LMFDB - Embedded newform 7448.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} - x^{5} + 66x^{4} + 4x^{3} - 76x^{2} + 8x + 16 $ x^8 - 15x^6 - x^5 + 66x^4 + 4x^3 - 76x^2 + 8*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.85665$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.85665 q^{3} -0.827910 q^{5} +5.16044 q^{9} +O(q^{10})$	$q-2.85665 q^{3} -0.827910 q^{5} +5.16044 q^{9} +5.71522 q^{11} +0.0340264 q^{13} +2.36505 q^{15} +4.23533 q^{17} -1.00000 q^{19} +4.89259 q^{23} -4.31456 q^{25} -6.17162 q^{27} +9.29907 q^{29} +2.53865 q^{31} -16.3264 q^{33} +4.58585 q^{37} -0.0972015 q^{39} +2.00000 q^{41} +9.61156 q^{43} -4.27238 q^{45} +11.3224 q^{47} -12.0989 q^{51} +3.43316 q^{53} -4.73169 q^{55} +2.85665 q^{57} +7.35351 q^{59} -2.49250 q^{61} -0.0281708 q^{65} -6.98306 q^{67} -13.9764 q^{69} +12.0575 q^{71} -0.751292 q^{73} +12.3252 q^{75} -11.9652 q^{79} +2.14883 q^{81} +13.7727 q^{83} -3.50647 q^{85} -26.5642 q^{87} -5.39729 q^{89} -7.25203 q^{93} +0.827910 q^{95} -8.49448 q^{97} +29.4930 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{5} + 6 q^{9}+O(q^{10}) $ 8 * q + q^5 + 6 * q^9	$ 8 q + q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} - 4 q^{17} - 8 q^{19} + 25 q^{23} + 15 q^{25} - 3 q^{27} + 6 q^{29} - 8 q^{33} + 13 q^{37} - 11 q^{39} + 16 q^{41} + 17 q^{43} - 17 q^{45} + 24 q^{47} + 5 q^{51} + 2 q^{53} - 5 q^{55} - 2 q^{59} + 13 q^{61} - 26 q^{65} + 2 q^{67} + 11 q^{69} + 10 q^{71} - 5 q^{73} + 20 q^{75} + 16 q^{79} - 12 q^{81} + 43 q^{83} + 24 q^{85} - 20 q^{87} - 8 q^{89} + 2 q^{93} - q^{95} + 12 q^{97} + 37 q^{99}+O(q^{100}) $ 8 * q + q^5 + 6 * q^9 + 9 * q^11 + 8 * q^15 - 4 * q^17 - 8 * q^19 + 25 * q^23 + 15 * q^25 - 3 * q^27 + 6 * q^29 - 8 * q^33 + 13 * q^37 - 11 * q^39 + 16 * q^41 + 17 * q^43 - 17 * q^45 + 24 * q^47 + 5 * q^51 + 2 * q^53 - 5 * q^55 - 2 * q^59 + 13 * q^61 - 26 * q^65 + 2 * q^67 + 11 * q^69 + 10 * q^71 - 5 * q^73 + 20 * q^75 + 16 * q^79 - 12 * q^81 + 43 * q^83 + 24 * q^85 - 20 * q^87 - 8 * q^89 + 2 * q^93 - q^95 + 12 * q^97 + 37 * 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
$3$	−2.85665	−1.64929	−0.824643	−	0.565653i	$-0.808624\pi$
$3$	−2.85665	−1.64929	−0.824643	+	0.565653i	$0.808624\pi$
$4$	0	0
$5$	−0.827910	−0.370253	−0.185126	−	0.982715i	$-0.559269\pi$
$5$	−0.827910	−0.370253	−0.185126	+	0.982715i	$0.559269\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.16044	1.72015
$10$	0	0
$11$	5.71522	1.72320	0.861601	−	0.507586i	$-0.169462\pi$
$11$	5.71522	1.72320	0.861601	+	0.507586i	$0.169462\pi$
$12$	0	0
$13$	0.0340264	0.00943723	0.00471861	−	0.999989i	$-0.498498\pi$
$13$	0.0340264	0.00943723	0.00471861	+	0.999989i	$0.498498\pi$
$14$	0	0
$15$	2.36505	0.610653
$16$	0	0
$17$	4.23533	1.02722	0.513609	−	0.858024i	$-0.328308\pi$
$17$	4.23533	1.02722	0.513609	+	0.858024i	$0.328308\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.89259	1.02018	0.510088	−	0.860122i	$-0.329613\pi$
$23$	4.89259	1.02018	0.510088	+	0.860122i	$0.329613\pi$
$24$	0	0
$25$	−4.31456	−0.862913
$26$	0	0
$27$	−6.17162	−1.18773
$28$	0	0
