94$	−0.826634	−0.0852608
$95$	22.4468	2.30300
$96$	0	0
$97$	−0.533862	−0.0542055	−0.0271027	−	0.999633i	$-0.508628\pi$
$97$	−0.533862	−0.0542055	−0.0271027	+	0.999633i	$0.508628\pi$
$98$	1.80545	0.182378
$99$	0	0
$100$	3.41687	0.341687
$101$	−5.31761	−0.529122	−0.264561	−	0.964369i	$-0.585227\pi$
$101$	−5.31761	−0.529122	−0.264561	+	0.964369i	$0.585227\pi$
$102$	0	0
$103$	18.5606	1.82883	0.914414	−	0.404781i	$-0.132652\pi$
$103$	18.5606	1.82883	0.914414	+	0.404781i	$0.132652\pi$
$104$	3.09183	0.303179
$105$	0	0
$106$	9.37644	0.910720
$107$	−9.42725	−0.911367	−0.455683	−	0.890142i	$-0.650605\pi$
$107$	−9.42725	−0.911367	−0.455683	+	0.890142i	$0.650605\pi$
$108$	0	0
$109$	−1.11200	−0.106511	−0.0532553	−	0.998581i	$-0.516960\pi$
$109$	−1.11200	−0.106511	−0.0532553	+	0.998581i	$0.516960\pi$
$110$	0	0
$111$	0	0
$112$	−4.93258	−0.466085
$113$	9.02473	0.848975	0.424488	−	0.905434i	$-0.360454\pi$
$113$	9.02473	0.848975	0.424488	+	0.905434i	$0.360454\pi$
$114$	0	0
$115$	−6.74866	−0.629315
$116$	−9.52000	−0.883910
$117$	0	0
$118$	−22.1757	−2.04144
$119$	2.66163	0.243991
$120$	0	0
$121$	0	0
$122$	3.28209	0.297146
$123$	0	0
$124$	11.6464	1.04587
$125$	6.35260	0.568194
$126$	0	0
$127$	8.33542	0.739649	0.369824	−	0.929102i	$-0.379418\pi$
$127$	8.33542	0.739649	0.369824	+	0.929102i	$0.379418\pi$
$128$	9.96061	0.880402
$129$	0	0
$130$	11.5979	1.01721
$131$	7.97862	0.697095	0.348548	−	0.937291i	$-0.386675\pi$
$131$	7.97862	0.697095	0.348548	+	0.937291i	$0.386675\pi$
$132$	0	0
$133$	−8.08270	−0.700859
$134$	−2.56857	−0.221891
$135$	0	0
$136$	−3.55768	−0.305068
$137$	10.3316	0.882691	0.441345	−	0.897337i	$-0.354501\pi$
$137$	10.3316	0.882691	0.441345	+	0.897337i	$0.354501\pi$
$138$	0	0
$139$	5.53964	0.469866	0.234933	−	0.972012i	$-0.424513\pi$
$139$	5.53964	0.469866	0.234933	+	0.972012i	$0.424513\pi$
$140$	−3.49825	−0.295656
$141$	0	0
$142$	−1.00762	−0.0845576
$143$	0	0
$144$	0	0
$145$	20.9886	1.74301
$146$	17.4458	1.44382
$147$	0	0
$148$	14.1051	1.15944
$149$	21.9883	1.80136	0.900678	−	0.434488i	$-0.143071\pi$
$149$	21.9883	1.80136	0.900678	+	0.434488i	$0.143071\pi$
$150$	0	0
$151$	14.1658	1.15280	0.576400	−	0.817168i	$-0.304457\pi$
$151$	14.1658	1.15280	0.576400	+	0.817168i	$0.304457\pi$
$152$	10.8038	0.876303
$153$	0	0
$154$	0	0
$155$	−25.6766	−2.06239
$156$	0	0
$157$	−1.37897	−0.110054	−0.0550270	−	0.998485i	$-0.517524\pi$
$157$	−1.37897	−0.110054	−0.0550270	+	0.998485i	$0.517524\pi$
$158$	18.6322	1.48230
$159$	0	0
$160$	17.3078	1.36830
$161$	2.43007	0.191516
$162$	0	0
$163$	−17.0408	−1.33474	−0.667370	−	0.744727i	$-0.732580\pi$
$163$	−17.0408	−1.33474	−0.667370	+	0.744727i	$0.732580\pi$
$164$	−0.377764	−0.0294984
$165$	0	0
$166$	−15.3658	−1.19261
$167$	−7.96033	−0.615989	−0.307995	−	0.951388i	$-0.599658\pi$
$167$	−7.96033	−0.615989	−0.307995	+	0.951388i	$0.599658\pi$
$168$	0	0
$169$	−7.64953	−0.588425
$170$	−13.3454	−1.02355
$171$	0	0
$172$	9.18702	0.700504
$173$	−10.2300	−0.777774	−0.388887	−	0.921285i	$-0.627140\pi$
$173$	−10.2300	−0.777774	−0.388887	+	0.921285i	$0.627140\pi$
$174$	0	0
$175$	2.71254	0.205049
$176$	0	0
$177$	0	0
$178$	2.82762	0.211939
$179$	−1.75709	−0.131331	−0.0656656	−	0.997842i	$-0.520917\pi$
$179$	−1.75709	−0.131331	−0.0656656	+	0.997842i	$0.520917\pi$
$180$	0	0
$181$	12.5171	0.930386	0.465193	−	0.885209i	$-0.345985\pi$
$181$	12.5171	0.930386	0.465193	+	0.885209i	$0.345985\pi$
$182$	−4.17621	−0.309561
$183$	0	0
$184$	−3.24816	−0.239458
$185$	−31.0974	−2.28633
$186$	0	0
$187$	0	0
$188$	−0.576739	−0.0420630
$189$	0	0
$190$	40.5267	2.94011
$191$	19.6079	1.41878	0.709389	−	0.704818i	$-0.248971\pi$
$191$	19.6079	1.41878	0.709389	+	0.704818i	$0.248971\pi$
$192$	0	0
$193$	0.612533	0.0440911	0.0220455	−	0.999757i	$-0.492982\pi$
$193$	0.612533	0.0440911	0.0220455	+	0.999757i	$0.492982\pi$
$194$	−0.963862	−0.0692013
$195$	0	0
$196$	1.25966	0.0899754
$197$	−13.5754	−0.967204	−0.483602	−	0.875288i	$-0.660672\pi$
$197$	−13.5754	−0.967204	−0.483602	+	0.875288i	$0.660672\pi$
$198$	0	0
$199$	4.17314	0.295826	0.147913	−	0.989000i	$-0.452744\pi$
$199$	4.17314	0.295826	0.147913	+	0.989000i	$0.452744\pi$
$200$	−3.62574	−0.256378
$201$	0	0
$202$	−9.60069	−0.675502
$203$	−7.55762	−0.530441
$204$	0	0
$205$	0.832851	0.0581688
$206$	33.5102	2.33477
