LMFDB - Embedded newform 308.2.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [308,2,Mod(1,308)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(308, 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("308.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 308 = 2^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	308.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:	$2.45939238226$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.17741$ of defining polynomial
Character	$\chi$	$=$	308.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.17741 q^{3} -0.741113 q^{5} +1.00000 q^{7} +1.74111 q^{9} +O(q^{10})$	$q+2.17741 q^{3} -0.741113 q^{5} +1.00000 q^{7} +1.74111 q^{9} +1.00000 q^{11} +6.91852 q^{13} -1.61371 q^{15} -7.27334 q^{17} +4.35482 q^{19} +2.17741 q^{21} +3.09593 q^{23} -4.45075 q^{25} -2.74111 q^{27} -3.83705 q^{29} -10.5322 q^{31} +2.17741 q^{33} -0.741113 q^{35} +2.74111 q^{37} +15.0645 q^{39} +7.27334 q^{41} -6.35482 q^{43} -1.29036 q^{45} -3.43630 q^{47} +1.00000 q^{49} -15.8370 q^{51} +1.48223 q^{53} -0.741113 q^{55} +9.48223 q^{57} -10.1774 q^{59} +8.40075 q^{61} +1.74111 q^{63} -5.12741 q^{65} -7.09593 q^{67} +6.74111 q^{69} +4.57816 q^{71} +3.27334 q^{73} -9.69111 q^{75} +1.00000 q^{77} +8.87259 q^{79} -11.1919 q^{81} -14.5467 q^{83} +5.39037 q^{85} -8.35482 q^{87} -15.9685 q^{89} +6.91852 q^{91} -22.9330 q^{93} -3.22741 q^{95} +0.386294 q^{97} +1.74111 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - q^{5} + 3 q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q - q^3 - q^5 + 3 * q^7 + 4 * q^9	$ 3 q - q^{3} - q^{5} + 3 q^{7} + 4 q^{9} + 3 q^{11} + 12 q^{13} + 9 q^{15} + 2 q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 18 q^{25} - 7 q^{27} + 6 q^{29} - 9 q^{31} - q^{33} - q^{35} + 7 q^{37} - 2 q^{41} - 4 q^{43} - 34 q^{45} - 4 q^{47} + 3 q^{49} - 30 q^{51} + 2 q^{53} - q^{55} + 26 q^{57} - 23 q^{59} + 14 q^{61} + 4 q^{63} - 28 q^{65} - 5 q^{67} + 19 q^{69} - 5 q^{71} - 14 q^{73} - 48 q^{75} + 3 q^{77} + 14 q^{79} - q^{81} + 4 q^{83} + 6 q^{85} - 10 q^{87} - 19 q^{89} + 12 q^{91} - 35 q^{93} + 18 q^{95} + 15 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q - q^3 - q^5 + 3 * q^7 + 4 * q^9 + 3 * q^11 + 12 * q^13 + 9 * q^15 + 2 * q^17 - 2 * q^19 - q^21 - 7 * q^23 + 18 * q^25 - 7 * q^27 + 6 * q^29 - 9 * q^31 - q^33 - q^35 + 7 * q^37 - 2 * q^41 - 4 * q^43 - 34 * q^45 - 4 * q^47 + 3 * q^49 - 30 * q^51 + 2 * q^53 - q^55 + 26 * q^57 - 23 * q^59 + 14 * q^61 + 4 * q^63 - 28 * q^65 - 5 * q^67 + 19 * q^69 - 5 * q^71 - 14 * q^73 - 48 * q^75 + 3 * q^77 + 14 * q^79 - q^81 + 4 * q^83 + 6 * q^85 - 10 * q^87 - 19 * q^89 + 12 * q^91 - 35 * q^93 + 18 * q^95 + 15 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.17741	1.25713	0.628564	−	0.777758i	$-0.283643\pi$
$3$	2.17741	1.25713	0.628564	+	0.777758i	$0.283643\pi$
$4$	0	0
$5$	−0.741113	−0.331436	−0.165718	−	0.986173i	$-0.552994\pi$
$5$	−0.741113	−0.331436	−0.165718	+	0.986173i	$0.552994\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.74111	0.580371
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.91852	1.91885	0.959426	−	0.281959i	$-0.0909842\pi$
$13$	6.91852	1.91885	0.959426	+	0.281959i	$0.0909842\pi$
$14$	0	0
$15$	−1.61371	−0.416657
$16$	0	0
$17$	−7.27334	−1.76404	−0.882022	−	0.471208i	$-0.843818\pi$
$17$	−7.27334	−1.76404	−0.882022	+	0.471208i	$0.843818\pi$
$18$	0	0
$19$	4.35482	0.999064	0.499532	−	0.866295i	$-0.333505\pi$
$19$	4.35482	0.999064	0.499532	+	0.866295i	$0.333505\pi$
$20$	0	0
$21$	2.17741	0.475150
$22$	0	0
$23$	3.09593	0.645547	0.322773	−	0.946476i	$-0.395385\pi$
$23$	3.09593	0.645547	0.322773	+	0.946476i	$0.395385\pi$
$24$	0	0
$25$	−4.45075	−0.890150
$26$	0	0
$27$	−2.74111	−0.527527
$28$	0	0
$29$	−3.83705	−0.712521	−0.356261	−	0.934387i	$-0.615948\pi$
$29$	−3.83705	−0.712521	−0.356261	+	0.934387i	$0.615948\pi$
$30$	0	0
$31$	−10.5322	−1.89164	−0.945822	−	0.324685i	$-0.894742\pi$
$31$	−10.5322	−1.89164	−0.945822	+	0.324685i	$0.894742\pi$
$32$	0	0
$33$	2.17741	0.379038
$34$	0	0
$35$	−0.741113	−0.125271
$36$	0	0
$37$	2.74111	0.450636	0.225318	−	0.974285i	$-0.427658\pi$
$37$	2.74111	0.450636	0.225318	+	0.974285i	$0.427658\pi$
$38$	0	0
$39$	15.0645	2.41224
$40$	0	0
$41$	7.27334	1.13591	0.567953	−	0.823061i	$-0.307736\pi$
$41$	7.27334	1.13591	0.567953	+	0.823061i	$0.307736\pi$
$42$	0	0
$43$	−6.35482	−0.969101	−0.484550	−	0.874763i	$-0.661017\pi$
$43$	−6.35482	−0.969101	−0.484550	+	0.874763i	$0.661017\pi$
$44$	0	0
$45$	−1.29036	−0.192356
$46$	0	0
$47$	−3.43630	−0.501235	−0.250618	−	0.968086i	$-0.580634\pi$
$47$	−3.43630	−0.501235	−0.250618	+	0.968086i	$0.580634\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−15.8370	−2.21763
$52$	0	0
$53$	1.48223	0.203599	0.101800	−	0.994805i	$-0.467540\pi$
$53$	1.48223	0.203599	0.101800	+	0.994805i	$0.467540\pi$
$54$	0	0
$55$	−0.741113	−0.0999316
$56$	0	0
