LMFDB - Embedded newform 4304.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4304,2,Mod(1,4304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4304, 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("4304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4304 = 2^{4} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4304.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:	$34.3676130300$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 $ x^7 - x^6 - 15x^5 + 16x^4 + 49x^3 - 53x^2 - 44*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.19866$ of defining polynomial
Character	$\chi$	$=$	4304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.12827 q^{3} +3.84433 q^{5} +4.95403 q^{7} +1.52953 q^{9} +0.453515 q^{11} -0.0945931 q^{13} -8.18177 q^{15} +1.41349 q^{17} +2.49192 q^{19} -10.5435 q^{21} +2.25654 q^{23} +9.77888 q^{25} +3.12955 q^{27} -6.64864 q^{29} -5.36736 q^{31} -0.965201 q^{33} +19.0449 q^{35} +7.31117 q^{37} +0.201320 q^{39} -10.6653 q^{41} +12.2765 q^{43} +5.88003 q^{45} +1.43212 q^{47} +17.5424 q^{49} -3.00829 q^{51} +9.96897 q^{53} +1.74346 q^{55} -5.30347 q^{57} -7.89955 q^{59} +0.950317 q^{61} +7.57735 q^{63} -0.363647 q^{65} +8.06276 q^{67} -4.80253 q^{69} +3.06085 q^{71} +12.8264 q^{73} -20.8121 q^{75} +2.24672 q^{77} -13.7673 q^{79} -11.2491 q^{81} -10.1171 q^{83} +5.43392 q^{85} +14.1501 q^{87} -1.77803 q^{89} -0.468616 q^{91} +11.4232 q^{93} +9.57975 q^{95} -12.9137 q^{97} +0.693665 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{3} + 7 q^{5} - 6 q^{7} + 12 q^{9} + 3 q^{11} - 9 q^{13} - 8 q^{15} + 8 q^{17} + 11 q^{19} - 6 q^{21} - 12 q^{23} + 22 q^{25} + 14 q^{27} - 5 q^{29} - 14 q^{31} - 4 q^{33} + 4 q^{35} + 13 q^{37}+ \cdots + 37 q^{99}+O(q^{100}) $ 7 * q - q^3 + 7 * q^5 - 6 * q^7 + 12 * q^9 + 3 * q^11 - 9 * q^13 - 8 * q^15 + 8 * q^17 + 11 * q^19 - 6 * q^21 - 12 * q^23 + 22 * q^25 + 14 * q^27 - 5 * q^29 - 14 * q^31 - 4 * q^33 + 4 * q^35 + 13 * q^37 + 18 * q^39 + 12 * q^41 + 11 * q^43 + 3 * q^45 - 2 * q^47 + 15 * q^49 + 26 * q^51 + 19 * q^53 + 40 * q^55 - 12 * q^57 + 9 * q^59 - 3 * q^61 + 26 * q^63 - 10 * q^65 + 33 * q^67 - 64 * q^69 - 28 * q^71 - 14 * q^73 + 45 * q^75 + 10 * q^77 - 2 * q^79 + 15 * q^81 + 7 * q^83 - 16 * q^85 - 16 * q^87 + 18 * q^89 + 26 * q^91 + 6 * q^93 + 34 * q^95 - 4 * q^97 + 37 * q^99

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
$3$	−2.12827	−1.22876	−0.614379	−	0.789011i	$-0.710593\pi$
$3$	−2.12827	−1.22876	−0.614379	+	0.789011i	$0.710593\pi$
$4$	0	0
$5$	3.84433	1.71924	0.859619	−	0.510936i	$-0.170701\pi$
$5$	3.84433	1.71924	0.859619	+	0.510936i	$0.170701\pi$
$6$	0	0
$7$	4.95403	1.87245	0.936223	−	0.351407i	$-0.114297\pi$
$7$	4.95403	1.87245	0.936223	+	0.351407i	$0.114297\pi$
$8$	0	0
$9$	1.52953	0.509844
$10$	0	0
$11$	0.453515	0.136740	0.0683699	−	0.997660i	$-0.478220\pi$
$11$	0.453515	0.136740	0.0683699	+	0.997660i	$0.478220\pi$
$12$	0	0
$13$	−0.0945931	−0.0262354	−0.0131177	−	0.999914i	$-0.504176\pi$
$13$	−0.0945931	−0.0262354	−0.0131177	+	0.999914i	$0.504176\pi$
$14$	0	0
$15$	−8.18177	−2.11253
$16$	0	0
$17$	1.41349	0.342822	0.171411	−	0.985200i	$-0.445167\pi$
$17$	1.41349	0.342822	0.171411	+	0.985200i	$0.445167\pi$
$18$	0	0
$19$	2.49192	0.571685	0.285842	−	0.958277i	$-0.407727\pi$
$19$	2.49192	0.571685	0.285842	+	0.958277i	$0.407727\pi$
$20$	0	0
$21$	−10.5435	−2.30078
$22$	0	0
$23$	2.25654	0.470521	0.235261	−	0.971932i	$-0.424406\pi$
$23$	2.25654	0.470521	0.235261	+	0.971932i	$0.424406\pi$
$24$	0	0
$25$	9.77888	1.95578
$26$	0	0
$27$	3.12955	0.602282
$28$	0	0
$29$	−6.64864	−1.23462	−0.617311	−	0.786720i	$-0.711778\pi$
$29$	−6.64864	−1.23462	−0.617311	+	0.786720i	$0.711778\pi$
$30$	0	0
$31$	−5.36736	−0.964006	−0.482003	−	0.876170i	$-0.660091\pi$
$31$	−5.36736	−0.964006	−0.482003	+	0.876170i	$0.660091\pi$
$32$	0	0
$33$	−0.965201	−0.168020
$34$	0	0
$35$	19.0449	3.21918
$36$	0	0
$37$	7.31117	1.20195	0.600974	−	0.799268i	$-0.294779\pi$
$37$	7.31117	1.20195	0.600974	+	0.799268i	$0.294779\pi$
$38$	0	0
$39$	0.201320	0.0322369
$40$	0	0
$41$	−10.6653	−1.66563	−0.832817	−	0.553548i	$-0.813274\pi$
$41$	−10.6653	−1.66563	−0.832817	+	0.553548i	$0.813274\pi$
$42$	0	0
$43$	12.2765	1.87215	0.936076	−	0.351798i	$-0.114430\pi$
$43$	12.2765	1.87215	0.936076	+	0.351798i	$0.114430\pi$
$44$	0	0
$45$	5.88003	0.876543
$46$	0	0
$47$	1.43212	0.208897	0.104448	−	0.994530i	$-0.466692\pi$
$47$	1.43212	0.208897	0.104448	+	0.994530i	$0.466692\pi$
$48$	0	0
$49$	17.5424	2.50605
$50$	0	0
$51$	−3.00829	−0.421245
$52$	0	0
$53$	9.96897	1.36934	0.684672	−	0.728852i	$-0.259946\pi$
$53$	9.96897	1.36934	0.684672	+	0.728852i	$0.259946\pi$
$54$	0	0
$55$	1.74346	0.235088
$56$	0	0
$57$	−5.30347	−0.702462