$207$	0	0
$208$	11.4096	0.791113
$209$	0	0
$210$	0	0
$211$	−10.6630	−0.734073	−0.367037	−	0.930206i	$-0.619628\pi$
$211$	−10.6630	−0.734073	−0.367037	+	0.930206i	$0.619628\pi$
$212$	6.54190	0.449300
$213$	0	0
$214$	−17.0204	−1.16349
$215$	−20.2545	−1.38135
$216$	0	0
$217$	9.24567	0.627637
$218$	−2.00767	−0.135976
$219$	0	0
$220$	0	0
$221$	−6.15663	−0.414140
$222$	0	0
$223$	3.64321	0.243967	0.121984	−	0.992532i	$-0.461074\pi$
$223$	3.64321	0.243967	0.121984	+	0.992532i	$0.461074\pi$
$224$	−6.23222	−0.416408
$225$	0	0
$226$	16.2937	1.08384
$227$	20.9279	1.38904	0.694518	−	0.719475i	$-0.255618\pi$
$227$	20.9279	1.38904	0.694518	+	0.719475i	$0.255618\pi$
$228$	0	0
$229$	−16.5049	−1.09068	−0.545338	−	0.838216i	$-0.683599\pi$
$229$	−16.5049	−1.09068	−0.545338	+	0.838216i	$0.683599\pi$
$230$	−12.1844	−0.803414
$231$	0	0
$232$	10.1019	0.663225
$233$	7.28683	0.477376	0.238688	−	0.971096i	$-0.423283\pi$
$233$	7.28683	0.477376	0.238688	+	0.971096i	$0.423283\pi$
$234$	0	0
$235$	1.27153	0.0829454
$236$	−15.4719	−1.00714
$237$	0	0
$238$	4.80544	0.311490
$239$	−11.9626	−0.773798	−0.386899	−	0.922122i	$-0.626454\pi$
$239$	−11.9626	−0.773798	−0.386899	+	0.922122i	$0.626454\pi$
$240$	0	0
$241$	4.39102	0.282850	0.141425	−	0.989949i	$-0.454832\pi$
$241$	4.39102	0.282850	0.141425	+	0.989949i	$0.454832\pi$
$242$	0	0
$243$	0	0
$244$	2.28990	0.146596
$245$	−2.77715	−0.177425
$246$	0	0
$247$	18.6962	1.18961
$248$	−12.3583	−0.784751
$249$	0	0
$250$	11.4693	0.725383
$251$	13.5401	0.854643	0.427322	−	0.904100i	$-0.359457\pi$
$251$	13.5401	0.854643	0.427322	+	0.904100i	$0.359457\pi$
$252$	0	0
$253$	0	0
$254$	15.0492	0.944271
$255$	0	0
$256$	20.7570	1.29731
$257$	−0.121519	−0.00758013	−0.00379006	−	0.999993i	$-0.501206\pi$
$257$	−0.121519	−0.00758013	−0.00379006	+	0.999993i	$0.501206\pi$
$258$	0	0
$259$	11.1976	0.695786
$260$	8.09183	0.501834
$261$	0	0
$262$	14.4050	0.889944
$263$	5.88836	0.363092	0.181546	−	0.983382i	$-0.441890\pi$
$263$	5.88836	0.363092	0.181546	+	0.983382i	$0.441890\pi$
$264$	0	0
$265$	−14.4228	−0.885988
$266$	−14.5929	−0.894749
$267$	0	0
$268$	−1.79208	−0.109469
$269$	2.21155	0.134841	0.0674203	−	0.997725i	$-0.478523\pi$
$269$	2.21155	0.134841	0.0674203	+	0.997725i	$0.478523\pi$
$270$	0	0
$271$	7.59429	0.461321	0.230660	−	0.973034i	$-0.425911\pi$
$271$	7.59429	0.461321	0.230660	+	0.973034i	$0.425911\pi$
$272$	−13.1287	−0.796043
$273$	0	0
$274$	18.6533	1.12688
$275$	0	0
$276$	0	0
$277$	24.1488	1.45096	0.725481	−	0.688243i	$-0.241618\pi$
$277$	24.1488	1.45096	0.725481	+	0.688243i	$0.241618\pi$
$278$	10.0016	0.599854
$279$	0	0
$280$	3.71209	0.221840
$281$	27.2967	1.62838	0.814192	−	0.580595i	$-0.197180\pi$
$281$	27.2967	1.62838	0.814192	+	0.580595i	$0.197180\pi$
$282$	0	0
$283$	−25.6085	−1.52227	−0.761135	−	0.648594i	$-0.775357\pi$
$283$	−25.6085	−1.52227	−0.761135	+	0.648594i	$0.775357\pi$
$284$	−0.703012	−0.0417161
$285$	0	0
$286$	0	0
$287$	−0.299894	−0.0177022
$288$	0	0
$289$	−9.91574	−0.583279
$290$	37.8939	2.22521
$291$	0	0
$292$	12.1718	0.712303
$293$	−11.6993	−0.683482	−0.341741	−	0.939794i	$-0.611017\pi$
$293$	−11.6993	−0.683482	−0.341741	+	0.939794i	$0.611017\pi$
$294$	0	0
$295$	34.1107	1.98600
$296$	−14.9674	−0.869960
$297$	0	0
$298$	39.6989	2.29969
$299$	−5.62101	−0.325072
$300$	0	0
$301$	7.29328	0.420378
$302$	25.5757	1.47172
$303$	0	0
$304$	39.8685	2.28662
$305$	−5.04851	−0.289077
$306$	0	0
$307$	−28.0637	−1.60168	−0.800839	−	0.598880i	$-0.795613\pi$
$307$	−28.0637	−1.60168	−0.800839	+	0.598880i	$0.795613\pi$
$308$	0	0
$309$	0	0
$310$	−46.3578	−2.63295
$311$	−28.7137	−1.62820	−0.814102	−	0.580722i	$-0.802770\pi$
$311$	−28.7137	−1.62820	−0.814102	+	0.580722i	$0.802770\pi$
$312$	0	0
$313$	−29.4483	−1.66451	−0.832257	−	0.554390i	$-0.812952\pi$
$313$	−29.4483	−1.66451	−0.832257	+	0.554390i	$0.812952\pi$
$314$	−2.48967	−0.140500
$315$	0	0
$316$	12.9996	0.731286
$317$	0.0336526	0.00189012	0.000945058	−	1.00000i	$-0.499699\pi$
$317$	0.0336526	0.00189012	0.000945058		1.00000i	$0.499699\pi$
$318$	0	0
$319$	0	0
$320$	3.85139	0.215299
$321$	0	0
$322$	4.38737	0.244499
$323$	−21.5131	−1.19702
$324$	0	0
$325$	−6.27441	−0.348042
$326$	−30.7664	−1.70399
$327$	0	0
$328$	0.400856	0.0221335
$329$	−0.457855	−0.0252423
$330$	0	0