$57$	9.48223	1.25595
$58$	0	0
$59$	−10.1774	−1.32499	−0.662493	−	0.749068i	$-0.730502\pi$
$59$	−10.1774	−1.32499	−0.662493	+	0.749068i	$0.730502\pi$
$60$	0	0
$61$	8.40075	1.07561	0.537803	−	0.843071i	$-0.319254\pi$
$61$	8.40075	1.07561	0.537803	+	0.843071i	$0.319254\pi$
$62$	0	0
$63$	1.74111	0.219360
$64$	0	0
$65$	−5.12741	−0.635977
$66$	0	0
$67$	−7.09593	−0.866906	−0.433453	−	0.901176i	$-0.642705\pi$
$67$	−7.09593	−0.866906	−0.433453	+	0.901176i	$0.642705\pi$
$68$	0	0
$69$	6.74111	0.811535
$70$	0	0
$71$	4.57816	0.543327	0.271664	−	0.962392i	$-0.412426\pi$
$71$	4.57816	0.543327	0.271664	+	0.962392i	$0.412426\pi$
$72$	0	0
$73$	3.27334	0.383116	0.191558	−	0.981481i	$-0.438646\pi$
$73$	3.27334	0.383116	0.191558	+	0.981481i	$0.438646\pi$
$74$	0	0
$75$	−9.69111	−1.11903
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	8.87259	0.998245	0.499122	−	0.866532i	$-0.333656\pi$
$79$	8.87259	0.998245	0.499122	+	0.866532i	$0.333656\pi$
$80$	0	0
$81$	−11.1919	−1.24354
$82$	0	0
$83$	−14.5467	−1.59671	−0.798353	−	0.602190i	$-0.794295\pi$
$83$	−14.5467	−1.59671	−0.798353	+	0.602190i	$0.794295\pi$
$84$	0	0
$85$	5.39037	0.584667
$86$	0	0
$87$	−8.35482	−0.895731
$88$	0	0
$89$	−15.9685	−1.69266	−0.846330	−	0.532659i	$-0.821193\pi$
$89$	−15.9685	−1.69266	−0.846330	+	0.532659i	$0.821193\pi$
$90$	0	0
$91$	6.91852	0.725258
$92$	0	0
$93$	−22.9330	−2.37804
$94$	0	0
$95$	−3.22741	−0.331126
$96$	0	0
$97$	0.386294	0.0392222	0.0196111	−	0.999808i	$-0.493757\pi$
$97$	0.386294	0.0392222	0.0196111	+	0.999808i	$0.493757\pi$
$98$	0	0
$99$	1.74111	0.174988
$100$	0	0
$101$	−1.08148	−0.107611	−0.0538055	−	0.998551i	$-0.517135\pi$
$101$	−1.08148	−0.107611	−0.0538055	+	0.998551i	$0.517135\pi$
$102$	0	0
$103$	−5.27334	−0.519598	−0.259799	−	0.965663i	$-0.583656\pi$
$103$	−5.27334	−0.519598	−0.259799	+	0.965663i	$0.583656\pi$
$104$	0	0
$105$	−1.61371	−0.157482
$106$	0	0
$107$	1.48223	0.143292	0.0716461	−	0.997430i	$-0.477175\pi$
$107$	1.48223	0.143292	0.0716461	+	0.997430i	$0.477175\pi$
$108$	0	0
$109$	15.0645	1.44291	0.721457	−	0.692460i	$-0.243473\pi$
$109$	15.0645	1.44291	0.721457	+	0.692460i	$0.243473\pi$
$110$	0	0
$111$	5.96853	0.566507
$112$	0	0
$113$	14.4863	1.36276	0.681378	−	0.731931i	$-0.261381\pi$
$113$	14.4863	1.36276	0.681378	+	0.731931i	$0.261381\pi$
$114$	0	0
$115$	−2.29444	−0.213957
$116$	0	0
$117$	12.0459	1.11365
$118$	0	0
$119$	−7.27334	−0.666746
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	15.8370	1.42798
$124$	0	0
$125$	7.00407	0.626463
$126$	0	0
$127$	1.80814	0.160446	0.0802230	−	0.996777i	$-0.474437\pi$
$127$	1.80814	0.160446	0.0802230	+	0.996777i	$0.474437\pi$
$128$	0	0
$129$	−13.8370	−1.21828
$130$	0	0
$131$	−10.5178	−0.918942	−0.459471	−	0.888193i	$-0.651961\pi$
$131$	−10.5178	−0.918942	−0.459471	+	0.888193i	$0.651961\pi$
$132$	0	0
$133$	4.35482	0.377611
$134$	0	0
$135$	2.03147	0.174841
$136$	0	0
$137$	12.9330	1.10494	0.552469	−	0.833533i	$-0.313686\pi$
$137$	12.9330	1.10494	0.552469	+	0.833533i	$0.313686\pi$
$138$	0	0
$139$	7.31927	0.620812	0.310406	−	0.950604i	$-0.399535\pi$
$139$	7.31927	0.620812	0.310406	+	0.950604i	$0.399535\pi$
$140$	0	0
$141$	−7.48223	−0.630117
$142$	0	0
$143$	6.91852	0.578556
$144$	0	0
$145$	2.84368	0.236155
$146$	0	0
$147$	2.17741	0.179590
$148$	0	0
$149$	13.6741	1.12023	0.560113	−	0.828417i	$-0.310758\pi$
$149$	13.6741	1.12023	0.560113	+	0.828417i	$0.310758\pi$
$150$	0	0
$151$	8.87259	0.722041	0.361021	−	0.932558i	$-0.382428\pi$
$151$	8.87259	0.722041	0.361021	+	0.932558i	$0.382428\pi$
$152$	0	0
$153$	−12.6637	−1.02380
$154$	0	0
$155$	7.80557	0.626959
$156$	0	0
$157$	7.25889	0.579322	0.289661	−	0.957129i	$-0.406457\pi$
$157$	7.25889	0.579322	0.289661	+	0.957129i	$0.406457\pi$
$158$	0	0
$159$	3.22741	0.255950
$160$	0	0
$161$	3.09593	0.243994
$162$	0	0
$163$	2.87259	0.224999	0.112499	−	0.993652i	$-0.464114\pi$
$163$	2.87259	0.224999	0.112499	+	0.993652i	$0.464114\pi$
$164$	0	0
$165$	−1.61371	−0.125627
$166$	0	0
$167$	5.80814	0.449447	0.224724	−	0.974423i	$-0.427852\pi$
$167$	5.80814	0.449447	0.224724	+	0.974423i	$0.427852\pi$
$168$	0	0
$169$	34.8660	2.68200
$170$	0	0
$171$	7.58223	0.579828
$172$	0	0
$173$	4.04593	0.307606	0.153803	−	0.988102i	$-0.450848\pi$
$173$	4.04593	0.307606	0.153803	+	0.988102i	$0.450848\pi$
$174$	0	0
$175$	−4.45075	−0.336445
$176$	0	0
$177$	−22.1604	−1.66568
$178$	0	0
$179$	23.4508	1.75279	0.876396	−	0.481592i	$-0.159941\pi$
$179$	23.4508	1.75279	0.876396	+	0.481592i	$0.159941\pi$
$180$	0	0
$181$	−20.7700	−1.54382	−0.771912	−	0.635730i	$-0.780699\pi$
$181$	−20.7700	−1.54382	−0.771912	+	0.635730i	$0.780699\pi$
$182$	0	0
$183$	18.2919	1.35217
$184$	0	0
$185$	−2.03147	−0.149357
$186$	0	0
$187$	−7.27334	−0.531879
$188$	0	0
$189$	−2.74111	−0.199387
$190$	0	0
$191$	−5.25889	−0.380520	−0.190260	−	0.981734i	$-0.560933\pi$
$191$	−5.25889	−0.380520	−0.190260	+	0.981734i	$0.560933\pi$
$192$	0	0