$58$	0	0
$59$	−7.89955	−1.02843	−0.514217	−	0.857660i	$-0.671917\pi$
$59$	−7.89955	−1.02843	−0.514217	+	0.857660i	$0.671917\pi$
$60$	0	0
$61$	0.950317	0.121676	0.0608378	−	0.998148i	$-0.480623\pi$
$61$	0.950317	0.121676	0.0608378	+	0.998148i	$0.480623\pi$
$62$	0	0
$63$	7.57735	0.954656
$64$	0	0
$65$	−0.363647	−0.0451049
$66$	0	0
$67$	8.06276	0.985023	0.492512	−	0.870306i	$-0.336079\pi$
$67$	8.06276	0.985023	0.492512	+	0.870306i	$0.336079\pi$
$68$	0	0
$69$	−4.80253	−0.578156
$70$	0	0
$71$	3.06085	0.363256	0.181628	−	0.983367i	$-0.441863\pi$
$71$	3.06085	0.363256	0.181628	+	0.983367i	$0.441863\pi$
$72$	0	0
$73$	12.8264	1.50121	0.750607	−	0.660749i	$-0.229761\pi$
$73$	12.8264	1.50121	0.750607	+	0.660749i	$0.229761\pi$
$74$	0	0
$75$	−20.8121	−2.40317
$76$	0	0
$77$	2.24672	0.256038
$78$	0	0
$79$	−13.7673	−1.54894	−0.774469	−	0.632611i	$-0.781983\pi$
$79$	−13.7673	−1.54894	−0.774469	+	0.632611i	$0.781983\pi$
$80$	0	0
$81$	−11.2491	−1.24990
$82$	0	0
$83$	−10.1171	−1.11049	−0.555246	−	0.831686i	$-0.687376\pi$
$83$	−10.1171	−1.11049	−0.555246	+	0.831686i	$0.687376\pi$
$84$	0	0
$85$	5.43392	0.589392
$86$	0	0
$87$	14.1501	1.51705
$88$	0	0
$89$	−1.77803	−0.188471	−0.0942354	−	0.995550i	$-0.530041\pi$
$89$	−1.77803	−0.188471	−0.0942354	+	0.995550i	$0.530041\pi$
$90$	0	0
$91$	−0.468616	−0.0491244
$92$	0	0
$93$	11.4232	1.18453
$94$	0	0
$95$	9.57975	0.982862
$96$	0	0
$97$	−12.9137	−1.31119	−0.655593	−	0.755115i	$-0.727581\pi$
$97$	−12.9137	−1.31119	−0.655593	+	0.755115i	$0.727581\pi$
$98$	0	0
$99$	0.693665	0.0697160
$100$	0	0
$101$	−6.03448	−0.600453	−0.300226	−	0.953868i	$-0.597062\pi$
$101$	−6.03448	−0.600453	−0.300226	+	0.953868i	$0.597062\pi$
$102$	0	0
$103$	1.09297	0.107694	0.0538468	−	0.998549i	$-0.482852\pi$
$103$	1.09297	0.107694	0.0538468	+	0.998549i	$0.482852\pi$
$104$	0	0
$105$	−40.5327	−3.95559
$106$	0	0
$107$	−5.58489	−0.539911	−0.269956	−	0.962873i	$-0.587009\pi$
$107$	−5.58489	−0.539911	−0.269956	+	0.962873i	$0.587009\pi$
$108$	0	0
$109$	−6.76759	−0.648218	−0.324109	−	0.946020i	$-0.605064\pi$
$109$	−6.76759	−0.648218	−0.324109	+	0.946020i	$0.605064\pi$
$110$	0	0
$111$	−15.5601	−1.47690
$112$	0	0
$113$	−0.653864	−0.0615103	−0.0307552	−	0.999527i	$-0.509791\pi$
$113$	−0.653864	−0.0615103	−0.0307552	+	0.999527i	$0.509791\pi$
$114$	0	0
$115$	8.67489	0.808937
$116$	0	0
$117$	−0.144683	−0.0133760
$118$	0	0
$119$	7.00247	0.641915
$120$	0	0
$121$	−10.7943	−0.981302
$122$	0	0
$123$	22.6986	2.04666
$124$	0	0
$125$	18.3716	1.64321
$126$	0	0
$127$	−8.29619	−0.736168	−0.368084	−	0.929792i	$-0.619986\pi$
$127$	−8.29619	−0.736168	−0.368084	+	0.929792i	$0.619986\pi$
$128$	0	0
$129$	−26.1277	−2.30042
$130$	0	0
$131$	15.8736	1.38688	0.693440	−	0.720514i	$-0.256094\pi$
$131$	15.8736	1.38688	0.693440	+	0.720514i	$0.256094\pi$
$132$	0	0
$133$	12.3450	1.07045
$134$	0	0
$135$	12.0310	1.03547
$136$	0	0
$137$	−5.16032	−0.440876	−0.220438	−	0.975401i	$-0.570749\pi$
$137$	−5.16032	−0.440876	−0.220438	+	0.975401i	$0.570749\pi$
$138$	0	0
$139$	−15.2369	−1.29237	−0.646187	−	0.763179i	$-0.723637\pi$
$139$	−15.2369	−1.29237	−0.646187	+	0.763179i	$0.723637\pi$
$140$	0	0
$141$	−3.04794	−0.256683
$142$	0	0
$143$	−0.0428993	−0.00358742
$144$	0	0
$145$	−25.5596	−2.12261
$146$	0	0
$147$	−37.3349	−3.07933
$148$	0	0
$149$	16.0979	1.31879	0.659397	−	0.751795i	$-0.270812\pi$
$149$	16.0979	1.31879	0.659397	+	0.751795i	$0.270812\pi$
$150$	0	0
$151$	7.57215	0.616212	0.308106	−	0.951352i	$-0.400305\pi$
$151$	7.57215	0.616212	0.308106	+	0.951352i	$0.400305\pi$
$152$	0	0
$153$	2.16198	0.174786
$154$	0	0
$155$	−20.6339	−1.65735
$156$	0	0
$157$	21.8948	1.74739	0.873696	−	0.486472i	$-0.161717\pi$
$157$	21.8948	1.74739	0.873696	+	0.486472i	$0.161717\pi$
$158$	0	0
$159$	−21.2167	−1.68259
$160$	0	0
$161$	11.1790	0.881025
$162$	0	0
$163$	6.02116	0.471614	0.235807	−	0.971800i	$-0.424227\pi$
$163$	6.02116	0.471614	0.235807	+	0.971800i	$0.424227\pi$
$164$	0	0
$165$	−3.71055	−0.288866
$166$	0	0
$167$	−19.6784	−1.52276	−0.761382	−	0.648304i	$-0.775479\pi$
$167$	−19.6784	−1.52276	−0.761382	+	0.648304i	$0.775479\pi$
$168$	0	0
$169$	−12.9911	−0.999312
$170$	0	0
$171$	3.81147	0.291470
$172$	0	0
$173$	12.4812	0.948931	0.474466	−	0.880274i	$-0.342641\pi$
$173$	12.4812	0.948931	0.474466	+	0.880274i	$0.342641\pi$
$174$	0	0
$175$	48.4448	3.66209
$176$	0	0
$177$	16.8124	1.26370
$178$	0	0
$179$	−7.84995	−0.586733	−0.293366	−	0.956000i	$-0.594776\pi$
$179$	−7.84995	−0.586733	−0.293366	+	0.956000i	$0.594776\pi$
$180$	0	0
$181$	−17.9797	−1.33642	−0.668209	−	0.743973i	$-0.732939\pi$
$181$	−17.9797	−1.33642	−0.668209	+	0.743973i	$0.732939\pi$
$182$	0	0
$183$	−2.02253	−0.149510
$184$	0	0
$185$	28.1065	2.06643
$186$	0	0
$187$	0.641038	0.0468774