$331$	16.4575	0.904588	0.452294	−	0.891869i	$-0.350606\pi$
$331$	16.4575	0.904588	0.452294	+	0.891869i	$0.350606\pi$
$332$	−10.7206	−0.588371
$333$	0	0
$334$	−14.3720	−0.786401
$335$	3.95097	0.215865
$336$	0	0
$337$	−13.2223	−0.720262	−0.360131	−	0.932902i	$-0.617268\pi$
$337$	−13.2223	−0.720262	−0.360131	+	0.932902i	$0.617268\pi$
$338$	−13.8108	−0.751211
$339$	0	0
$340$	−9.31103	−0.504962
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−9.74860	−0.525609
$345$	0	0
$346$	−18.4698	−0.992943
$347$	17.1944	0.923045	0.461523	−	0.887128i	$-0.347303\pi$
$347$	17.1944	0.923045	0.461523	+	0.887128i	$0.347303\pi$
$348$	0	0
$349$	25.2943	1.35397	0.676987	−	0.735995i	$-0.263285\pi$
$349$	25.2943	1.35397	0.676987	+	0.735995i	$0.263285\pi$
$350$	4.89737	0.261775
$351$	0	0
$352$	0	0
$353$	−6.60477	−0.351536	−0.175768	−	0.984432i	$-0.556241\pi$
$353$	−6.60477	−0.351536	−0.175768	+	0.984432i	$0.556241\pi$
$354$	0	0
$355$	1.54992	0.0822613
$356$	1.97282	0.104559
$357$	0	0
$358$	−3.17234	−0.167664
$359$	36.4260	1.92249	0.961246	−	0.275692i	$-0.0889070\pi$
$359$	36.4260	1.92249	0.961246	+	0.275692i	$0.0889070\pi$
$360$	0	0
$361$	46.3300	2.43842
$362$	22.5990	1.18777
$363$	0	0
$364$	−2.91372	−0.152720
$365$	−26.8351	−1.40461
$366$	0	0
$367$	9.80019	0.511566	0.255783	−	0.966734i	$-0.417667\pi$
$367$	9.80019	0.511566	0.255783	+	0.966734i	$0.417667\pi$
$368$	−11.9865	−0.624840
$369$	0	0
$370$	−56.1449	−2.91883
$371$	5.19340	0.269628
$372$	0	0
$373$	−11.7043	−0.606027	−0.303013	−	0.952986i	$-0.597993\pi$
$373$	−11.7043	−0.606027	−0.303013	+	0.952986i	$0.597993\pi$
$374$	0	0
$375$	0	0
$376$	0.611994	0.0315612
$377$	17.4816	0.900348
$378$	0	0
$379$	22.0823	1.13429	0.567146	−	0.823617i	$-0.308047\pi$
$379$	22.0823	1.13429	0.567146	+	0.823617i	$0.308047\pi$
$380$	28.2753	1.45049
$381$	0	0
$382$	35.4011	1.81128
$383$	21.7965	1.11375	0.556874	−	0.830597i	$-0.312001\pi$
$383$	21.7965	1.11375	0.556874	+	0.830597i	$0.312001\pi$
$384$	0	0
$385$	0	0
$386$	1.10590	0.0562887
$387$	0	0
$388$	−0.672483	−0.0341401
$389$	18.6044	0.943280	0.471640	−	0.881791i	$-0.343662\pi$
$389$	18.6044	0.943280	0.471640	+	0.881791i	$0.343662\pi$
$390$	0	0
$391$	6.46793	0.327097
$392$	−1.33666	−0.0675113
$393$	0	0
$394$	−24.5096	−1.23478
$395$	−28.6601	−1.44205
$396$	0	0
$397$	−17.4492	−0.875750	−0.437875	−	0.899036i	$-0.644269\pi$
$397$	−17.4492	−0.875750	−0.437875	+	0.899036i	$0.644269\pi$
$398$	7.53440	0.377665
$399$	0	0
$400$	−13.3798	−0.668992
$401$	37.4603	1.87068	0.935339	−	0.353754i	$-0.115095\pi$
$401$	37.4603	1.87068	0.935339	+	0.353754i	$0.115095\pi$
$402$	0	0
$403$	−21.3862	−1.06532
$404$	−6.69836	−0.333256
$405$	0	0
$406$	−13.6449	−0.677186
$407$	0	0
$408$	0	0
$409$	−1.93760	−0.0958083	−0.0479041	−	0.998852i	$-0.515254\pi$
$409$	−1.93760	−0.0958083	−0.0479041	+	0.998852i	$0.515254\pi$
$410$	1.50367	0.0742611
$411$	0	0
$412$	23.3799	1.15185
$413$	−12.2826	−0.604389
$414$	0	0
$415$	23.6356	1.16023
$416$	14.4158	0.706793
$417$	0	0
$418$	0	0
$419$	17.2226	0.841378	0.420689	−	0.907205i	$-0.361788\pi$
$419$	17.2226	0.841378	0.420689	+	0.907205i	$0.361788\pi$
$420$	0	0
$421$	7.23823	0.352770	0.176385	−	0.984321i	$-0.443560\pi$
$421$	7.23823	0.352770	0.176385	+	0.984321i	$0.443560\pi$
$422$	−19.2516	−0.937153
$423$	0	0
$424$	−6.94179	−0.337123
$425$	7.21978	0.350211
$426$	0	0
$427$	1.81788	0.0879732
$428$	−11.8751	−0.574004
$429$	0	0
$430$	−36.5685	−1.76349
$431$	−32.4557	−1.56334	−0.781668	−	0.623695i	$-0.785631\pi$
$431$	−32.4557	−1.56334	−0.781668	+	0.623695i	$0.785631\pi$
$432$	0	0
$433$	22.7744	1.09447	0.547235	−	0.836979i	$-0.315680\pi$
$433$	22.7744	1.09447	0.547235	+	0.836979i	$0.315680\pi$
$434$	16.6926	0.801271
$435$	0	0
$436$	−1.40074	−0.0670833
$437$	−19.6415	−0.939581
$438$	0	0
$439$	−30.4359	−1.45263	−0.726314	−	0.687363i	$-0.758768\pi$
$439$	−30.4359	−1.45263	−0.726314	+	0.687363i	$0.758768\pi$
$440$	0	0
$441$	0	0
$442$	−11.1155	−0.528711
$443$	19.9801	0.949285	0.474643	−	0.880179i	$-0.342577\pi$
$443$	19.9801	0.949285	0.474643	+	0.880179i	$0.342577\pi$
$444$	0	0
$445$	−4.34945	−0.206184
$446$	6.57764	0.311460
$447$	0	0
$448$	−1.38682	−0.0655209
$449$	−11.8269	−0.558148	−0.279074	−	0.960270i	$-0.590027\pi$
$449$	−11.8269	−0.558148	−0.279074	+	0.960270i	$0.590027\pi$