$193$	−0.964452	−0.0694228	−0.0347114	−	0.999397i	$-0.511051\pi$
$193$	−0.964452	−0.0694228	−0.0347114	+	0.999397i	$0.511051\pi$
$194$	0	0
$195$	−11.1645	−0.799504
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−8.23779	−0.583962	−0.291981	−	0.956424i	$-0.594314\pi$
$199$	−8.23779	−0.583962	−0.291981	+	0.956424i	$0.594314\pi$
$200$	0	0
$201$	−15.4508	−1.08981
$202$	0	0
$203$	−3.83705	−0.269308
$204$	0	0
$205$	−5.39037	−0.376480
$206$	0	0
$207$	5.39037	0.374656
$208$	0	0
$209$	4.35482	0.301229
$210$	0	0
$211$	9.48223	0.652783	0.326392	−	0.945235i	$-0.394167\pi$
$211$	9.48223	0.652783	0.326392	+	0.945235i	$0.394167\pi$
$212$	0	0
$213$	9.96853	0.683032
$214$	0	0
$215$	4.70964	0.321195
$216$	0	0
$217$	−10.5322	−0.714974
$218$	0	0
$219$	7.12741	0.481625
$220$	0	0
$221$	−50.3208	−3.38494
$222$	0	0
$223$	18.5611	1.24295	0.621473	−	0.783436i	$-0.286535\pi$
$223$	18.5611	1.24295	0.621473	+	0.783436i	$0.286535\pi$
$224$	0	0
$225$	−7.74926	−0.516617
$226$	0	0
$227$	−2.96445	−0.196758	−0.0983788	−	0.995149i	$-0.531366\pi$
$227$	−2.96445	−0.196758	−0.0983788	+	0.995149i	$0.531366\pi$
$228$	0	0
$229$	1.86852	0.123475	0.0617376	−	0.998092i	$-0.480336\pi$
$229$	1.86852	0.123475	0.0617376	+	0.998092i	$0.480336\pi$
$230$	0	0
$231$	2.17741	0.143263
$232$	0	0
$233$	−15.8370	−1.03752	−0.518760	−	0.854920i	$-0.673606\pi$
$233$	−15.8370	−1.03752	−0.518760	+	0.854920i	$0.673606\pi$
$234$	0	0
$235$	2.54668	0.166127
$236$	0	0
$237$	19.3193	1.25492
$238$	0	0
$239$	−12.3837	−0.801037	−0.400518	−	0.916289i	$-0.631170\pi$
$239$	−12.3837	−0.801037	−0.400518	+	0.916289i	$0.631170\pi$
$240$	0	0
$241$	4.75557	0.306333	0.153167	−	0.988200i	$-0.451053\pi$
$241$	4.75557	0.306333	0.153167	+	0.988200i	$0.451053\pi$
$242$	0	0
$243$	−16.1459	−1.03576
$244$	0	0
$245$	−0.741113	−0.0473480
$246$	0	0
$247$	30.1289	1.91706
$248$	0	0
$249$	−31.6741	−2.00726
$250$	0	0
$251$	−7.63073	−0.481647	−0.240824	−	0.970569i	$-0.577418\pi$
$251$	−7.63073	−0.481647	−0.240824	+	0.970569i	$0.577418\pi$
$252$	0	0
$253$	3.09593	0.194640
$254$	0	0
$255$	11.7370	0.735002
$256$	0	0
$257$	22.7096	1.41659	0.708294	−	0.705917i	$-0.249465\pi$
$257$	22.7096	1.41659	0.708294	+	0.705917i	$0.249465\pi$
$258$	0	0
$259$	2.74111	0.170324
$260$	0	0
$261$	−6.68073	−0.413527
$262$	0	0
$263$	13.4822	0.831350	0.415675	−	0.909513i	$-0.363545\pi$
$263$	13.4822	0.831350	0.415675	+	0.909513i	$0.363545\pi$
$264$	0	0
$265$	−1.09850	−0.0674801
$266$	0	0
$267$	−34.7700	−2.12789
$268$	0	0
$269$	−20.5467	−1.25275	−0.626377	−	0.779521i	$-0.715463\pi$
$269$	−20.5467	−1.25275	−0.626377	+	0.779521i	$0.715463\pi$
$270$	0	0
$271$	−18.5467	−1.12663	−0.563315	−	0.826242i	$-0.690474\pi$
$271$	−18.5467	−1.12663	−0.563315	+	0.826242i	$0.690474\pi$
$272$	0	0
$273$	15.0645	0.911742
$274$	0	0
$275$	−4.45075	−0.268390
$276$	0	0
$277$	−13.2274	−0.794758	−0.397379	−	0.917655i	$-0.630080\pi$
$277$	−13.2274	−0.794758	−0.397379	+	0.917655i	$0.630080\pi$
$278$	0	0
$279$	−18.3378	−1.09786
$280$	0	0
$281$	−17.3193	−1.03318	−0.516591	−	0.856233i	$-0.672799\pi$
$281$	−17.3193	−1.03318	−0.516591	+	0.856233i	$0.672799\pi$
$282$	0	0
$283$	−11.6741	−0.693953	−0.346976	−	0.937874i	$-0.612792\pi$
$283$	−11.6741	−0.693953	−0.346976	+	0.937874i	$0.612792\pi$
$284$	0	0
$285$	−7.02740	−0.416267
$286$	0	0
$287$	7.27334	0.429332
$288$	0	0
$289$	35.9015	2.11185
$290$	0	0
$291$	0.841119	0.0493073
$292$	0	0
$293$	30.8556	1.80260	0.901301	−	0.433193i	$-0.142613\pi$
$293$	30.8556	1.80260	0.901301	+	0.433193i	$0.142613\pi$
$294$	0	0
$295$	7.54261	0.439148
$296$	0	0
$297$	−2.74111	−0.159056
$298$	0	0
$299$	21.4193	1.23871
$300$	0	0
$301$	−6.35482	−0.366286
$302$	0	0
$303$	−2.35482	−0.135281
$304$	0	0
$305$	−6.22590	−0.356494
$306$	0	0
$307$	−18.9645	−1.08236	−0.541179	−	0.840907i	$-0.682022\pi$
$307$	−18.9645	−1.08236	−0.541179	+	0.840907i	$0.682022\pi$
$308$	0	0
$309$	−11.4822	−0.653201
$310$	0	0
$311$	17.1815	0.974273	0.487136	−	0.873326i	$-0.338041\pi$
$311$	17.1815	0.974273	0.487136	+	0.873326i	$0.338041\pi$
$312$	0	0
$313$	10.4863	0.592721	0.296360	−	0.955076i	$-0.404227\pi$
$313$	10.4863	0.592721	0.296360	+	0.955076i	$0.404227\pi$
$314$	0	0
$315$	−1.29036	−0.0727036
$316$	0	0
$317$	−18.9330	−1.06338	−0.531691	−	0.846938i	$-0.678443\pi$
$317$	−18.9330	−1.06338	−0.531691	+	0.846938i	$0.678443\pi$
$318$	0	0
$319$	−3.83705	−0.214833
$320$	0	0
$321$	3.22741	0.180137
$322$	0	0
$323$	−31.6741	−1.76239
$324$	0	0
$325$	−30.7926	−1.70807
$326$	0	0
$327$	32.8015	1.81393
$328$	0	0
$329$	−3.43630	−0.189449
$330$	0	0
$331$	−0.486300	−0.0267295	−0.0133647	−	0.999911i	$-0.504254\pi$
$331$	−0.486300	−0.0267295	−0.0133647	+	0.999911i	$0.504254\pi$
$332$	0	0
$333$	4.77259	0.261536
$334$	0	0
$335$	5.25889	0.287324
$336$	0	0
$337$	20.8726	1.13700	0.568501	−	0.822682i	$-0.307523\pi$
$337$	20.8726	1.13700	0.568501	+	0.822682i	$0.307523\pi$
$338$	0	0
$339$	31.5426	1.71316