$188$	0	0
$189$	15.5039	1.12774
$190$	0	0
$191$	12.3054	0.890386	0.445193	−	0.895435i	$-0.353135\pi$
$191$	12.3054	0.890386	0.445193	+	0.895435i	$0.353135\pi$
$192$	0	0
$193$	5.82207	0.419081	0.209541	−	0.977800i	$-0.432803\pi$
$193$	5.82207	0.419081	0.209541	+	0.977800i	$0.432803\pi$
$194$	0	0
$195$	0.773939	0.0554229
$196$	0	0
$197$	−21.6619	−1.54334	−0.771671	−	0.636021i	$-0.780579\pi$
$197$	−21.6619	−1.54334	−0.771671	+	0.636021i	$0.780579\pi$
$198$	0	0
$199$	−5.71843	−0.405369	−0.202684	−	0.979244i	$-0.564967\pi$
$199$	−5.71843	−0.405369	−0.202684	+	0.979244i	$0.564967\pi$
$200$	0	0
$201$	−17.1597	−1.21035
$202$	0	0
$203$	−32.9375	−2.31176
$204$	0	0
$205$	−41.0008	−2.86362
$206$	0	0
$207$	3.45145	0.239892
$208$	0	0
$209$	1.13012	0.0781721
$210$	0	0
$211$	20.0839	1.38264	0.691318	−	0.722551i	$-0.257030\pi$
$211$	20.0839	1.38264	0.691318	+	0.722551i	$0.257030\pi$
$212$	0	0
$213$	−6.51431	−0.446353
$214$	0	0
$215$	47.1950	3.21867
$216$	0	0
$217$	−26.5900	−1.80505
$218$	0	0
$219$	−27.2980	−1.84463
$220$	0	0
$221$	−0.133706	−0.00899406
$222$	0	0
$223$	5.45166	0.365070	0.182535	−	0.983199i	$-0.441570\pi$
$223$	5.45166	0.365070	0.182535	+	0.983199i	$0.441570\pi$
$224$	0	0
$225$	14.9571	0.997141
$226$	0	0
$227$	−14.8158	−0.983360	−0.491680	−	0.870776i	$-0.663617\pi$
$227$	−14.8158	−0.983360	−0.491680	+	0.870776i	$0.663617\pi$
$228$	0	0
$229$	21.9863	1.45289	0.726447	−	0.687223i	$-0.241170\pi$
$229$	21.9863	1.45289	0.726447	+	0.687223i	$0.241170\pi$
$230$	0	0
$231$	−4.78163	−0.314608
$232$	0	0
$233$	17.7722	1.16429	0.582147	−	0.813084i	$-0.302213\pi$
$233$	17.7722	1.16429	0.582147	+	0.813084i	$0.302213\pi$
$234$	0	0
$235$	5.50555	0.359143
$236$	0	0
$237$	29.3005	1.90327
$238$	0	0
$239$	12.0820	0.781517	0.390759	−	0.920493i	$-0.372213\pi$
$239$	12.0820	0.781517	0.390759	+	0.920493i	$0.372213\pi$
$240$	0	0
$241$	11.3215	0.729285	0.364642	−	0.931148i	$-0.381191\pi$
$241$	11.3215	0.729285	0.364642	+	0.931148i	$0.381191\pi$
$242$	0	0
$243$	14.5525	0.933545
$244$	0	0
$245$	67.4387	4.30850
$246$	0	0
$247$	−0.235718	−0.0149984
$248$	0	0
$249$	21.5318	1.36453
$250$	0	0
$251$	−8.60802	−0.543333	−0.271667	−	0.962391i	$-0.587575\pi$
$251$	−8.60802	−0.543333	−0.271667	+	0.962391i	$0.587575\pi$
$252$	0	0
$253$	1.02337	0.0643390
$254$	0	0
$255$	−11.5649	−0.724219
$256$	0	0
$257$	−14.3976	−0.898096	−0.449048	−	0.893508i	$-0.648237\pi$
$257$	−14.3976	−0.898096	−0.449048	+	0.893508i	$0.648237\pi$
$258$	0	0
$259$	36.2197	2.25058
$260$	0	0
$261$	−10.1693	−0.629464
$262$	0	0
$263$	22.3410	1.37761	0.688803	−	0.724949i	$-0.258137\pi$
$263$	22.3410	1.37761	0.688803	+	0.724949i	$0.258137\pi$
$264$	0	0
$265$	38.3240	2.35423
$266$	0	0
$267$	3.78413	0.231585
$268$	0	0
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	−16.7534	−1.01770	−0.508848	−	0.860856i	$-0.669929\pi$
$271$	−16.7534	−1.01770	−0.508848	+	0.860856i	$0.669929\pi$
$272$	0	0
$273$	0.997342	0.0603619
$274$	0	0
$275$	4.43487	0.267432
$276$	0	0
$277$	14.7850	0.888345	0.444172	−	0.895941i	$-0.353498\pi$
$277$	14.7850	0.888345	0.444172	+	0.895941i	$0.353498\pi$
$278$	0	0
$279$	−8.20955	−0.491493
$280$	0	0
$281$	25.0439	1.49400	0.746998	−	0.664827i	$-0.231495\pi$
$281$	25.0439	1.49400	0.746998	+	0.664827i	$0.231495\pi$
$282$	0	0
$283$	−24.6030	−1.46249	−0.731247	−	0.682113i	$-0.761061\pi$
$283$	−24.6030	−1.46249	−0.731247	+	0.682113i	$0.761061\pi$
$284$	0	0
$285$	−20.3883	−1.20770
$286$	0	0
$287$	−52.8360	−3.11881
$288$	0	0
$289$	−15.0020	−0.882473
$290$	0	0
$291$	27.4838	1.61113
$292$	0	0
$293$	−11.4618	−0.669607	−0.334804	−	0.942288i	$-0.608670\pi$
$293$	−11.4618	−0.669607	−0.334804	+	0.942288i	$0.608670\pi$
$294$	0	0
$295$	−30.3685	−1.76812
$296$	0	0
$297$	1.41930	0.0823560
$298$	0	0
$299$	−0.213453	−0.0123443
$300$	0	0
$301$	60.8182	3.50550
$302$	0	0
$303$	12.8430	0.737811
$304$	0	0
$305$	3.65333	0.209189
$306$	0	0
$307$	17.7679	1.01407	0.507033	−	0.861927i	$-0.330742\pi$
$307$	17.7679	1.01407	0.507033	+	0.861927i	$0.330742\pi$
$308$	0	0
$309$	−2.32614	−0.132329
$310$	0	0
$311$	−7.62146	−0.432173	−0.216087	−	0.976374i	$-0.569329\pi$
$311$	−7.62146	−0.432173	−0.216087	+	0.976374i	$0.569329\pi$
$312$	0	0
$313$	7.97031	0.450509	0.225254	−	0.974300i	$-0.427679\pi$
$313$	7.97031	0.450509	0.225254	+	0.974300i	$0.427679\pi$
$314$	0	0
$315$	29.1298	1.64128
$316$	0	0
$317$	1.71640	0.0964028	0.0482014	−	0.998838i	$-0.484651\pi$
$317$	1.71640	0.0964028	0.0482014	+	0.998838i	$0.484651\pi$
$318$	0	0
$319$	−3.01525	−0.168822
$320$	0	0
$321$	11.8861	0.663420
$322$	0	0
$323$	3.52230	0.195986
$324$	0	0
$325$	−0.925014	−0.0513106
$326$	0	0
$327$	14.4033	0.796502
$328$	0	0
$329$	7.09477	0.391147
$330$	0	0