$450$	0	0
$451$	0	0
$452$	11.3681	0.534708
$453$	0	0
$454$	37.7844	1.77331
$455$	6.42384	0.301154
$456$	0	0
$457$	1.96299	0.0918248	0.0459124	−	0.998945i	$-0.485380\pi$
$457$	1.96299	0.0918248	0.0459124	+	0.998945i	$0.485380\pi$
$458$	−29.7988	−1.39241
$459$	0	0
$460$	−8.50098	−0.396360
$461$	−15.5498	−0.724226	−0.362113	−	0.932134i	$-0.617945\pi$
$461$	−15.5498	−0.724226	−0.362113	+	0.932134i	$0.617945\pi$
$462$	0	0
$463$	−3.73913	−0.173772	−0.0868860	−	0.996218i	$-0.527692\pi$
$463$	−3.73913	−0.173772	−0.0868860	+	0.996218i	$0.527692\pi$
$464$	37.2786	1.73061
$465$	0	0
$466$	13.1560	0.609441
$467$	−10.1969	−0.471857	−0.235929	−	0.971770i	$-0.575813\pi$
$467$	−10.1969	−0.471857	−0.235929	+	0.971770i	$0.575813\pi$
$468$	0	0
$469$	−1.42267	−0.0656930
$470$	2.29568	0.105892
$471$	0	0
$472$	16.4177	0.755684
$473$	0	0
$474$	0	0
$475$	−21.9247	−1.00597
$476$	3.35273	0.153672
$477$	0	0
$478$	−21.5980	−0.987867
$479$	22.6431	1.03459	0.517296	−	0.855807i	$-0.326939\pi$
$479$	22.6431	1.03459	0.517296	+	0.855807i	$0.326939\pi$
$480$	0	0
$481$	−25.9013	−1.18100
$482$	7.92777	0.361100
$483$	0	0
$484$	0	0
$485$	1.48261	0.0673220
$486$	0	0
$487$	−23.1079	−1.04712	−0.523559	−	0.851989i	$-0.675396\pi$
$487$	−23.1079	−1.04712	−0.523559	+	0.851989i	$0.675396\pi$
$488$	−2.42987	−0.109995
$489$	0	0
$490$	−5.01400	−0.226510
$491$	−32.7793	−1.47931	−0.739653	−	0.672988i	$-0.765011\pi$
$491$	−32.7793	−1.47931	−0.739653	+	0.672988i	$0.765011\pi$
$492$	0	0
$493$	−20.1156	−0.905960
$494$	33.7550	1.51871
$495$	0	0
$496$	−45.6050	−2.04772
$497$	−0.558099	−0.0250342
$498$	0	0
$499$	17.7759	0.795757	0.397878	−	0.917438i	$-0.369747\pi$
$499$	17.7759	0.795757	0.397878	+	0.917438i	$0.369747\pi$
$500$	8.00209	0.357864
$501$	0	0
$502$	24.4460	1.09108
$503$	−15.0014	−0.668879	−0.334439	−	0.942417i	$-0.608547\pi$
$503$	−15.0014	−0.668879	−0.334439	+	0.942417i	$0.608547\pi$
$504$	0	0
$505$	14.7678	0.657158
$506$	0	0
$507$	0	0
$508$	10.4998	0.465852
$509$	24.3906	1.08109	0.540547	−	0.841314i	$-0.318217\pi$
$509$	24.3906	1.08109	0.540547	+	0.841314i	$0.318217\pi$
$510$	0	0
$511$	9.66283	0.427459
$512$	17.5546	0.775811
$513$	0	0
$514$	−0.219396	−0.00967715
$515$	−51.5454	−2.27136
$516$	0	0
$517$	0	0
$518$	20.2168	0.888273
$519$	0	0
$520$	−8.58647	−0.376542
$521$	−8.07185	−0.353634	−0.176817	−	0.984244i	$-0.556580\pi$
$521$	−8.07185	−0.353634	−0.176817	+	0.984244i	$0.556580\pi$
$522$	0	0
$523$	0.370242	0.0161895	0.00809477	−	0.999967i	$-0.497423\pi$
$523$	0.370242	0.0161895	0.00809477	+	0.999967i	$0.497423\pi$
$524$	10.0503	0.439050
$525$	0	0
$526$	10.6311	0.463540
$527$	24.6085	1.07196
$528$	0	0
$529$	−17.0948	−0.743251
$530$	−26.0398	−1.13109
$531$	0	0
$532$	−10.1814	−0.441420
$533$	0.693689	0.0300470
$534$	0	0
$535$	26.1809	1.13190
$536$	1.90163	0.0821377
$537$	0	0
$538$	3.99285	0.172144
$539$	0	0
$540$	0	0
$541$	2.96946	0.127667	0.0638336	−	0.997961i	$-0.479667\pi$
$541$	2.96946	0.127667	0.0638336	+	0.997961i	$0.479667\pi$
$542$	13.7111	0.588943
$543$	0	0
$544$	−16.5878	−0.711198
$545$	3.08819	0.132284
$546$	0	0
$547$	−16.6650	−0.712545	−0.356273	−	0.934382i	$-0.615953\pi$
$547$	−16.6650	−0.712545	−0.356273	+	0.934382i	$0.615953\pi$
$548$	13.0143	0.555943
$549$	0	0
$550$	0	0
$551$	61.0860	2.60235
$552$	0	0
$553$	10.3200	0.438850
$554$	43.5995	1.85237
$555$	0	0
$556$	6.97804	0.295935
$557$	−30.1565	−1.27777	−0.638885	−	0.769302i	$-0.720604\pi$
$557$	−30.1565	−1.27777	−0.638885	+	0.769302i	$0.720604\pi$
$558$	0	0
$559$	−16.8702	−0.713531
$560$	13.6985	0.578867
$561$	0	0
$562$	49.2829	2.07887
$563$	−4.22354	−0.178001	−0.0890004	−	0.996032i	$-0.528367\pi$
$563$	−4.22354	−0.178001	−0.0890004	+	0.996032i	$0.528367\pi$
$564$	0	0
$565$	−25.0630	−1.05441
$566$	−46.2350	−1.94340
$567$	0	0
$568$	0.745986	0.0313009
$569$	25.2216	1.05735	0.528673	−	0.848826i	$-0.322690\pi$
$569$	25.2216	1.05735	0.528673	+	0.848826i	$0.322690\pi$
$570$	0	0
$571$	42.9041	1.79548	0.897740	−	0.440525i	$-0.145208\pi$
$571$	42.9041	1.79548	0.897740	+	0.440525i	$0.145208\pi$
$572$	0	0
$573$	0	0
$574$	−0.541445	−0.0225995
$575$	6.59167	0.274892
$576$	0	0
$577$	−28.8359	−1.20045	−0.600227	−	0.799830i	$-0.704923\pi$
$577$	−28.8359	−1.20045	−0.600227	+	0.799830i	$0.704923\pi$
$578$	−17.9024	−0.744642