$340$	0	0
$341$	−10.5322	−0.570352
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−4.99593	−0.268972
$346$	0	0
$347$	20.5467	1.10300	0.551502	−	0.834174i	$-0.314055\pi$
$347$	20.5467	1.10300	0.551502	+	0.834174i	$0.314055\pi$
$348$	0	0
$349$	−30.9474	−1.65658	−0.828289	−	0.560301i	$-0.810685\pi$
$349$	−30.9474	−1.65658	−0.828289	+	0.560301i	$0.810685\pi$
$350$	0	0
$351$	−18.9645	−1.01225
$352$	0	0
$353$	23.3797	1.24437	0.622187	−	0.782869i	$-0.286244\pi$
$353$	23.3797	1.24437	0.622187	+	0.782869i	$0.286244\pi$
$354$	0	0
$355$	−3.39293	−0.180078
$356$	0	0
$357$	−15.8370	−0.838185
$358$	0	0
$359$	19.0645	1.00618	0.503092	−	0.864233i	$-0.332196\pi$
$359$	19.0645	1.00618	0.503092	+	0.864233i	$0.332196\pi$
$360$	0	0
$361$	−0.0355483	−0.00187096
$362$	0	0
$363$	2.17741	0.114284
$364$	0	0
$365$	−2.42592	−0.126978
$366$	0	0
$367$	18.9789	0.990691	0.495345	−	0.868696i	$-0.335041\pi$
$367$	18.9789	0.990691	0.495345	+	0.868696i	$0.335041\pi$
$368$	0	0
$369$	12.6637	0.659246
$370$	0	0
$371$	1.48223	0.0769533
$372$	0	0
$373$	7.80814	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$373$	7.80814	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$374$	0	0
$375$	15.2507	0.787545
$376$	0	0
$377$	−26.5467	−1.36722
$378$	0	0
$379$	−29.9974	−1.54086	−0.770432	−	0.637522i	$-0.779960\pi$
$379$	−29.9974	−1.54086	−0.770432	+	0.637522i	$0.779960\pi$
$380$	0	0
$381$	3.93705	0.201701
$382$	0	0
$383$	−34.8241	−1.77943	−0.889714	−	0.456518i	$-0.849096\pi$
$383$	−34.8241	−1.77943	−0.889714	+	0.456518i	$0.849096\pi$
$384$	0	0
$385$	−0.741113	−0.0377706
$386$	0	0
$387$	−11.0645	−0.562438
$388$	0	0
$389$	20.4152	1.03509	0.517546	−	0.855655i	$-0.326846\pi$
$389$	20.4152	1.03509	0.517546	+	0.855655i	$0.326846\pi$
$390$	0	0
$391$	−22.5178	−1.13877
$392$	0	0
$393$	−22.9015	−1.15523
$394$	0	0
$395$	−6.57559	−0.330854
$396$	0	0
$397$	15.7452	0.790228	0.395114	−	0.918632i	$-0.370705\pi$
$397$	15.7452	0.790228	0.395114	+	0.918632i	$0.370705\pi$
$398$	0	0
$399$	9.48223	0.474705
$400$	0	0
$401$	22.7096	1.13407	0.567033	−	0.823695i	$-0.308091\pi$
$401$	22.7096	1.13407	0.567033	+	0.823695i	$0.308091\pi$
$402$	0	0
$403$	−72.8675	−3.62979
$404$	0	0
$405$	8.29444	0.412154
$406$	0	0
$407$	2.74111	0.135872
$408$	0	0
$409$	0.308890	0.0152736	0.00763682	−	0.999971i	$-0.497569\pi$
$409$	0.308890	0.0152736	0.00763682	+	0.999971i	$0.497569\pi$
$410$	0	0
$411$	28.1604	1.38905
$412$	0	0
$413$	−10.1774	−0.500798
$414$	0	0
$415$	10.7807	0.529205
$416$	0	0
$417$	15.9371	0.780441
$418$	0	0
$419$	32.1459	1.57043	0.785216	−	0.619222i	$-0.212552\pi$
$419$	32.1459	1.57043	0.785216	+	0.619222i	$0.212552\pi$
$420$	0	0
$421$	−2.51777	−0.122709	−0.0613544	−	0.998116i	$-0.519542\pi$
$421$	−2.51777	−0.122709	−0.0613544	+	0.998116i	$0.519542\pi$
$422$	0	0
$423$	−5.98298	−0.290902
$424$	0	0
$425$	32.3718	1.57026
$426$	0	0
$427$	8.40075	0.406541
$428$	0	0
$429$	15.0645	0.727319
$430$	0	0
$431$	−35.7741	−1.72318	−0.861589	−	0.507607i	$-0.830530\pi$
$431$	−35.7741	−1.72318	−0.861589	+	0.507607i	$0.830530\pi$
$432$	0	0
$433$	12.7700	0.613688	0.306844	−	0.951760i	$-0.400727\pi$
$433$	12.7700	0.613688	0.306844	+	0.951760i	$0.400727\pi$
$434$	0	0
$435$	6.19186	0.296877
$436$	0	0
$437$	13.4822	0.644942
$438$	0	0
$439$	−24.3548	−1.16239	−0.581196	−	0.813764i	$-0.697415\pi$
$439$	−24.3548	−1.16239	−0.581196	+	0.813764i	$0.697415\pi$
$440$	0	0
$441$	1.74111	0.0829101
$442$	0	0
$443$	−0.932977	−0.0443271	−0.0221635	−	0.999754i	$-0.507055\pi$
$443$	−0.932977	−0.0443271	−0.0221635	+	0.999754i	$0.507055\pi$
$444$	0	0
$445$	11.8345	0.561008
$446$	0	0
$447$	29.7741	1.40827
$448$	0	0
$449$	−29.6426	−1.39892	−0.699461	−	0.714671i	$-0.746576\pi$
$449$	−29.6426	−1.39892	−0.699461	+	0.714671i	$0.746576\pi$
$450$	0	0
$451$	7.27334	0.342488
$452$	0	0
$453$	19.3193	0.907699
$454$	0	0
$455$	−5.12741	−0.240377
$456$	0	0
$457$	14.2630	0.667193	0.333597	−	0.942716i	$-0.391738\pi$
$457$	14.2630	0.667193	0.333597	+	0.942716i	$0.391738\pi$
$458$	0	0
$459$	19.9371	0.930582
$460$	0	0
$461$	9.85406	0.458950	0.229475	−	0.973315i	$-0.426299\pi$
$461$	9.85406	0.458950	0.229475	+	0.973315i	$0.426299\pi$
$462$	0	0
$463$	−14.7700	−0.686421	−0.343211	−	0.939258i	$-0.611514\pi$
$463$	−14.7700	−0.686421	−0.343211	+	0.939258i	$0.611514\pi$
$464$	0	0
$465$	16.9959	0.788167
$466$	0	0
$467$	−5.85150	−0.270775	−0.135388	−	0.990793i	$-0.543228\pi$
$467$	−5.85150	−0.270775	−0.135388	+	0.990793i	$0.543228\pi$
$468$	0	0
$469$	−7.09593	−0.327660
$470$	0	0
$471$	15.8056	0.728282
$472$	0	0
$473$	−6.35482	−0.292195
$474$	0	0
$475$	−19.3822	−0.889317
$476$	0	0
$477$	2.58072	0.118163
$478$	0	0
$479$	9.45332	0.431933	0.215967	−	0.976401i	$-0.430710\pi$
$479$	9.45332	0.431933	0.215967	+	0.976401i	$0.430710\pi$
$480$	0	0
$481$	18.9645	0.864705
$482$	0	0
$483$	6.74111	0.306731
$484$	0	0
$485$	−0.286287	−0.0129996
$486$	0	0