$331$	13.7537	0.755970	0.377985	−	0.925812i	$-0.376617\pi$
$331$	13.7537	0.755970	0.377985	+	0.925812i	$0.376617\pi$
$332$	0	0
$333$	11.1827	0.612806
$334$	0	0
$335$	30.9959	1.69349
$336$	0	0
$337$	18.7032	1.01883	0.509413	−	0.860522i	$-0.329863\pi$
$337$	18.7032	1.01883	0.509413	+	0.860522i	$0.329863\pi$
$338$	0	0
$339$	1.39160	0.0755812
$340$	0	0
$341$	−2.43417	−0.131818
$342$	0	0
$343$	52.2272	2.82000
$344$	0	0
$345$	−18.4625	−0.993988
$346$	0	0
$347$	−12.9029	−0.692662	−0.346331	−	0.938112i	$-0.612573\pi$
$347$	−12.9029	−0.692662	−0.346331	+	0.938112i	$0.612573\pi$
$348$	0	0
$349$	−4.29547	−0.229931	−0.114966	−	0.993369i	$-0.536676\pi$
$349$	−4.29547	−0.229931	−0.114966	+	0.993369i	$0.536676\pi$
$350$	0	0
$351$	−0.296034	−0.0158011
$352$	0	0
$353$	−22.3707	−1.19067	−0.595337	−	0.803476i	$-0.702982\pi$
$353$	−22.3707	−1.19067	−0.595337	+	0.803476i	$0.702982\pi$
$354$	0	0
$355$	11.7669	0.624523
$356$	0	0
$357$	−14.9031	−0.788758
$358$	0	0
$359$	−32.0628	−1.69221	−0.846105	−	0.533016i	$-0.821058\pi$
$359$	−32.0628	−1.69221	−0.846105	+	0.533016i	$0.821058\pi$
$360$	0	0
$361$	−12.7903	−0.673176
$362$	0	0
$363$	22.9732	1.20578
$364$	0	0
$365$	49.3088	2.58094
$366$	0	0
$367$	4.79576	0.250337	0.125168	−	0.992136i	$-0.460053\pi$
$367$	4.79576	0.250337	0.125168	+	0.992136i	$0.460053\pi$
$368$	0	0
$369$	−16.3129	−0.849214
$370$	0	0
$371$	49.3865	2.56402
$372$	0	0
$373$	32.9108	1.70406	0.852028	−	0.523497i	$-0.175373\pi$
$373$	32.9108	1.70406	0.852028	+	0.523497i	$0.175373\pi$
$374$	0	0
$375$	−39.0997	−2.01910
$376$	0	0
$377$	0.628915	0.0323908
$378$	0	0
$379$	8.98570	0.461565	0.230782	−	0.973005i	$-0.425871\pi$
$379$	8.98570	0.461565	0.230782	+	0.973005i	$0.425871\pi$
$380$	0	0
$381$	17.6565	0.904572
$382$	0	0
$383$	−0.787688	−0.0402490	−0.0201245	−	0.999797i	$-0.506406\pi$
$383$	−0.787688	−0.0402490	−0.0201245	+	0.999797i	$0.506406\pi$
$384$	0	0
$385$	8.63715	0.440190
$386$	0	0
$387$	18.7773	0.954506
$388$	0	0
$389$	−4.15609	−0.210722	−0.105361	−	0.994434i	$-0.533600\pi$
$389$	−4.15609	−0.210722	−0.105361	+	0.994434i	$0.533600\pi$
$390$	0	0
$391$	3.18960	0.161305
$392$	0	0
$393$	−33.7833	−1.70414
$394$	0	0
$395$	−52.9259	−2.66299
$396$	0	0
$397$	1.90587	0.0956527	0.0478263	−	0.998856i	$-0.484771\pi$
$397$	1.90587	0.0956527	0.0478263	+	0.998856i	$0.484771\pi$
$398$	0	0
$399$	−26.2735	−1.31532
$400$	0	0
$401$	−11.0163	−0.550128	−0.275064	−	0.961426i	$-0.588699\pi$
$401$	−11.0163	−0.550128	−0.275064	+	0.961426i	$0.588699\pi$
$402$	0	0
$403$	0.507715	0.0252911
$404$	0	0
$405$	−43.2454	−2.14888
$406$	0	0
$407$	3.31572	0.164354
$408$	0	0
$409$	−5.30078	−0.262107	−0.131053	−	0.991375i	$-0.541836\pi$
$409$	−5.30078	−0.262107	−0.131053	+	0.991375i	$0.541836\pi$
$410$	0	0
$411$	10.9826	0.541730
$412$	0	0
$413$	−39.1346	−1.92569
$414$	0	0
$415$	−38.8934	−1.90920
$416$	0	0
$417$	32.4281	1.58801
$418$	0	0
$419$	25.5590	1.24864	0.624319	−	0.781169i	$-0.285376\pi$
$419$	25.5590	1.24864	0.624319	+	0.781169i	$0.285376\pi$
$420$	0	0
$421$	−24.8795	−1.21255	−0.606276	−	0.795254i	$-0.707337\pi$
$421$	−24.8795	−1.21255	−0.606276	+	0.795254i	$0.707337\pi$
$422$	0	0
$423$	2.19048	0.106505
$424$	0	0
$425$	13.8224	0.670483
$426$	0	0
$427$	4.70789	0.227831
$428$	0	0
$429$	0.0913014	0.00440807
$430$	0	0
$431$	−19.4524	−0.936988	−0.468494	−	0.883467i	$-0.655203\pi$
$431$	−19.4524	−0.936988	−0.468494	+	0.883467i	$0.655203\pi$
$432$	0	0
$433$	−16.7217	−0.803593	−0.401797	−	0.915729i	$-0.631614\pi$
$433$	−16.7217	−0.803593	−0.401797	+	0.915729i	$0.631614\pi$
$434$	0	0
$435$	54.3977	2.60817
$436$	0	0
$437$	5.62311	0.268990
$438$	0	0
$439$	14.8215	0.707390	0.353695	−	0.935361i	$-0.384925\pi$
$439$	14.8215	0.707390	0.353695	+	0.935361i	$0.384925\pi$
$440$	0	0
$441$	26.8316	1.27770
$442$	0	0
$443$	24.8635	1.18130	0.590651	−	0.806927i	$-0.298871\pi$
$443$	24.8635	1.18130	0.590651	+	0.806927i	$0.298871\pi$
$444$	0	0
$445$	−6.83533	−0.324026
$446$	0	0
$447$	−34.2607	−1.62048
$448$	0	0
$449$	22.0294	1.03963	0.519817	−	0.854278i	$-0.326000\pi$
$449$	22.0294	1.03963	0.519817	+	0.854278i	$0.326000\pi$
$450$	0	0
$451$	−4.83685	−0.227759
$452$	0	0
$453$	−16.1156	−0.757175
$454$	0	0
$455$	−1.80152	−0.0844564
$456$	0	0
$457$	−27.1457	−1.26982	−0.634911	−	0.772585i	$-0.718963\pi$
$457$	−27.1457	−1.26982	−0.634911	+	0.772585i	$0.718963\pi$
$458$	0	0
$459$	4.42359	0.206475
$460$	0	0
$461$	27.6585	1.28819	0.644093	−	0.764947i	$-0.277235\pi$
$461$	27.6585	1.28819	0.644093	+	0.764947i	$0.277235\pi$
$462$	0	0
$463$	−31.2336	−1.45155	−0.725774	−	0.687933i	$-0.758518\pi$
$463$	−31.2336	−1.45155	−0.725774	+	0.687933i	$0.758518\pi$
$464$	0	0
$465$	43.9145	2.03649
$466$	0	0
$467$	13.4254	0.621255	0.310628	−	0.950532i	$-0.399461\pi$