$579$	0	0
$580$	26.4384	1.09780
$581$	−8.51075	−0.353086
$582$	0	0
$583$	0	0
$584$	−12.9159	−0.534463
$585$	0	0
$586$	−21.1226	−0.872566
$587$	37.7758	1.55918	0.779588	−	0.626293i	$-0.215429\pi$
$587$	37.7758	1.55918	0.779588	+	0.626293i	$0.215429\pi$
$588$	0	0
$589$	−74.7299	−3.07919
$590$	61.5852	2.53542
$591$	0	0
$592$	−55.2331	−2.27007
$593$	−10.8394	−0.445119	−0.222559	−	0.974919i	$-0.571441\pi$
$593$	−10.8394	−0.445119	−0.222559	+	0.974919i	$0.571441\pi$
$594$	0	0
$595$	−7.39173	−0.303031
$596$	27.6977	1.13454
$597$	0	0
$598$	−10.1485	−0.415002
$599$	−43.4225	−1.77420	−0.887098	−	0.461582i	$-0.847282\pi$
$599$	−43.4225	−1.77420	−0.887098	+	0.461582i	$0.847282\pi$
$600$	0	0
$601$	−0.476391	−0.0194324	−0.00971619	−	0.999953i	$-0.503093\pi$
$601$	−0.476391	−0.0194324	−0.00971619	+	0.999953i	$0.503093\pi$
$602$	13.1677	0.536674
$603$	0	0
$604$	17.8441	0.726065
$605$	0	0
$606$	0	0
$607$	4.90638	0.199144	0.0995720	−	0.995030i	$-0.468253\pi$
$607$	4.90638	0.199144	0.0995720	+	0.995030i	$0.468253\pi$
$608$	50.3732	2.04290
$609$	0	0
$610$	−9.11484	−0.369049
$611$	1.05907	0.0428453
$612$	0	0
$613$	6.14615	0.248241	0.124120	−	0.992267i	$-0.460389\pi$
$613$	6.14615	0.248241	0.124120	+	0.992267i	$0.460389\pi$
$614$	−50.6676	−2.04478
$615$	0	0
$616$	0	0
$617$	11.2456	0.452730	0.226365	−	0.974043i	$-0.427316\pi$
$617$	11.2456	0.452730	0.226365	+	0.974043i	$0.427316\pi$
$618$	0	0
$619$	−33.3779	−1.34157	−0.670786	−	0.741651i	$-0.734043\pi$
$619$	−33.3779	−1.34157	−0.670786	+	0.741651i	$0.734043\pi$
$620$	−32.3436	−1.29895
$621$	0	0
$622$	−51.8412	−2.07864
$623$	1.56616	0.0627468
$624$	0	0
$625$	−31.2048	−1.24819
$626$	−53.1674	−2.12500
$627$	0	0
$628$	−1.73703	−0.0693151
$629$	29.8039	1.18836
$630$	0	0
$631$	37.0130	1.47346	0.736731	−	0.676186i	$-0.236368\pi$
$631$	37.0130	1.47346	0.736731	+	0.676186i	$0.236368\pi$
$632$	−13.7943	−0.548706
$633$	0	0
$634$	0.0607581	0.00241301
$635$	−23.1487	−0.918628
$636$	0	0
$637$	−2.31311	−0.0916487
$638$	0	0
$639$	0	0
$640$	−27.6621	−1.09344
$641$	18.0178	0.711662	0.355831	−	0.934550i	$-0.384198\pi$
$641$	18.0178	0.711662	0.355831	+	0.934550i	$0.384198\pi$
$642$	0	0
$643$	6.89603	0.271953	0.135976	−	0.990712i	$-0.456583\pi$
$643$	6.89603	0.271953	0.135976	+	0.990712i	$0.456583\pi$
$644$	3.06105	0.120622
$645$	0	0
$646$	−38.8409	−1.52818
$647$	−37.6477	−1.48008	−0.740041	−	0.672561i	$-0.765194\pi$
$647$	−37.6477	−1.48008	−0.740041	+	0.672561i	$0.765194\pi$
$648$	0	0
$649$	0	0
$650$	−11.3281	−0.444326
$651$	0	0
$652$	−21.4656	−0.840656
$653$	6.97518	0.272960	0.136480	−	0.990643i	$-0.456421\pi$
$653$	6.97518	0.272960	0.136480	+	0.990643i	$0.456421\pi$
$654$	0	0
$655$	−22.1578	−0.865777
$656$	1.47925	0.0577551
$657$	0	0
$658$	−0.826634	−0.0322256
$659$	−0.567182	−0.0220943	−0.0110471	−	0.999939i	$-0.503516\pi$
$659$	−0.567182	−0.0220943	−0.0110471	+	0.999939i	$0.503516\pi$
$660$	0	0
$661$	0.107429	0.00417852	0.00208926	−	0.999998i	$-0.499335\pi$
$661$	0.107429	0.00417852	0.00208926	+	0.999998i	$0.499335\pi$
$662$	29.7133	1.15484
$663$	0	0
$664$	11.3759	0.441472
$665$	22.4468	0.870451
$666$	0	0
$667$	−18.3655	−0.711116
$668$	−10.0273	−0.387967
$669$	0	0
$670$	7.13329	0.275583
$671$	0	0
$672$	0	0
$673$	−20.1725	−0.777591	−0.388795	−	0.921324i	$-0.627109\pi$
$673$	−20.1725	−0.777591	−0.388795	+	0.921324i	$0.627109\pi$
$674$	−23.8721	−0.919520
$675$	0	0
$676$	−9.63577	−0.370607
$677$	10.0671	0.386910	0.193455	−	0.981109i	$-0.438031\pi$
$677$	10.0671	0.386910	0.193455	+	0.981109i	$0.438031\pi$
$678$	0	0
$679$	−0.533862	−0.0204877
$680$	9.88019	0.378888
$681$	0	0
$682$	0	0
$683$	32.4580	1.24197	0.620985	−	0.783822i	$-0.286733\pi$
$683$	32.4580	1.24197	0.620985	+	0.783822i	$0.286733\pi$
$684$	0	0
$685$	−28.6925	−1.09628
$686$	1.80545	0.0689325
$687$	0	0
$688$	−35.9747	−1.37152
$689$	−12.0129	−0.457655
$690$	0	0
$691$	47.1070	1.79204	0.896018	−	0.444018i	$-0.146447\pi$
$691$	47.1070	1.79204	0.896018	+	0.444018i	$0.146447\pi$
$692$	−12.8863	−0.489864
$693$	0	0
$694$	31.0437	1.17840
$695$	−15.3844	−0.583564
$696$	0	0
$697$	−0.798207	−0.0302342
$698$	45.6677	1.72855
$699$	0	0
$700$	3.41687	0.129146
$701$	20.4498	0.772379	0.386190	−	0.922419i	$-0.373791\pi$
$701$	20.4498	0.772379	0.386190	+	0.922419i	$0.373791\pi$