$487$	−32.9619	−1.49365	−0.746823	−	0.665023i	$-0.768422\pi$
$487$	−32.9619	−1.49365	−0.746823	+	0.665023i	$0.768422\pi$
$488$	0	0
$489$	6.25481	0.282852
$490$	0	0
$491$	−5.58223	−0.251923	−0.125961	−	0.992035i	$-0.540202\pi$
$491$	−5.58223	−0.251923	−0.125961	+	0.992035i	$0.540202\pi$
$492$	0	0
$493$	27.9081	1.25692
$494$	0	0
$495$	−1.29036	−0.0579974
$496$	0	0
$497$	4.57816	0.205358
$498$	0	0
$499$	−39.9660	−1.78912	−0.894561	−	0.446946i	$-0.852512\pi$
$499$	−39.9660	−1.78912	−0.894561	+	0.446946i	$0.852512\pi$
$500$	0	0
$501$	12.6467	0.565012
$502$	0	0
$503$	41.8949	1.86800	0.934000	−	0.357273i	$-0.116294\pi$
$503$	41.8949	1.86800	0.934000	+	0.357273i	$0.116294\pi$
$504$	0	0
$505$	0.801497	0.0356661
$506$	0	0
$507$	75.9175	3.37161
$508$	0	0
$509$	−30.1315	−1.33555	−0.667777	−	0.744361i	$-0.732754\pi$
$509$	−30.1315	−1.33555	−0.667777	+	0.744361i	$0.732754\pi$
$510$	0	0
$511$	3.27334	0.144804
$512$	0	0
$513$	−11.9371	−0.527034
$514$	0	0
$515$	3.90814	0.172213
$516$	0	0
$517$	−3.43630	−0.151128
$518$	0	0
$519$	8.80965	0.386701
$520$	0	0
$521$	−33.7426	−1.47829	−0.739146	−	0.673546i	$-0.764770\pi$
$521$	−33.7426	−1.47829	−0.739146	+	0.673546i	$0.764770\pi$
$522$	0	0
$523$	8.38373	0.366595	0.183297	−	0.983058i	$-0.441323\pi$
$523$	8.38373	0.366595	0.183297	+	0.983058i	$0.441323\pi$
$524$	0	0
$525$	−9.69111	−0.422955
$526$	0	0
$527$	76.6045	3.33694
$528$	0	0
$529$	−13.4152	−0.583270
$530$	0	0
$531$	−17.7200	−0.768983
$532$	0	0
$533$	50.3208	2.17963
$534$	0	0
$535$	−1.09850	−0.0474922
$536$	0	0
$537$	51.0619	2.20348
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−8.51777	−0.366208	−0.183104	−	0.983094i	$-0.558614\pi$
$541$	−8.51777	−0.366208	−0.183104	+	0.983094i	$0.558614\pi$
$542$	0	0
$543$	−45.2248	−1.94078
$544$	0	0
$545$	−11.1645	−0.478233
$546$	0	0
$547$	41.8030	1.78737	0.893684	−	0.448697i	$-0.148112\pi$
$547$	41.8030	1.78737	0.893684	+	0.448697i	$0.148112\pi$
$548$	0	0
$549$	14.6267	0.624250
$550$	0	0
$551$	−16.7096	−0.711855
$552$	0	0
$553$	8.87259	0.377301
$554$	0	0
$555$	−4.42335	−0.187761
$556$	0	0
$557$	−7.74519	−0.328174	−0.164087	−	0.986446i	$-0.552468\pi$
$557$	−7.74519	−0.328174	−0.164087	+	0.986446i	$0.552468\pi$
$558$	0	0
$559$	−43.9660	−1.85956
$560$	0	0
$561$	−15.8370	−0.668641
$562$	0	0
$563$	−0.417768	−0.0176068	−0.00880341	−	0.999961i	$-0.502802\pi$
$563$	−0.417768	−0.0176068	−0.00880341	+	0.999961i	$0.502802\pi$
$564$	0	0
$565$	−10.7360	−0.451666
$566$	0	0
$567$	−11.1919	−0.470014
$568$	0	0
$569$	−24.8726	−1.04271	−0.521357	−	0.853339i	$-0.674574\pi$
$569$	−24.8726	−1.04271	−0.521357	+	0.853339i	$0.674574\pi$
$570$	0	0
$571$	15.6741	0.655940	0.327970	−	0.944688i	$-0.393635\pi$
$571$	15.6741	0.655940	0.327970	+	0.944688i	$0.393635\pi$
$572$	0	0
$573$	−11.4508	−0.478362
$574$	0	0
$575$	−13.7792	−0.574633
$576$	0	0
$577$	−2.57816	−0.107330	−0.0536651	−	0.998559i	$-0.517090\pi$
$577$	−2.57816	−0.107330	−0.0536651	+	0.998559i	$0.517090\pi$
$578$	0	0
$579$	−2.10001	−0.0872733
$580$	0	0
$581$	−14.5467	−0.603498
$582$	0	0
$583$	1.48223	0.0613875
$584$	0	0
$585$	−8.92739	−0.369102
$586$	0	0
$587$	4.23779	0.174912	0.0874562	−	0.996168i	$-0.472126\pi$
$587$	4.23779	0.174912	0.0874562	+	0.996168i	$0.472126\pi$
$588$	0	0
$589$	−45.8660	−1.88987
$590$	0	0
$591$	−13.0645	−0.537400
$592$	0	0
$593$	9.82003	0.403260	0.201630	−	0.979462i	$-0.435376\pi$
$593$	9.82003	0.403260	0.201630	+	0.979462i	$0.435376\pi$
$594$	0	0
$595$	5.39037	0.220984
$596$	0	0
$597$	−17.9371	−0.734115
$598$	0	0
$599$	15.9081	0.649989	0.324995	−	0.945716i	$-0.394638\pi$
$599$	15.9081	0.649989	0.324995	+	0.945716i	$0.394638\pi$
$600$	0	0
$601$	−31.6282	−1.29014	−0.645070	−	0.764124i	$-0.723172\pi$
$601$	−31.6282	−1.29014	−0.645070	+	0.764124i	$0.723172\pi$
$602$	0	0
$603$	−12.3548	−0.503127
$604$	0	0
$605$	−0.741113	−0.0301305
$606$	0	0
$607$	0.0629484	0.00255500	0.00127750	−	0.999999i	$-0.499593\pi$
$607$	0.0629484	0.00255500	0.00127750	+	0.999999i	$0.499593\pi$
$608$	0	0
$609$	−8.35482	−0.338554
$610$	0	0
$611$	−23.7741	−0.961797
$612$	0	0
$613$	8.99336	0.363238	0.181619	−	0.983369i	$-0.441866\pi$
$613$	8.99336	0.363238	0.181619	+	0.983369i	$0.441866\pi$
$614$	0	0
$615$	−11.7370	−0.473283
$616$	0	0
$617$	44.3208	1.78429	0.892144	−	0.451752i	$-0.149201\pi$
$617$	44.3208	1.78429	0.892144	+	0.451752i	$0.149201\pi$
$618$	0	0
$619$	−15.6596	−0.629414	−0.314707	−	0.949189i	$-0.601906\pi$
$619$	−15.6596	−0.629414	−0.314707	+	0.949189i	$0.601906\pi$
$620$	0	0
$621$	−8.48630	−0.340543
$622$	0	0
$623$	−15.9685	−0.639765
$624$	0	0
$625$	17.0629	0.682518
$626$	0	0
$627$	9.48223	0.378684
$628$	0	0
$629$	−19.9371	−0.794942
$630$	0	0
$631$	7.71371	0.307078	0.153539	−	0.988143i	$-0.450933\pi$
$631$	7.71371	0.307078	0.153539	+	0.988143i	$0.450933\pi$
$632$	0	0
$633$	20.6467	0.820632
$634$	0	0
$635$	−1.34003	−0.0531776
$636$	0	0
$637$	6.91852	0.274122
$638$	0	0
$639$	7.97109	0.315331