$467$	13.4254	0.621255	0.310628	+	0.950532i	$0.399461\pi$
$468$	0	0
$469$	39.9431	1.84440
$470$	0	0
$471$	−46.5980	−2.14712
$472$	0	0
$473$	5.56758	0.255998
$474$	0	0
$475$	24.3682	1.11809
$476$	0	0
$477$	15.2479	0.698152
$478$	0	0
$479$	−2.09566	−0.0957533	−0.0478767	−	0.998853i	$-0.515245\pi$
$479$	−2.09566	−0.0957533	−0.0478767	+	0.998853i	$0.515245\pi$
$480$	0	0
$481$	−0.691586	−0.0315336
$482$	0	0
$483$	−23.7918	−1.08257
$484$	0	0
$485$	−49.6445	−2.25424
$486$	0	0
$487$	38.1661	1.72947	0.864736	−	0.502226i	$-0.167485\pi$
$487$	38.1661	1.72947	0.864736	+	0.502226i	$0.167485\pi$
$488$	0	0
$489$	−12.8147	−0.579499
$490$	0	0
$491$	−6.36732	−0.287353	−0.143676	−	0.989625i	$-0.545892\pi$
$491$	−6.36732	−0.287353	−0.143676	+	0.989625i	$0.545892\pi$
$492$	0	0
$493$	−9.39778	−0.423255
$494$	0	0
$495$	2.66668	0.119858
$496$	0	0
$497$	15.1635	0.680176
$498$	0	0
$499$	−33.6939	−1.50835	−0.754173	−	0.656676i	$-0.771962\pi$
$499$	−33.6939	−1.50835	−0.754173	+	0.656676i	$0.771962\pi$
$500$	0	0
$501$	41.8810	1.87111
$502$	0	0
$503$	29.6505	1.32205	0.661026	−	0.750363i	$-0.270121\pi$
$503$	29.6505	1.32205	0.661026	+	0.750363i	$0.270121\pi$
$504$	0	0
$505$	−23.1985	−1.03232
$506$	0	0
$507$	27.6485	1.22791
$508$	0	0
$509$	−3.41765	−0.151485	−0.0757423	−	0.997127i	$-0.524133\pi$
$509$	−3.41765	−0.151485	−0.0757423	+	0.997127i	$0.524133\pi$
$510$	0	0
$511$	63.5422	2.81094
$512$	0	0
$513$	7.79858	0.344316
$514$	0	0
$515$	4.20174	0.185151
$516$	0	0
$517$	0.649488	0.0285645
$518$	0	0
$519$	−26.5635	−1.16601
$520$	0	0
$521$	28.2730	1.23866	0.619330	−	0.785130i	$-0.287404\pi$
$521$	28.2730	1.23866	0.619330	+	0.785130i	$0.287404\pi$
$522$	0	0
$523$	17.2424	0.753956	0.376978	−	0.926222i	$-0.376963\pi$
$523$	17.2424	0.753956	0.376978	+	0.926222i	$0.376963\pi$
$524$	0	0
$525$	−103.104	−4.49981
$526$	0	0
$527$	−7.58670	−0.330482
$528$	0	0
$529$	−17.9080	−0.778610
$530$	0	0
$531$	−12.0826	−0.524341
$532$	0	0
$533$	1.00886	0.0436986
$534$	0	0
$535$	−21.4702	−0.928236
$536$	0	0
$537$	16.7068	0.720952
$538$	0	0
$539$	7.95572	0.342677
$540$	0	0
$541$	−2.32570	−0.0999896	−0.0499948	−	0.998749i	$-0.515920\pi$
$541$	−2.32570	−0.0999896	−0.0499948	+	0.998749i	$0.515920\pi$
$542$	0	0
$543$	38.2656	1.64213
$544$	0	0
$545$	−26.0169	−1.11444
$546$	0	0
$547$	−26.5955	−1.13714	−0.568570	−	0.822635i	$-0.692503\pi$
$547$	−26.5955	−1.13714	−0.568570	+	0.822635i	$0.692503\pi$
$548$	0	0
$549$	1.45354	0.0620356
$550$	0	0
$551$	−16.5679	−0.705814
$552$	0	0
$553$	−68.2034	−2.90030
$554$	0	0
$555$	−59.8183	−2.53915
$556$	0	0
$557$	19.7222	0.835658	0.417829	−	0.908526i	$-0.362791\pi$
$557$	19.7222	0.835658	0.417829	+	0.908526i	$0.362791\pi$
$558$	0	0
$559$	−1.16127	−0.0491166
$560$	0	0
$561$	−1.36430	−0.0576009
$562$	0	0
$563$	25.9388	1.09319	0.546595	−	0.837397i	$-0.315924\pi$
$563$	25.9388	1.09319	0.546595	+	0.837397i	$0.315924\pi$
$564$	0	0
$565$	−2.51367	−0.105751
$566$	0	0
$567$	−55.7285	−2.34038
$568$	0	0
$569$	−44.9748	−1.88544	−0.942720	−	0.333584i	$-0.891742\pi$
$569$	−44.9748	−1.88544	−0.942720	+	0.333584i	$0.891742\pi$
$570$	0	0
$571$	44.4599	1.86059	0.930294	−	0.366814i	$-0.119552\pi$
$571$	44.4599	1.86059	0.930294	+	0.366814i	$0.119552\pi$
$572$	0	0
$573$	−26.1892	−1.09407
$574$	0	0
$575$	22.0664	0.920234
$576$	0	0
$577$	−36.6191	−1.52447	−0.762236	−	0.647299i	$-0.775898\pi$
$577$	−36.6191	−1.52447	−0.762236	+	0.647299i	$0.775898\pi$
$578$	0	0
$579$	−12.3909	−0.514949
$580$	0	0
$581$	−50.1202	−2.07934
$582$	0	0
$583$	4.52107	0.187244
$584$	0	0
$585$	−0.556210	−0.0229965
$586$	0	0
$587$	−20.5256	−0.847184	−0.423592	−	0.905853i	$-0.639231\pi$
$587$	−20.5256	−0.847184	−0.423592	+	0.905853i	$0.639231\pi$
$588$	0	0
$589$	−13.3750	−0.551108
$590$	0	0
$591$	46.1023	1.89639
$592$	0	0
$593$	−1.99866	−0.0820753	−0.0410377	−	0.999158i	$-0.513066\pi$
$593$	−1.99866	−0.0820753	−0.0410377	+	0.999158i	$0.513066\pi$
$594$	0	0
$595$	26.9198	1.10360
$596$	0	0
$597$	12.1704	0.498100
$598$	0	0
$599$	−21.2473	−0.868141	−0.434070	−	0.900879i	$-0.642923\pi$
$599$	−21.2473	−0.868141	−0.434070	+	0.900879i	$0.642923\pi$
$600$	0	0
$601$	−22.2574	−0.907899	−0.453949	−	0.891028i	$-0.649985\pi$
$601$	−22.2574	−0.907899	−0.453949	+	0.891028i	$0.649985\pi$
$602$	0	0
$603$	12.3323	0.502208
$604$	0	0
$605$	−41.4970	−1.68709
$606$	0	0
$607$	−7.20720	−0.292531	−0.146266	−	0.989245i	$-0.546725\pi$
$607$	−7.20720	−0.292531	−0.146266	+	0.989245i	$0.546725\pi$
$608$	0	0
$609$	70.0999	2.84059
$610$	0	0
$611$	−0.135469	−0.00548048
$612$	0	0
$613$	15.9436	0.643956	0.321978	−	0.946747i	$-0.395652\pi$
$613$	15.9436	0.643956	0.321978	+	0.946747i	$0.395652\pi$
$614$	0	0
$615$	87.2608	3.51870
$616$	0	0
$617$	18.0019	0.724729	0.362364	−	0.932037i	$-0.381970\pi$