$702$	0	0
$703$	−90.5070	−3.41353
$704$	0	0
$705$	0	0
$706$	−11.9246	−0.448788
$707$	−5.31761	−0.199989
$708$	0	0
$709$	−9.94643	−0.373546	−0.186773	−	0.982403i	$-0.559803\pi$
$709$	−9.94643	−0.373546	−0.186773	+	0.982403i	$0.559803\pi$
$710$	2.79831	0.105019
$711$	0	0
$712$	−2.09341	−0.0784540
$713$	22.4676	0.841418
$714$	0	0
$715$	0	0
$716$	−2.21333	−0.0827160
$717$	0	0
$718$	65.7654	2.45434
$719$	−6.83696	−0.254975	−0.127488	−	0.991840i	$-0.540691\pi$
$719$	−6.83696	−0.254975	−0.127488	+	0.991840i	$0.540691\pi$
$720$	0	0
$721$	18.5606	0.691232
$722$	83.6466	3.11300
$723$	0	0
$724$	15.7672	0.585983
$725$	−20.5004	−0.761365
$726$	0	0
$727$	49.3919	1.83184	0.915922	−	0.401357i	$-0.131461\pi$
$727$	49.3919	1.83184	0.915922	+	0.401357i	$0.131461\pi$
$728$	3.09183	0.114591
$729$	0	0
$730$	−48.4495	−1.79320
$731$	19.4120	0.717978
$732$	0	0
$733$	−39.7282	−1.46739	−0.733697	−	0.679477i	$-0.762207\pi$
$733$	−39.7282	−1.46739	−0.733697	+	0.679477i	$0.762207\pi$
$734$	17.6938	0.653089
$735$	0	0
$736$	−15.1447	−0.558242
$737$	0	0
$738$	0	0
$739$	−31.1915	−1.14740	−0.573698	−	0.819067i	$-0.694492\pi$
$739$	−31.1915	−1.14740	−0.573698	+	0.819067i	$0.694492\pi$
$740$	−39.1721	−1.43999
$741$	0	0
$742$	9.37644	0.344220
$743$	30.9041	1.13376	0.566880	−	0.823800i	$-0.308150\pi$
$743$	30.9041	1.13376	0.566880	+	0.823800i	$0.308150\pi$
$744$	0	0
$745$	−61.0649	−2.23724
$746$	−21.1316	−0.773682
$747$	0	0
$748$	0	0
$749$	−9.42725	−0.344464
$750$	0	0
$751$	−17.1976	−0.627549	−0.313775	−	0.949497i	$-0.601594\pi$
$751$	−17.1976	−0.627549	−0.313775	+	0.949497i	$0.601594\pi$
$752$	2.25840	0.0823555
$753$	0	0
$754$	31.5622	1.14943
$755$	−39.3406	−1.43175
$756$	0	0
$757$	5.67500	0.206261	0.103131	−	0.994668i	$-0.467114\pi$
$757$	5.67500	0.206261	0.103131	+	0.994668i	$0.467114\pi$
$758$	39.8686	1.44809
$759$	0	0
$760$	−30.0037	−1.08835
$761$	8.73463	0.316630	0.158315	−	0.987389i	$-0.449394\pi$
$761$	8.73463	0.316630	0.158315	+	0.987389i	$0.449394\pi$
$762$	0	0
$763$	−1.11200	−0.0402572
$764$	24.6992	0.893585
$765$	0	0
$766$	39.3525	1.42186
$767$	28.4111	1.02587
$768$	0	0
$769$	−7.76277	−0.279933	−0.139966	−	0.990156i	$-0.544699\pi$
$769$	−7.76277	−0.279933	−0.139966	+	0.990156i	$0.544699\pi$
$770$	0	0
$771$	0	0
$772$	0.771580	0.0277698
$773$	−32.9829	−1.18631	−0.593156	−	0.805087i	$-0.702118\pi$
$773$	−32.9829	−1.18631	−0.593156	+	0.805087i	$0.702118\pi$
$774$	0	0
$775$	25.0793	0.900874
$776$	0.713590	0.0256164
$777$	0	0
$778$	33.5893	1.20424
$779$	2.42396	0.0868472
$780$	0	0
$781$	0	0
$782$	11.6775	0.417588
$783$	0	0
$784$	−4.93258	−0.176164
$785$	3.82961	0.136685
$786$	0	0
$787$	−16.4069	−0.584845	−0.292422	−	0.956289i	$-0.594461\pi$
$787$	−16.4069	−0.584845	−0.292422	+	0.956289i	$0.594461\pi$
$788$	−17.1003	−0.609172
$789$	0	0
$790$	−51.7444	−1.84098
$791$	9.02473	0.320882
$792$	0	0
$793$	−4.20495	−0.149322
$794$	−31.5037	−1.11802
$795$	0	0
$796$	5.25672	0.186319
$797$	5.86473	0.207740	0.103870	−	0.994591i	$-0.466877\pi$
$797$	5.86473	0.207740	0.103870	+	0.994591i	$0.466877\pi$
$798$	0	0
$799$	−1.21864	−0.0431123
$800$	−16.9052	−0.597688
$801$	0	0
$802$	67.6327	2.38819
$803$	0	0
$804$	0	0
$805$	−6.74866	−0.237859
$806$	−38.6118	−1.36004
$807$	0	0
$808$	7.10782	0.250052
$809$	−17.9373	−0.630643	−0.315322	−	0.948985i	$-0.602112\pi$
$809$	−17.9373	−0.630643	−0.315322	+	0.948985i	$0.602112\pi$
$810$	0	0
$811$	27.2249	0.955996	0.477998	−	0.878361i	$-0.341363\pi$
$811$	27.2249	0.955996	0.477998	+	0.878361i	$0.341363\pi$
$812$	−9.52000	−0.334087
$813$	0	0
$814$	0	0
$815$	47.3248	1.65772
$816$	0	0
$817$	−58.9494	−2.06238
$818$	−3.49825	−0.122313
$819$	0	0
$820$	1.04911	0.0366363
$821$	4.06642	0.141919	0.0709595	−	0.997479i	$-0.477394\pi$
$821$	4.06642	0.141919	0.0709595	+	0.997479i	$0.477394\pi$
$822$	0	0
$823$	−32.5150	−1.13340	−0.566700	−	0.823924i	$-0.691780\pi$
$823$	−32.5150	−1.13340	−0.566700	+	0.823924i	$0.691780\pi$
$824$	−24.8091	−0.864266
$825$	0	0
$826$	−22.1757	−0.771592
$827$	18.6162	0.647349	0.323674	−	0.946169i	$-0.395082\pi$
$827$	18.6162	0.647349	0.323674	+	0.946169i	$0.395082\pi$
$828$	0	0
$829$	48.9304	1.69942	0.849712	−	0.527247i	$-0.176776\pi$
$829$	48.9304	1.69942	0.849712	+	0.527247i	$0.176776\pi$
$830$	42.6730	1.48120