$640$	0	0
$641$	18.7411	0.740229	0.370115	−	0.928986i	$-0.379318\pi$
$641$	18.7411	0.740229	0.370115	+	0.928986i	$0.379318\pi$
$642$	0	0
$643$	−13.3759	−0.527495	−0.263747	−	0.964592i	$-0.584958\pi$
$643$	−13.3759	−0.527495	−0.263747	+	0.964592i	$0.584958\pi$
$644$	0	0
$645$	10.2548	0.403783
$646$	0	0
$647$	−24.7241	−0.972004	−0.486002	−	0.873958i	$-0.661545\pi$
$647$	−24.7241	−0.972004	−0.486002	+	0.873958i	$0.661545\pi$
$648$	0	0
$649$	−10.1774	−0.399498
$650$	0	0
$651$	−22.9330	−0.898814
$652$	0	0
$653$	8.67816	0.339603	0.169801	−	0.985478i	$-0.445687\pi$
$653$	8.67816	0.339603	0.169801	+	0.985478i	$0.445687\pi$
$654$	0	0
$655$	7.79486	0.304570
$656$	0	0
$657$	5.69926	0.222349
$658$	0	0
$659$	−2.61778	−0.101974	−0.0509871	−	0.998699i	$-0.516237\pi$
$659$	−2.61778	−0.101974	−0.0509871	+	0.998699i	$0.516237\pi$
$660$	0	0
$661$	−10.6700	−0.415016	−0.207508	−	0.978233i	$-0.566535\pi$
$661$	−10.6700	−0.415016	−0.207508	+	0.978233i	$0.566535\pi$
$662$	0	0
$663$	−109.569	−4.25531
$664$	0	0
$665$	−3.22741	−0.125154
$666$	0	0
$667$	−11.8792	−0.459966
$668$	0	0
$669$	40.4152	1.56254
$670$	0	0
$671$	8.40075	0.324307
$672$	0	0
$673$	−18.7385	−0.722318	−0.361159	−	0.932504i	$-0.617619\pi$
$673$	−18.7385	−0.722318	−0.361159	+	0.932504i	$0.617619\pi$
$674$	0	0
$675$	12.2000	0.469579
$676$	0	0
$677$	−19.6571	−0.755483	−0.377741	−	0.925911i	$-0.623299\pi$
$677$	−19.6571	−0.755483	−0.377741	+	0.925911i	$0.623299\pi$
$678$	0	0
$679$	0.386294	0.0148246
$680$	0	0
$681$	−6.45483	−0.247349
$682$	0	0
$683$	−7.67409	−0.293641	−0.146820	−	0.989163i	$-0.546904\pi$
$683$	−7.67409	−0.293641	−0.146820	+	0.989163i	$0.546904\pi$
$684$	0	0
$685$	−9.58480	−0.366216
$686$	0	0
$687$	4.06853	0.155224
$688$	0	0
$689$	10.2548	0.390677
$690$	0	0
$691$	−22.2693	−0.847163	−0.423581	−	0.905858i	$-0.639227\pi$
$691$	−22.2693	−0.847163	−0.423581	+	0.905858i	$0.639227\pi$
$692$	0	0
$693$	1.74111	0.0661394
$694$	0	0
$695$	−5.42441	−0.205759
$696$	0	0
$697$	−52.9015	−2.00379
$698$	0	0
$699$	−34.4837	−1.30429
$700$	0	0
$701$	10.0289	0.378787	0.189393	−	0.981901i	$-0.439348\pi$
$701$	10.0289	0.378787	0.189393	+	0.981901i	$0.439348\pi$
$702$	0	0
$703$	11.9371	0.450214
$704$	0	0
$705$	5.54517	0.208843
$706$	0	0
$707$	−1.08148	−0.0406731
$708$	0	0
$709$	−16.7411	−0.628726	−0.314363	−	0.949303i	$-0.601791\pi$
$709$	−16.7411	−0.628726	−0.314363	+	0.949303i	$0.601791\pi$
$710$	0	0
$711$	15.4482	0.579352
$712$	0	0
$713$	−32.6071	−1.22114
$714$	0	0
$715$	−5.12741	−0.191754
$716$	0	0
$717$	−26.9645	−1.00701
$718$	0	0
$719$	0.724094	0.0270041	0.0135021	−	0.999909i	$-0.495702\pi$
$719$	0.724094	0.0270041	0.0135021	+	0.999909i	$0.495702\pi$
$720$	0	0
$721$	−5.27334	−0.196390
$722$	0	0
$723$	10.3548	0.385100
$724$	0	0
$725$	17.0777	0.634251
$726$	0	0
$727$	25.8515	0.958779	0.479390	−	0.877602i	$-0.340858\pi$
$727$	25.8515	0.958779	0.479390	+	0.877602i	$0.340858\pi$
$728$	0	0
$729$	−1.58072	−0.0585453
$730$	0	0
$731$	46.2208	1.70954
$732$	0	0
$733$	49.3733	1.82365	0.911823	−	0.410583i	$-0.134675\pi$
$733$	49.3733	1.82365	0.911823	+	0.410583i	$0.134675\pi$
$734$	0	0
$735$	−1.61371	−0.0595225
$736$	0	0
$737$	−7.09593	−0.261382
$738$	0	0
$739$	14.5756	0.536172	0.268086	−	0.963395i	$-0.413609\pi$
$739$	14.5756	0.536172	0.268086	+	0.963395i	$0.413609\pi$
$740$	0	0
$741$	65.6030	2.40999
$742$	0	0
$743$	5.92890	0.217510	0.108755	−	0.994069i	$-0.465314\pi$
$743$	5.92890	0.217510	0.108755	+	0.994069i	$0.465314\pi$
$744$	0	0
$745$	−10.1340	−0.371283
$746$	0	0
$747$	−25.3274	−0.926682
$748$	0	0
$749$	1.48223	0.0541594
$750$	0	0
$751$	12.5782	0.458984	0.229492	−	0.973311i	$-0.426294\pi$
$751$	12.5782	0.458984	0.229492	+	0.973311i	$0.426294\pi$
$752$	0	0
$753$	−16.6152	−0.605492
$754$	0	0
$755$	−6.57559	−0.239310
$756$	0	0
$757$	32.4548	1.17959	0.589795	−	0.807553i	$-0.299208\pi$
$757$	32.4548	1.17959	0.589795	+	0.807553i	$0.299208\pi$
$758$	0	0
$759$	6.74111	0.244687
$760$	0	0
$761$	−1.49925	−0.0543476	−0.0271738	−	0.999631i	$-0.508651\pi$
$761$	−1.49925	−0.0543476	−0.0271738	+	0.999631i	$0.508651\pi$
$762$	0	0
$763$	15.0645	0.545370
$764$	0	0
$765$	9.38524	0.339324
$766$	0	0
$767$	−70.4126	−2.54245
$768$	0	0
$769$	−38.5008	−1.38837	−0.694186	−	0.719795i	$-0.744236\pi$
$769$	−38.5008	−1.38837	−0.694186	+	0.719795i	$0.744236\pi$
$770$	0	0
$771$	49.4482	1.78083
$772$	0	0
$773$	49.6401	1.78543	0.892714	−	0.450623i	$-0.148798\pi$
$773$	49.6401	1.78543	0.892714	+	0.450623i	$0.148798\pi$
$774$	0	0
$775$	46.8763	1.68385
$776$	0	0
$777$	5.96853	0.214120
$778$	0	0
$779$	31.6741	1.13484
$780$	0	0
$781$	4.57816	0.163819
$782$	0	0
$783$	10.5178	0.375875
$784$	0	0
$785$	−5.37965	−0.192008
$786$	0	0
$787$	−38.1000	−1.35812	−0.679059	−	0.734083i	$-0.737612\pi$
$787$	−38.1000	−1.35812	−0.679059	+	0.734083i	$0.737612\pi$
$788$	0	0
$789$	29.3563	1.04511
$790$	0	0
$791$	14.4863	0.515074
$792$	0	0