$617$	18.0019	0.724729	0.362364	+	0.932037i	$0.381970\pi$
$618$	0	0
$619$	17.9006	0.719487	0.359744	−	0.933051i	$-0.382864\pi$
$619$	17.9006	0.719487	0.359744	+	0.933051i	$0.382864\pi$
$620$	0	0
$621$	7.06196	0.283387
$622$	0	0
$623$	−8.80840	−0.352901
$624$	0	0
$625$	21.7321	0.869285
$626$	0	0
$627$	−2.40520	−0.0960545
$628$	0	0
$629$	10.3343	0.412054
$630$	0	0
$631$	−25.6888	−1.02265	−0.511327	−	0.859386i	$-0.670846\pi$
$631$	−25.6888	−1.02265	−0.511327	+	0.859386i	$0.670846\pi$
$632$	0	0
$633$	−42.7441	−1.69892
$634$	0	0
$635$	−31.8933	−1.26565
$636$	0	0
$637$	−1.65939	−0.0657473
$638$	0	0
$639$	4.68166	0.185204
$640$	0	0
$641$	25.4553	1.00542	0.502712	−	0.864454i	$-0.332336\pi$
$641$	25.4553	1.00542	0.502712	+	0.864454i	$0.332336\pi$
$642$	0	0
$643$	−22.7411	−0.896823	−0.448411	−	0.893827i	$-0.648010\pi$
$643$	−22.7411	−0.896823	−0.448411	+	0.893827i	$0.648010\pi$
$644$	0	0
$645$	−100.444	−3.95497
$646$	0	0
$647$	−28.5975	−1.12428	−0.562142	−	0.827041i	$-0.690023\pi$
$647$	−28.5975	−1.12428	−0.562142	+	0.827041i	$0.690023\pi$
$648$	0	0
$649$	−3.58256	−0.140628
$650$	0	0
$651$	56.5907	2.21797
$652$	0	0
$653$	−5.54791	−0.217107	−0.108553	−	0.994091i	$-0.534622\pi$
$653$	−5.54791	−0.217107	−0.108553	+	0.994091i	$0.534622\pi$
$654$	0	0
$655$	61.0233	2.38438
$656$	0	0
$657$	19.6184	0.765385
$658$	0	0
$659$	27.7914	1.08260	0.541299	−	0.840830i	$-0.317933\pi$
$659$	27.7914	1.08260	0.541299	+	0.840830i	$0.317933\pi$
$660$	0	0
$661$	1.92881	0.0750221	0.0375111	−	0.999296i	$-0.488057\pi$
$661$	1.92881	0.0750221	0.0375111	+	0.999296i	$0.488057\pi$
$662$	0	0
$663$	0.284563	0.0110515
$664$	0	0
$665$	47.4584	1.84036
$666$	0	0
$667$	−15.0029	−0.580915
$668$	0	0
$669$	−11.6026	−0.448583
$670$	0	0
$671$	0.430982	0.0166379
$672$	0	0
$673$	19.2283	0.741195	0.370597	−	0.928794i	$-0.379153\pi$
$673$	19.2283	0.741195	0.370597	+	0.928794i	$0.379153\pi$
$674$	0	0
$675$	30.6035	1.17793
$676$	0	0
$677$	3.56493	0.137011	0.0685057	−	0.997651i	$-0.478177\pi$
$677$	3.56493	0.137011	0.0685057	+	0.997651i	$0.478177\pi$
$678$	0	0
$679$	−63.9747	−2.45512
$680$	0	0
$681$	31.5320	1.20831
$682$	0	0
$683$	−36.0055	−1.37771	−0.688856	−	0.724899i	$-0.741887\pi$
$683$	−36.0055	−1.37771	−0.688856	+	0.724899i	$0.741887\pi$
$684$	0	0
$685$	−19.8380	−0.757971
$686$	0	0
$687$	−46.7927	−1.78525
$688$	0	0
$689$	−0.942995	−0.0359253
$690$	0	0
$691$	−6.11836	−0.232754	−0.116377	−	0.993205i	$-0.537128\pi$
$691$	−6.11836	−0.232754	−0.116377	+	0.993205i	$0.537128\pi$
$692$	0	0
$693$	3.43644	0.130539
$694$	0	0
$695$	−58.5755	−2.22190
$696$	0	0
$697$	−15.0752	−0.571016
$698$	0	0
$699$	−37.8240	−1.43063
$700$	0	0
$701$	−27.4930	−1.03840	−0.519198	−	0.854654i	$-0.673769\pi$
$701$	−27.4930	−1.03840	−0.519198	+	0.854654i	$0.673769\pi$
$702$	0	0
$703$	18.2188	0.687136
$704$	0	0
$705$	−11.7173	−0.441299
$706$	0	0
$707$	−29.8949	−1.12432
$708$	0	0
$709$	−41.8276	−1.57087	−0.785433	−	0.618946i	$-0.787560\pi$
$709$	−41.8276	−1.57087	−0.785433	+	0.618946i	$0.787560\pi$
$710$	0	0
$711$	−21.0575	−0.789717
$712$	0	0
$713$	−12.1117	−0.453585
$714$	0	0
$715$	−0.164919	−0.00616763
$716$	0	0
$717$	−25.7137	−0.960295
$718$	0	0
$719$	36.4204	1.35825	0.679126	−	0.734022i	$-0.262359\pi$
$719$	36.4204	1.35825	0.679126	+	0.734022i	$0.262359\pi$
$720$	0	0
$721$	5.41461	0.201650
$722$	0	0
$723$	−24.0953	−0.896114
$724$	0	0
$725$	−65.0162	−2.41464
$726$	0	0
$727$	−1.23081	−0.0456481	−0.0228241	−	0.999739i	$-0.507266\pi$
$727$	−1.23081	−0.0456481	−0.0228241	+	0.999739i	$0.507266\pi$
$728$	0	0
$729$	2.77568	0.102803
$730$	0	0
$731$	17.3527	0.641814
$732$	0	0
$733$	27.5812	1.01873	0.509367	−	0.860550i	$-0.329880\pi$
$733$	27.5812	1.01873	0.509367	+	0.860550i	$0.329880\pi$
$734$	0	0
$735$	−143.528	−5.29410
$736$	0	0
$737$	3.65658	0.134692
$738$	0	0
$739$	−19.3896	−0.713257	−0.356629	−	0.934246i	$-0.616074\pi$
$739$	−19.3896	−0.713257	−0.356629	+	0.934246i	$0.616074\pi$
$740$	0	0
$741$	0.501672	0.0184294
$742$	0	0
$743$	3.02696	0.111048	0.0555242	−	0.998457i	$-0.482317\pi$
$743$	3.02696	0.111048	0.0555242	+	0.998457i	$0.482317\pi$
$744$	0	0
$745$	61.8858	2.26732
$746$	0	0
$747$	−15.4744	−0.566178
$748$	0	0
$749$	−27.6677	−1.01096
$750$	0	0
$751$	−5.10910	−0.186434	−0.0932168	−	0.995646i	$-0.529715\pi$
$751$	−5.10910	−0.186434	−0.0932168	+	0.995646i	$0.529715\pi$
$752$	0	0
$753$	18.3202	0.667625
$754$	0	0
$755$	29.1098	1.05942
$756$	0	0
$757$	−27.3805	−0.995161	−0.497580	−	0.867418i	$-0.665778\pi$
$757$	−27.3805	−0.995161	−0.497580	+	0.867418i	$0.665778\pi$
$758$	0	0
$759$	−2.17802	−0.0790570
$760$	0	0
$761$	−49.3210	−1.78789	−0.893943	−	0.448181i	$-0.852072\pi$
$761$	−49.3210	−1.78789	−0.893943	+	0.448181i	$0.852072\pi$