$831$	0	0
$832$	3.20786	0.111212
$833$	2.66163	0.0922199
$834$	0	0
$835$	22.1070	0.765045
$836$	0	0
$837$	0	0
$838$	31.0945	1.07414
$839$	2.90781	0.100389	0.0501944	−	0.998739i	$-0.484016\pi$
$839$	2.90781	0.100389	0.0501944	+	0.998739i	$0.484016\pi$
$840$	0	0
$841$	28.1176	0.969574
$842$	13.0683	0.450363
$843$	0	0
$844$	−13.4317	−0.462340
$845$	21.2439	0.730811
$846$	0	0
$847$	0	0
$848$	−25.6169	−0.879687
$849$	0	0
$850$	13.0350	0.447096
$851$	27.2110	0.932780
$852$	0	0
$853$	−26.5311	−0.908406	−0.454203	−	0.890898i	$-0.650076\pi$
$853$	−26.5311	−0.908406	−0.454203	+	0.890898i	$0.650076\pi$
$854$	3.28209	0.112311
$855$	0	0
$856$	12.6010	0.430693
$857$	32.5818	1.11297	0.556487	−	0.830856i	$-0.312149\pi$
$857$	32.5818	1.11297	0.556487	+	0.830856i	$0.312149\pi$
$858$	0	0
$859$	30.4695	1.03961	0.519803	−	0.854286i	$-0.326005\pi$
$859$	30.4695	1.03961	0.519803	+	0.854286i	$0.326005\pi$
$860$	−25.5137	−0.870010
$861$	0	0
$862$	−58.5972	−1.99583
$863$	−21.0577	−0.716814	−0.358407	−	0.933566i	$-0.616680\pi$
$863$	−21.0577	−0.716814	−0.358407	+	0.933566i	$0.616680\pi$
$864$	0	0
$865$	28.4103	0.965978
$866$	41.1182	1.39725
$867$	0	0
$868$	11.6464	0.395303
$869$	0	0
$870$	0	0
$871$	3.29080	0.111504
$872$	1.48636	0.0503347
$873$	0	0
$874$	−35.4618	−1.19951
$875$	6.35260	0.214757
$876$	0	0
$877$	−9.82296	−0.331698	−0.165849	−	0.986151i	$-0.553036\pi$
$877$	−9.82296	−0.331698	−0.165849	+	0.986151i	$0.553036\pi$
$878$	−54.9506	−1.85449
$879$	0	0
$880$	0	0
$881$	−16.4077	−0.552789	−0.276394	−	0.961044i	$-0.589140\pi$
$881$	−16.4077	−0.552789	−0.276394	+	0.961044i	$0.589140\pi$
$882$	0	0
$883$	27.7229	0.932949	0.466475	−	0.884535i	$-0.345524\pi$
$883$	27.7229	0.932949	0.466475	+	0.884535i	$0.345524\pi$
$884$	−7.75524	−0.260837
$885$	0	0
$886$	36.0732	1.21190
$887$	−33.1490	−1.11304	−0.556518	−	0.830836i	$-0.687863\pi$
$887$	−33.1490	−1.11304	−0.556518	+	0.830836i	$0.687863\pi$
$888$	0	0
$889$	8.33542	0.279561
$890$	−7.85272	−0.263224
$891$	0	0
$892$	4.58919	0.153657
$893$	3.70070	0.123839
$894$	0	0
$895$	4.87970	0.163110
$896$	9.96061	0.332761
$897$	0	0
$898$	−21.3530	−0.712558
$899$	−69.8752	−2.33047
$900$	0	0
$901$	13.8229	0.460508
$902$	0	0
$903$	0	0
$904$	−12.0630	−0.401208
$905$	−34.7617	−1.15552
$906$	0	0
$907$	2.69837	0.0895980	0.0447990	−	0.998996i	$-0.485735\pi$
$907$	2.69837	0.0895980	0.0447990	+	0.998996i	$0.485735\pi$
$908$	26.3620	0.874854
$909$	0	0
$910$	11.5979	0.384468
$911$	−30.2768	−1.00312	−0.501558	−	0.865124i	$-0.667239\pi$
$911$	−30.2768	−1.00312	−0.501558	+	0.865124i	$0.667239\pi$
$912$	0	0
$913$	0	0
$914$	3.54408	0.117228
$915$	0	0
$916$	−20.7905	−0.686938
$917$	7.97862	0.263477
$918$	0	0
$919$	−38.2913	−1.26311	−0.631557	−	0.775329i	$-0.717584\pi$
$919$	−38.2913	−1.26311	−0.631557	+	0.775329i	$0.717584\pi$
$920$	9.02063	0.297401
$921$	0	0
$922$	−28.0744	−0.924581
$923$	1.29094	0.0424919
$924$	0	0
$925$	30.3740	0.998692
$926$	−6.75081	−0.221845
$927$	0	0
$928$	47.1008	1.54616
$929$	1.97036	0.0646453	0.0323227	−	0.999477i	$-0.489710\pi$
$929$	1.97036	0.0646453	0.0323227	+	0.999477i	$0.489710\pi$
$930$	0	0
$931$	−8.08270	−0.264900
$932$	9.17890	0.300665
$933$	0	0
$934$	−18.4101	−0.602395
$935$	0	0
$936$	0	0
$937$	−37.2007	−1.21529	−0.607647	−	0.794207i	$-0.707886\pi$
$937$	−37.2007	−1.21529	−0.607647	+	0.794207i	$0.707886\pi$
$938$	−2.56857	−0.0838667
$939$	0	0
$940$	1.60169	0.0522413
$941$	−30.1836	−0.983957	−0.491979	−	0.870607i	$-0.663726\pi$
$941$	−30.1836	−0.983957	−0.491979	+	0.870607i	$0.663726\pi$
$942$	0	0
$943$	−0.728764	−0.0237318
$944$	60.5851	1.97188
$945$	0	0
$946$	0	0
$947$	−54.8775	−1.78328	−0.891639	−	0.452746i	$-0.850444\pi$
$947$	−54.8775	−1.78328	−0.891639	+	0.452746i	$0.850444\pi$
$948$	0	0
$949$	−22.3512	−0.725550
$950$	−39.5839	−1.28427
$951$	0	0
$952$	−3.55768	−0.115305
$953$	−25.5001	−0.826028	−0.413014	−	0.910725i	$-0.635524\pi$
$953$	−25.5001	−0.826028	−0.413014	+	0.910725i	$0.635524\pi$
$954$	0	0
$955$	−54.4540	−1.76209
$956$	−15.0688	−0.487360
$957$	0	0
$958$	40.8811	1.32081
$959$	10.3316	0.333626
$960$	0	0
$961$	54.4823	1.75749
$962$	−46.7636	−1.50772
$963$	0	0
$964$	5.53117	0.178147
$965$	−1.70109	−0.0547601
$966$	0	0
$967$	52.7171	1.69527	0.847633	−	0.530583i	$-0.178027\pi$