$793$	58.1208	2.06393
$794$	0	0
$795$	−2.39188	−0.0848311
$796$	0	0
$797$	42.9330	1.52076	0.760382	−	0.649476i	$-0.225012\pi$
$797$	42.9330	1.52076	0.760382	+	0.649476i	$0.225012\pi$
$798$	0	0
$799$	24.9934	0.884202
$800$	0	0
$801$	−27.8030	−0.982371
$802$	0	0
$803$	3.27334	0.115514
$804$	0	0
$805$	−2.29444	−0.0808682
$806$	0	0
$807$	−44.7385	−1.57487
$808$	0	0
$809$	24.4548	0.859786	0.429893	−	0.902880i	$-0.358551\pi$
$809$	24.4548	0.859786	0.429893	+	0.902880i	$0.358551\pi$
$810$	0	0
$811$	−13.1563	−0.461981	−0.230990	−	0.972956i	$-0.574197\pi$
$811$	−13.1563	−0.461981	−0.230990	+	0.972956i	$0.574197\pi$
$812$	0	0
$813$	−40.3837	−1.41632
$814$	0	0
$815$	−2.12892	−0.0745727
$816$	0	0
$817$	−27.6741	−0.968194
$818$	0	0
$819$	12.0459	0.420919
$820$	0	0
$821$	−41.0223	−1.43169	−0.715844	−	0.698261i	$-0.753958\pi$
$821$	−41.0223	−1.43169	−0.715844	+	0.698261i	$0.753958\pi$
$822$	0	0
$823$	5.58480	0.194674	0.0973369	−	0.995251i	$-0.468968\pi$
$823$	5.58480	0.194674	0.0973369	+	0.995251i	$0.468968\pi$
$824$	0	0
$825$	−9.69111	−0.337401
$826$	0	0
$827$	−12.9934	−0.451823	−0.225912	−	0.974148i	$-0.572536\pi$
$827$	−12.9934	−0.451823	−0.225912	+	0.974148i	$0.572536\pi$
$828$	0	0
$829$	10.2233	0.355071	0.177536	−	0.984114i	$-0.443188\pi$
$829$	10.2233	0.355071	0.177536	+	0.984114i	$0.443188\pi$
$830$	0	0
$831$	−28.8015	−0.999113
$832$	0	0
$833$	−7.27334	−0.252006
$834$	0	0
$835$	−4.30448	−0.148963
$836$	0	0
$837$	28.8700	0.997894
$838$	0	0
$839$	0.724094	0.0249985	0.0124992	−	0.999922i	$-0.496021\pi$
$839$	0.724094	0.0249985	0.0124992	+	0.999922i	$0.496021\pi$
$840$	0	0
$841$	−14.2771	−0.492313
$842$	0	0
$843$	−37.7111	−1.29884
$844$	0	0
$845$	−25.8396	−0.888910
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−25.4193	−0.872387
$850$	0	0
$851$	8.48630	0.290907
$852$	0	0
$853$	20.9267	0.716516	0.358258	−	0.933623i	$-0.383371\pi$
$853$	20.9267	0.716516	0.358258	+	0.933623i	$0.383371\pi$
$854$	0	0
$855$	−5.61929	−0.192176
$856$	0	0
$857$	15.2104	0.519577	0.259789	−	0.965666i	$-0.416347\pi$
$857$	15.2104	0.519577	0.259789	+	0.965666i	$0.416347\pi$
$858$	0	0
$859$	7.65964	0.261343	0.130672	−	0.991426i	$-0.458287\pi$
$859$	7.65964	0.261343	0.130672	+	0.991426i	$0.458287\pi$
$860$	0	0
$861$	15.8370	0.539725
$862$	0	0
$863$	−36.3837	−1.23852	−0.619258	−	0.785187i	$-0.712567\pi$
$863$	−36.3837	−1.23852	−0.619258	+	0.785187i	$0.712567\pi$
$864$	0	0
$865$	−2.99849	−0.101952
$866$	0	0
$867$	78.1723	2.65487
$868$	0	0
$869$	8.87259	0.300982
$870$	0	0
$871$	−49.0934	−1.66347
$872$	0	0
$873$	0.672581	0.0227634
$874$	0	0
$875$	7.00407	0.236781
$876$	0	0
$877$	−33.5822	−1.13399	−0.566996	−	0.823721i	$-0.691894\pi$
$877$	−33.5822	−1.13399	−0.566996	+	0.823721i	$0.691894\pi$
$878$	0	0
$879$	67.1852	2.26610
$880$	0	0
$881$	21.4797	0.723668	0.361834	−	0.932243i	$-0.382151\pi$
$881$	21.4797	0.723668	0.361834	+	0.932243i	$0.382151\pi$
$882$	0	0
$883$	−17.7452	−0.597173	−0.298587	−	0.954383i	$-0.596515\pi$
$883$	−17.7452	−0.597173	−0.298587	+	0.954383i	$0.596515\pi$
$884$	0	0
$885$	16.4234	0.552065
$886$	0	0
$887$	56.3548	1.89221	0.946105	−	0.323861i	$-0.104981\pi$
$887$	56.3548	1.89221	0.946105	+	0.323861i	$0.104981\pi$
$888$	0	0
$889$	1.80814	0.0606429
$890$	0	0
$891$	−11.1919	−0.374942
$892$	0	0
$893$	−14.9645	−0.500766
$894$	0	0
$895$	−17.3797	−0.580938
$896$	0	0
$897$	46.6385	1.55722
$898$	0	0
$899$	40.4126	1.34784
$900$	0	0
$901$	−10.7807	−0.359158
$902$	0	0
$903$	−13.8370	−0.460468
$904$	0	0
$905$	15.3929	0.511678
$906$	0	0
$907$	−38.2208	−1.26910	−0.634550	−	0.772882i	$-0.718815\pi$
$907$	−38.2208	−1.26910	−0.634550	+	0.772882i	$0.718815\pi$
$908$	0	0
$909$	−1.88297	−0.0624543
$910$	0	0
$911$	7.25632	0.240413	0.120206	−	0.992749i	$-0.461644\pi$
$911$	7.25632	0.240413	0.120206	+	0.992749i	$0.461644\pi$
$912$	0	0
$913$	−14.5467	−0.481425
$914$	0	0
$915$	−13.5563	−0.448159
$916$	0	0
$917$	−10.5178	−0.347328
$918$	0	0
$919$	−7.34818	−0.242394	−0.121197	−	0.992628i	$-0.538673\pi$
$919$	−7.34818	−0.242394	−0.121197	+	0.992628i	$0.538673\pi$
$920$	0	0
$921$	−41.2934	−1.36066
$922$	0	0
$923$	31.6741	1.04257
$924$	0	0
$925$	−12.2000	−0.401134
$926$	0	0
$927$	−9.18148	−0.301559
$928$	0	0
$929$	35.5111	1.16508	0.582541	−	0.812801i	$-0.302058\pi$
$929$	35.5111	1.16508	0.582541	+	0.812801i	$0.302058\pi$
$930$	0	0
$931$	4.35482	0.142723
$932$	0	0
$933$	37.4111	1.22479
$934$	0	0
$935$	5.39037	0.176284
$936$	0	0
$937$	−25.2815	−0.825910	−0.412955	−	0.910751i	$-0.635503\pi$
$937$	−25.2815	−0.825910	−0.412955	+	0.910751i	$0.635503\pi$
$938$	0	0
$939$	22.8330	0.745126
$940$	0	0
$941$	−10.3800	−0.338378	−0.169189	−	0.985584i	$-0.554115\pi$
$941$	−10.3800	−0.338378	−0.169189	+	0.985584i	$0.554115\pi$
$942$	0	0
$943$	22.5178	0.733280
$944$	0	0
$945$	2.03147	0.0660839
$946$	0	0
$947$	−52.2812	−1.69891	−0.849455	−	0.527662i	$-0.823069\pi$
$947$	−52.2812	−1.69891	−0.849455	+	0.527662i	$0.823069\pi$