$762$	0	0
$763$	−33.5268	−1.21375
$764$	0	0
$765$	8.31136	0.300498
$766$	0	0
$767$	0.747243	0.0269814
$768$	0	0
$769$	11.8488	0.427277	0.213639	−	0.976913i	$-0.431468\pi$
$769$	11.8488	0.427277	0.213639	+	0.976913i	$0.431468\pi$
$770$	0	0
$771$	30.6419	1.10354
$772$	0	0
$773$	14.0516	0.505401	0.252700	−	0.967545i	$-0.418681\pi$
$773$	14.0516	0.505401	0.252700	+	0.967545i	$0.418681\pi$
$774$	0	0
$775$	−52.4867	−1.88538
$776$	0	0
$777$	−77.0853	−2.76542
$778$	0	0
$779$	−26.5770	−0.952218
$780$	0	0
$781$	1.38814	0.0496715
$782$	0	0
$783$	−20.8073	−0.743591
$784$	0	0
$785$	84.1707	3.00418
$786$	0	0
$787$	22.7651	0.811489	0.405744	−	0.913987i	$-0.367012\pi$
$787$	22.7651	0.811489	0.405744	+	0.913987i	$0.367012\pi$
$788$	0	0
$789$	−47.5477	−1.69274
$790$	0	0
$791$	−3.23926	−0.115175
$792$	0	0
$793$	−0.0898934	−0.00319221
$794$	0	0
$795$	−81.5638	−2.89277
$796$	0	0
$797$	32.9141	1.16588	0.582938	−	0.812517i	$-0.301903\pi$
$797$	32.9141	1.16588	0.582938	+	0.812517i	$0.301903\pi$
$798$	0	0
$799$	2.02429	0.0716143
$800$	0	0
$801$	−2.71955	−0.0960907
$802$	0	0
$803$	5.81695	0.205276
$804$	0	0
$805$	42.9756	1.51469
$806$	0	0
$807$	2.12827	0.0749186
$808$	0	0
$809$	−35.8700	−1.26112	−0.630560	−	0.776140i	$-0.717175\pi$
$809$	−35.8700	−1.26112	−0.630560	+	0.776140i	$0.717175\pi$
$810$	0	0
$811$	16.6343	0.584109	0.292055	−	0.956402i	$-0.405661\pi$
$811$	16.6343	0.584109	0.292055	+	0.956402i	$0.405661\pi$
$812$	0	0
$813$	35.6558	1.25050
$814$	0	0
$815$	23.1473	0.810816
$816$	0	0
$817$	30.5921	1.07028
$818$	0	0
$819$	−0.716764	−0.0250458
$820$	0	0
$821$	32.4924	1.13399	0.566997	−	0.823720i	$-0.308105\pi$
$821$	32.4924	1.13399	0.566997	+	0.823720i	$0.308105\pi$
$822$	0	0
$823$	29.9885	1.04533	0.522666	−	0.852537i	$-0.324937\pi$
$823$	29.9885	1.04533	0.522666	+	0.852537i	$0.324937\pi$
$824$	0	0
$825$	−9.43859	−0.328610
$826$	0	0
$827$	−31.1200	−1.08215	−0.541075	−	0.840975i	$-0.681982\pi$
$827$	−31.1200	−1.08215	−0.541075	+	0.840975i	$0.681982\pi$
$828$	0	0
$829$	−15.8365	−0.550025	−0.275013	−	0.961441i	$-0.588682\pi$
$829$	−15.8365	−0.550025	−0.275013	+	0.961441i	$0.588682\pi$
$830$	0	0
$831$	−31.4665	−1.09156
$832$	0	0
$833$	24.7960	0.859129
$834$	0	0
$835$	−75.6504	−2.61799
$836$	0	0
$837$	−16.7974	−0.580604
$838$	0	0
$839$	12.2519	0.422981	0.211491	−	0.977380i	$-0.432168\pi$
$839$	12.2519	0.422981	0.211491	+	0.977380i	$0.432168\pi$
$840$	0	0
$841$	15.2044	0.524289
$842$	0	0
$843$	−53.3002	−1.83576
$844$	0	0
$845$	−49.9419	−1.71805
$846$	0	0
$847$	−53.4754	−1.83744
$848$	0	0
$849$	52.3617	1.79705
$850$	0	0
$851$	16.4979	0.565542
$852$	0	0
$853$	−9.41699	−0.322431	−0.161216	−	0.986919i	$-0.551541\pi$
$853$	−9.41699	−0.322431	−0.161216	+	0.986919i	$0.551541\pi$
$854$	0	0
$855$	14.6525	0.501107
$856$	0	0
$857$	19.2781	0.658529	0.329264	−	0.944238i	$-0.393199\pi$
$857$	19.2781	0.658529	0.329264	+	0.944238i	$0.393199\pi$
$858$	0	0
$859$	48.7953	1.66487	0.832437	−	0.554120i	$-0.186945\pi$
$859$	48.7953	1.66487	0.832437	+	0.554120i	$0.186945\pi$
$860$	0	0
$861$	112.449	3.83226
$862$	0	0
$863$	−27.3052	−0.929481	−0.464741	−	0.885447i	$-0.653852\pi$
$863$	−27.3052	−0.929481	−0.464741	+	0.885447i	$0.653852\pi$
$864$	0	0
$865$	47.9820	1.63144
$866$	0	0
$867$	31.9284	1.08435
$868$	0	0
$869$	−6.24366	−0.211802
$870$	0	0
$871$	−0.762681	−0.0258425
$872$	0	0
$873$	−19.7519	−0.668500
$874$	0	0
$875$	91.0134	3.07681
$876$	0	0
$877$	−14.3551	−0.484737	−0.242368	−	0.970184i	$-0.577924\pi$
$877$	−14.3551	−0.484737	−0.242368	+	0.970184i	$0.577924\pi$
$878$	0	0
$879$	24.3939	0.822785
$880$	0	0
$881$	−20.4566	−0.689201	−0.344600	−	0.938749i	$-0.611986\pi$
$881$	−20.4566	−0.689201	−0.344600	+	0.938749i	$0.611986\pi$
$882$	0	0
$883$	−13.2088	−0.444510	−0.222255	−	0.974989i	$-0.571342\pi$
$883$	−13.2088	−0.444510	−0.222255	+	0.974989i	$0.571342\pi$
$884$	0	0
$885$	64.6323	2.17259
$886$	0	0
$887$	21.4469	0.720115	0.360058	−	0.932930i	$-0.382757\pi$
$887$	21.4469	0.720115	0.360058	+	0.932930i	$0.382757\pi$
$888$	0	0
$889$	−41.0996	−1.37844
$890$	0	0
$891$	−5.10164	−0.170911
$892$	0	0
$893$	3.56873	0.119423
$894$	0	0
$895$	−30.1778	−1.00873
$896$	0	0
$897$	0.454286	0.0151682
$898$	0	0
$899$	35.6856	1.19018
$900$	0	0
$901$	14.0910	0.469441
$902$	0	0
$903$	−129.438	−4.30741
$904$	0	0
$905$	−69.1198	−2.29762
$906$	0	0
$907$	53.7295	1.78406	0.892029	−	0.451977i	$-0.149281\pi$
$907$	53.7295	1.78406	0.892029	+	0.451977i	$0.149281\pi$
$908$	0	0
$909$	−9.22993	−0.306137
$910$	0	0
$911$	17.7091	0.586728	0.293364	−	0.956001i	$-0.405225\pi$
$911$	17.7091	0.586728	0.293364	+	0.956001i	$0.405225\pi$
$912$	0	0
$913$	−4.58824	−0.151848
$914$	0	0
$915$	−7.77528	−0.257043
$916$	0	0
$917$	78.6381	2.59686