$967$	52.7171	1.69527	0.847633	+	0.530583i	$0.178027\pi$
$968$	0	0
$969$	0	0
$970$	2.67679	0.0859465
$971$	−33.0543	−1.06076	−0.530382	−	0.847759i	$-0.677951\pi$
$971$	−33.0543	−1.06076	−0.530382	+	0.847759i	$0.677951\pi$
$972$	0	0
$973$	5.53964	0.177593
$974$	−41.7202	−1.33680
$975$	0	0
$976$	−8.96682	−0.287021
$977$	31.6423	1.01233	0.506164	−	0.862437i	$-0.331063\pi$
$977$	31.6423	1.01233	0.506164	+	0.862437i	$0.331063\pi$
$978$	0	0
$979$	0	0
$980$	−3.49825	−0.111747
$981$	0	0
$982$	−59.1814	−1.88855
$983$	−14.0077	−0.446778	−0.223389	−	0.974729i	$-0.571712\pi$
$983$	−14.0077	−0.446778	−0.223389	+	0.974729i	$0.571712\pi$
$984$	0	0
$985$	37.7007	1.20125
$986$	−36.3177	−1.15659
$987$	0	0
$988$	23.5507	0.749249
$989$	17.7232	0.563564
$990$	0	0
$991$	21.2818	0.676038	0.338019	−	0.941139i	$-0.390243\pi$
$991$	21.2818	0.676038	0.338019	+	0.941139i	$0.390243\pi$
$992$	−57.6210	−1.82947
$993$	0	0
$994$	−1.00762	−0.0319598
$995$	−11.5894	−0.367409
$996$	0	0
$997$	21.9094	0.693876	0.346938	−	0.937888i	$-0.387221\pi$
$997$	21.9094	0.693876	0.346938	+	0.937888i	$0.387221\pi$
$998$	32.0935	1.01590
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cy.1.8		10
3.2	odd	2		2541.2.a.br.1.3		10
11.7	odd	10		693.2.m.j.379.2		20
11.8	odd	10		693.2.m.j.64.2		20
11.10	odd	2		7623.2.a.cx.1.3		10
33.8	even	10		231.2.j.g.64.4	✓	20
33.29	even	10		231.2.j.g.148.4	yes	20
33.32	even	2		2541.2.a.bq.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.g.64.4	✓	20	33.8	even	10
231.2.j.g.148.4	yes	20	33.29	even	10
693.2.m.j.64.2		20	11.8	odd	10
693.2.m.j.379.2		20	11.7	odd	10
2541.2.a.bq.1.8		10	33.32	even	2
2541.2.a.br.1.3		10	3.2	odd	2
7623.2.a.cx.1.3		10	11.10	odd	2
7623.2.a.cy.1.8		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+1.80545 q^{2} +1.25966 q^{4} -2.77715 q^{5} +1.00000 q^{7} -1.33666 q^{8} +O(q^{10})\)	\(q+1.80545 q^{2} +1.25966 q^{4} -2.77715 q^{5} +1.00000 q^{7} -1.33666 q^{8} -5.01400 q^{10} -2.31311 q^{13} +1.80545 q^{14} -4.93258 q^{16} +2.66163 q^{17} -8.08270 q^{19} -3.49825 q^{20} +2.43007 q^{23} +2.71254 q^{25} -4.17621 q^{26} +1.25966 q^{28} -7.55762 q^{29} +9.24567 q^{31} -6.23222 q^{32} +4.80544 q^{34} -2.77715 q^{35} +11.1976 q^{37} -14.5929 q^{38} +3.71209 q^{40} -0.299894 q^{41} +7.29328 q^{43} +4.38737 q^{46} -0.457855 q^{47} +1.00000 q^{49} +4.89737 q^{50} -2.91372 q^{52} +5.19340 q^{53} -1.33666 q^{56} -13.6449 q^{58} -12.2826 q^{59} +1.81788 q^{61} +16.6926 q^{62} -1.38682 q^{64} +6.42384 q^{65} -1.42267 q^{67} +3.35273 q^{68} -5.01400 q^{70} -0.558099 q^{71} +9.66283 q^{73} +20.2168 q^{74} -10.1814 q^{76} +10.3200 q^{79} +13.6985 q^{80} -0.541445 q^{82} -8.51075 q^{83} -7.39173 q^{85} +13.1677 q^{86} +1.56616 q^{89} -2.31311 q^{91} +3.06105 q^{92} -0.826634 q^{94} +22.4468 q^{95} -0.533862 q^{97} +1.80545 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8}+O(q^{10}) \) 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8	\( 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8} - 6 q^{10} + 6 q^{13} + 38 q^{16} - 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} + 18 q^{28} + 14 q^{29} + 26 q^{31} + 41 q^{32} + 21 q^{34} - 5 q^{35} + 24 q^{37} - 8 q^{38} - 5 q^{40} - 19 q^{41} - 6 q^{43} - q^{46} - 15 q^{47} + 10 q^{49} + q^{50} - 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} - 11 q^{62} + 53 q^{64} + 29 q^{65} + 38 q^{67} - 87 q^{68} - 6 q^{70} - 26 q^{71} - q^{73} + 39 q^{74} - 2 q^{76} + 5 q^{79} - 6 q^{80} + 5 q^{82} - 6 q^{83} - q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} + 42 q^{95} + 24 q^{97}+O(q^{100}) \) 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8 - 6 * q^10 + 6 * q^13 + 38 * q^16 - 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 + 18 * q^28 + 14 * q^29 + 26 * q^31 + 41 * q^32 + 21 * q^34 - 5 * q^35 + 24 * q^37 - 8 * q^38 - 5 * q^40 - 19 * q^41 - 6 * q^43 - q^46 - 15 * q^47 + 10 * q^49 + q^50 - 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 - 11 * q^62 + 53 * q^64 + 29 * q^65 + 38 * q^67 - 87 * q^68 - 6 * q^70 - 26 * q^71 - q^73 + 39 * q^74 - 2 * q^76 + 5 * q^79 - 6 * q^80 + 5 * q^82 - 6 * q^83 - q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 + 42 * q^95 + 24 * q^97

Introduction

L-functions

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