$948$	0	0
$949$	22.6467	0.735143
$950$	0	0
$951$	−41.2248	−1.33681
$952$	0	0
$953$	42.6467	1.38146	0.690731	−	0.723112i	$-0.257289\pi$
$953$	42.6467	1.38146	0.690731	+	0.723112i	$0.257289\pi$
$954$	0	0
$955$	3.89743	0.126118
$956$	0	0
$957$	−8.35482	−0.270073
$958$	0	0
$959$	12.9330	0.417628
$960$	0	0
$961$	79.9278	2.57832
$962$	0	0
$963$	2.58072	0.0831626
$964$	0	0
$965$	0.714768	0.0230092
$966$	0	0
$967$	17.9081	0.575887	0.287944	−	0.957647i	$-0.407028\pi$
$967$	17.9081	0.575887	0.287944	+	0.957647i	$0.407028\pi$
$968$	0	0
$969$	−68.9675	−2.21555
$970$	0	0
$971$	−11.2759	−0.361861	−0.180931	−	0.983496i	$-0.557911\pi$
$971$	−11.2759	−0.361861	−0.180931	+	0.983496i	$0.557911\pi$
$972$	0	0
$973$	7.31927	0.234645
$974$	0	0
$975$	−67.0482	−2.14726
$976$	0	0
$977$	43.0537	1.37741	0.688706	−	0.725041i	$-0.258179\pi$
$977$	43.0537	1.37741	0.688706	+	0.725041i	$0.258179\pi$
$978$	0	0
$979$	−15.9685	−0.510356
$980$	0	0
$981$	26.2289	0.837425
$982$	0	0
$983$	−23.2759	−0.742386	−0.371193	−	0.928556i	$-0.621051\pi$
$983$	−23.2759	−0.742386	−0.371193	+	0.928556i	$0.621051\pi$
$984$	0	0
$985$	4.44668	0.141683
$986$	0	0
$987$	−7.48223	−0.238162
$988$	0	0
$989$	−19.6741	−0.625600
$990$	0	0
$991$	44.8015	1.42317	0.711583	−	0.702602i	$-0.247978\pi$
$991$	44.8015	1.42317	0.711583	+	0.702602i	$0.247978\pi$
$992$	0	0
$993$	−1.05887	−0.0336024
$994$	0	0
$995$	6.10514	0.193546
$996$	0	0
$997$	−20.8474	−0.660245	−0.330122	−	0.943938i	$-0.607090\pi$
$997$	−20.8474	−0.660245	−0.330122	+	0.943938i	$0.607090\pi$
$998$	0	0
$999$	−7.51370	−0.237723

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	308.2.a.c.1.3	✓	3
3.2	odd	2		2772.2.a.s.1.2		3
4.3	odd	2		1232.2.a.r.1.1		3
5.2	odd	4		7700.2.e.p.1849.2		6
5.3	odd	4		7700.2.e.p.1849.5		6
5.4	even	2		7700.2.a.y.1.1		3
7.2	even	3		2156.2.i.m.1145.1		6
7.3	odd	6		2156.2.i.k.177.3		6
7.4	even	3		2156.2.i.m.177.1		6
7.5	odd	6		2156.2.i.k.1145.3		6
7.6	odd	2		2156.2.a.j.1.1		3
8.3	odd	2		4928.2.a.bx.1.3		3
8.5	even	2		4928.2.a.ca.1.1		3
11.10	odd	2		3388.2.a.o.1.3		3
28.27	even	2		8624.2.a.cj.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.a.c.1.3	✓	3	1.1	even	1	trivial
1232.2.a.r.1.1		3	4.3	odd	2
2156.2.a.j.1.1		3	7.6	odd	2
2156.2.i.k.177.3		6	7.3	odd	6
2156.2.i.k.1145.3		6	7.5	odd	6
2156.2.i.m.177.1		6	7.4	even	3
2156.2.i.m.1145.1		6	7.2	even	3
2772.2.a.s.1.2		3	3.2	odd	2
3388.2.a.o.1.3		3	11.10	odd	2
4928.2.a.bx.1.3		3	8.3	odd	2
4928.2.a.ca.1.1		3	8.5	even	2
7700.2.a.y.1.1		3	5.4	even	2
7700.2.e.p.1849.2		6	5.2	odd	4
7700.2.e.p.1849.5		6	5.3	odd	4
8624.2.a.cj.1.3		3	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	308.a (trivial)

\(f(q)\)	\(=\)	\(q+2.17741 q^{3} -0.741113 q^{5} +1.00000 q^{7} +1.74111 q^{9} +O(q^{10})\)	\(q+2.17741 q^{3} -0.741113 q^{5} +1.00000 q^{7} +1.74111 q^{9} +1.00000 q^{11} +6.91852 q^{13} -1.61371 q^{15} -7.27334 q^{17} +4.35482 q^{19} +2.17741 q^{21} +3.09593 q^{23} -4.45075 q^{25} -2.74111 q^{27} -3.83705 q^{29} -10.5322 q^{31} +2.17741 q^{33} -0.741113 q^{35} +2.74111 q^{37} +15.0645 q^{39} +7.27334 q^{41} -6.35482 q^{43} -1.29036 q^{45} -3.43630 q^{47} +1.00000 q^{49} -15.8370 q^{51} +1.48223 q^{53} -0.741113 q^{55} +9.48223 q^{57} -10.1774 q^{59} +8.40075 q^{61} +1.74111 q^{63} -5.12741 q^{65} -7.09593 q^{67} +6.74111 q^{69} +4.57816 q^{71} +3.27334 q^{73} -9.69111 q^{75} +1.00000 q^{77} +8.87259 q^{79} -11.1919 q^{81} -14.5467 q^{83} +5.39037 q^{85} -8.35482 q^{87} -15.9685 q^{89} +6.91852 q^{91} -22.9330 q^{93} -3.22741 q^{95} +0.386294 q^{97} +1.74111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - q^{5} + 3 q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q - q^3 - q^5 + 3 * q^7 + 4 * q^9	\( 3 q - q^{3} - q^{5} + 3 q^{7} + 4 q^{9} + 3 q^{11} + 12 q^{13} + 9 q^{15} + 2 q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 18 q^{25} - 7 q^{27} + 6 q^{29} - 9 q^{31} - q^{33} - q^{35} + 7 q^{37} - 2 q^{41} - 4 q^{43} - 34 q^{45} - 4 q^{47} + 3 q^{49} - 30 q^{51} + 2 q^{53} - q^{55} + 26 q^{57} - 23 q^{59} + 14 q^{61} + 4 q^{63} - 28 q^{65} - 5 q^{67} + 19 q^{69} - 5 q^{71} - 14 q^{73} - 48 q^{75} + 3 q^{77} + 14 q^{79} - q^{81} + 4 q^{83} + 6 q^{85} - 10 q^{87} - 19 q^{89} + 12 q^{91} - 35 q^{93} + 18 q^{95} + 15 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q - q^3 - q^5 + 3 * q^7 + 4 * q^9 + 3 * q^11 + 12 * q^13 + 9 * q^15 + 2 * q^17 - 2 * q^19 - q^21 - 7 * q^23 + 18 * q^25 - 7 * q^27 + 6 * q^29 - 9 * q^31 - q^33 - q^35 + 7 * q^37 - 2 * q^41 - 4 * q^43 - 34 * q^45 - 4 * q^47 + 3 * q^49 - 30 * q^51 + 2 * q^53 - q^55 + 26 * q^57 - 23 * q^59 + 14 * q^61 + 4 * q^63 - 28 * q^65 - 5 * q^67 + 19 * q^69 - 5 * q^71 - 14 * q^73 - 48 * q^75 + 3 * q^77 + 14 * q^79 - q^81 + 4 * q^83 + 6 * q^85 - 10 * q^87 - 19 * q^89 + 12 * q^91 - 35 * q^93 + 18 * q^95 + 15 * q^97 + 4 * q^99

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