$918$	0	0
$919$	−59.7641	−1.97144	−0.985718	−	0.168406i	$-0.946138\pi$
$919$	−59.7641	−1.97144	−0.985718	+	0.168406i	$0.946138\pi$
$920$	0	0
$921$	−37.8148	−1.24604
$922$	0	0
$923$	−0.289535	−0.00953015
$924$	0	0
$925$	71.4950	2.35074
$926$	0	0
$927$	1.67173	0.0549070
$928$	0	0
$929$	−53.9545	−1.77019	−0.885095	−	0.465410i	$-0.845907\pi$
$929$	−53.9545	−1.77019	−0.885095	+	0.465410i	$0.845907\pi$
$930$	0	0
$931$	43.7141	1.43267
$932$	0	0
$933$	16.2205	0.531036
$934$	0	0
$935$	2.46436	0.0805933
$936$	0	0
$937$	−23.7929	−0.777280	−0.388640	−	0.921390i	$-0.627055\pi$
$937$	−23.7929	−0.777280	−0.388640	+	0.921390i	$0.627055\pi$
$938$	0	0
$939$	−16.9630	−0.553566
$940$	0	0
$941$	−30.7434	−1.00221	−0.501103	−	0.865388i	$-0.667072\pi$
$941$	−30.7434	−1.00221	−0.501103	+	0.865388i	$0.667072\pi$
$942$	0	0
$943$	−24.0666	−0.783716
$944$	0	0
$945$	59.6020	1.93885
$946$	0	0
$947$	−44.2821	−1.43897	−0.719487	−	0.694506i	$-0.755623\pi$
$947$	−44.2821	−1.43897	−0.719487	+	0.694506i	$0.755623\pi$
$948$	0	0
$949$	−1.21329	−0.0393849
$950$	0	0
$951$	−3.65297	−0.118456
$952$	0	0
$953$	−9.40636	−0.304702	−0.152351	−	0.988326i	$-0.548684\pi$
$953$	−9.40636	−0.304702	−0.152351	+	0.988326i	$0.548684\pi$
$954$	0	0
$955$	47.3059	1.53078
$956$	0	0
$957$	6.41727	0.207441
$958$	0	0
$959$	−25.5644	−0.825517
$960$	0	0
$961$	−2.19148	−0.0706930
$962$	0	0
$963$	−8.54227	−0.275271
$964$	0	0
$965$	22.3819	0.720500
$966$	0	0
$967$	56.8679	1.82875	0.914373	−	0.404872i	$-0.132684\pi$
$967$	56.8679	1.82875	0.914373	+	0.404872i	$0.132684\pi$
$968$	0	0
$969$	−7.49640	−0.240819
$970$	0	0
$971$	21.1723	0.679452	0.339726	−	0.940524i	$-0.389666\pi$
$971$	21.1723	0.679452	0.339726	+	0.940524i	$0.389666\pi$
$972$	0	0
$973$	−75.4838	−2.41990
$974$	0	0
$975$	1.96868	0.0630482
$976$	0	0
$977$	6.11162	0.195528	0.0977641	−	0.995210i	$-0.468831\pi$
$977$	6.11162	0.195528	0.0977641	+	0.995210i	$0.468831\pi$
$978$	0	0
$979$	−0.806362	−0.0257714
$980$	0	0
$981$	−10.3513	−0.330490
$982$	0	0
$983$	−42.4508	−1.35397	−0.676985	−	0.735997i	$-0.736714\pi$
$983$	−42.4508	−1.35397	−0.676985	+	0.735997i	$0.736714\pi$
$984$	0	0
$985$	−83.2753	−2.65337
$986$	0	0
$987$	−15.0996	−0.480625
$988$	0	0
$989$	27.7025	0.880887
$990$	0	0
$991$	52.9269	1.68128	0.840640	−	0.541594i	$-0.182179\pi$
$991$	52.9269	1.68128	0.840640	+	0.541594i	$0.182179\pi$
$992$	0	0
$993$	−29.2715	−0.928904
$994$	0	0
$995$	−21.9835	−0.696925
$996$	0	0
$997$	−43.4208	−1.37515	−0.687575	−	0.726113i	$-0.741325\pi$
$997$	−43.4208	−1.37515	−0.687575	+	0.726113i	$0.741325\pi$
$998$	0	0
$999$	22.8807	0.723912

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4304.2.a.h.1.2		7
4.3	odd	2		538.2.a.e.1.6	✓	7
12.11	even	2		4842.2.a.n.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.e.1.6	✓	7	4.3	odd	2
4304.2.a.h.1.2		7	1.1	even	1	trivial
4842.2.a.n.1.2		7	12.11	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:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12827 q^{3} +3.84433 q^{5} +4.95403 q^{7} +1.52953 q^{9} +0.453515 q^{11} -0.0945931 q^{13} -8.18177 q^{15} +1.41349 q^{17} +2.49192 q^{19} -10.5435 q^{21} +2.25654 q^{23} +9.77888 q^{25} +3.12955 q^{27} -6.64864 q^{29} -5.36736 q^{31} -0.965201 q^{33} +19.0449 q^{35} +7.31117 q^{37} +0.201320 q^{39} -10.6653 q^{41} +12.2765 q^{43} +5.88003 q^{45} +1.43212 q^{47} +17.5424 q^{49} -3.00829 q^{51} +9.96897 q^{53} +1.74346 q^{55} -5.30347 q^{57} -7.89955 q^{59} +0.950317 q^{61} +7.57735 q^{63} -0.363647 q^{65} +8.06276 q^{67} -4.80253 q^{69} +3.06085 q^{71} +12.8264 q^{73} -20.8121 q^{75} +2.24672 q^{77} -13.7673 q^{79} -11.2491 q^{81} -10.1171 q^{83} +5.43392 q^{85} +14.1501 q^{87} -1.77803 q^{89} -0.468616 q^{91} +11.4232 q^{93} +9.57975 q^{95} -12.9137 q^{97} +0.693665 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{3} + 7 q^{5} - 6 q^{7} + 12 q^{9} + 3 q^{11} - 9 q^{13} - 8 q^{15} + 8 q^{17} + 11 q^{19} - 6 q^{21} - 12 q^{23} + 22 q^{25} + 14 q^{27} - 5 q^{29} - 14 q^{31} - 4 q^{33} + 4 q^{35} + 13 q^{37}+ \cdots + 37 q^{99}+O(q^{100}) \) 7 * q - q^3 + 7 * q^5 - 6 * q^7 + 12 * q^9 + 3 * q^11 - 9 * q^13 - 8 * q^15 + 8 * q^17 + 11 * q^19 - 6 * q^21 - 12 * q^23 + 22 * q^25 + 14 * q^27 - 5 * q^29 - 14 * q^31 - 4 * q^33 + 4 * q^35 + 13 * q^37 + 18 * q^39 + 12 * q^41 + 11 * q^43 + 3 * q^45 - 2 * q^47 + 15 * q^49 + 26 * q^51 + 19 * q^53 + 40 * q^55 - 12 * q^57 + 9 * q^59 - 3 * q^61 + 26 * q^63 - 10 * q^65 + 33 * q^67 - 64 * q^69 - 28 * q^71 - 14 * q^73 + 45 * q^75 + 10 * q^77 - 2 * q^79 + 15 * q^81 + 7 * q^83 - 16 * q^85 - 16 * q^87 + 18 * q^89 + 26 * q^91 + 6 * q^93 + 34 * q^95 - 4 * q^97 + 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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