$29$	9.29907	1.72679	0.863397	−	0.504525i	$-0.168332\pi$
$29$	9.29907	1.72679	0.863397	+	0.504525i	$0.168332\pi$
$30$	0	0
$31$	2.53865	0.455955	0.227977	−	0.973666i	$-0.426789\pi$
$31$	2.53865	0.455955	0.227977	+	0.973666i	$0.426789\pi$
$32$	0	0
$33$	−16.3264	−2.84206
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.58585	0.753909	0.376954	−	0.926232i	$-0.376971\pi$
$37$	4.58585	0.753909	0.376954	+	0.926232i	$0.376971\pi$
$38$	0	0
$39$	−0.0972015	−0.0155647
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	9.61156	1.46575	0.732875	−	0.680364i	$-0.238178\pi$
$43$	9.61156	1.46575	0.732875	+	0.680364i	$0.238178\pi$
$44$	0	0
$45$	−4.27238	−0.636889
$46$	0	0
$47$	11.3224	1.65154	0.825769	−	0.564008i	$-0.190741\pi$
$47$	11.3224	1.65154	0.825769	+	0.564008i	$0.190741\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−12.0989	−1.69418
$52$	0	0
$53$	3.43316	0.471581	0.235791	−	0.971804i	$-0.424232\pi$
$53$	3.43316	0.471581	0.235791	+	0.971804i	$0.424232\pi$
$54$	0	0
$55$	−4.73169	−0.638020
$56$	0	0
$57$	2.85665	0.378372
$58$	0	0
$59$	7.35351	0.957345	0.478673	−	0.877994i	$-0.341118\pi$
$59$	7.35351	0.957345	0.478673	+	0.877994i	$0.341118\pi$
$60$	0	0
$61$	−2.49250	−0.319132	−0.159566	−	0.987187i	$-0.551009\pi$
$61$	−2.49250	−0.319132	−0.159566	+	0.987187i	$0.551009\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.0281708	−0.00349416
$66$	0	0
$67$	−6.98306	−0.853117	−0.426559	−	0.904460i	$-0.640274\pi$
$67$	−6.98306	−0.853117	−0.426559	+	0.904460i	$0.640274\pi$
$68$	0	0
$69$	−13.9764	−1.68256
$70$	0	0
$71$	12.0575	1.43097	0.715484	−	0.698630i	$-0.246206\pi$
$71$	12.0575	1.43097	0.715484	+	0.698630i	$0.246206\pi$
$72$	0	0
$73$	−0.751292	−0.0879320	−0.0439660	−	0.999033i	$-0.513999\pi$
$73$	−0.751292	−0.0879320	−0.0439660	+	0.999033i	$0.513999\pi$
$74$	0	0
$75$	12.3252	1.42319
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.9652	−1.34619	−0.673097	−	0.739554i	$-0.735037\pi$
$79$	−11.9652	−1.34619	−0.673097	+	0.739554i	$0.735037\pi$
$80$	0	0
$81$	2.14883	0.238759
$82$	0	0
$83$	13.7727	1.51175	0.755875	−	0.654716i	$-0.227212\pi$
$83$	13.7727	1.51175	0.755875	+	0.654716i	$0.227212\pi$
$84$	0	0
$85$	−3.50647	−0.380330
$86$	0	0
$87$	−26.5642	−2.84798
$88$	0	0
$89$	−5.39729	−0.572111	−0.286056	−	0.958213i	$-0.592344\pi$
$89$	−5.39729	−0.572111	−0.286056	+	0.958213i	$0.592344\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.25203	−0.752000
$94$	0	0
$95$	0.827910	0.0849418
$96$	0	0
$97$	−8.49448	−0.862484	−0.431242	−	0.902236i	$-0.641924\pi$
$97$	−8.49448	−0.862484	−0.431242	+	0.902236i	$0.641924\pi$
$98$	0	0
$99$	29.4930	2.96416
$100$	0	0
$101$	−12.3885	−1.23270	−0.616351	−	0.787471i	$-0.711390\pi$
$101$	−12.3885	−1.23270	−0.616351	+	0.787471i	$0.711390\pi$
$102$	0	0
$103$	−14.2271	−1.40184	−0.700920	−	0.713240i	$-0.747227\pi$
$103$	−14.2271	−1.40184	−0.700920	+	0.713240i	$0.747227\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.408398	−0.0394813	−0.0197407	−	0.999805i	$-0.506284\pi$
$107$	−0.408398	−0.0394813	−0.0197407	+	0.999805i	$0.506284\pi$
$108$	0	0
$109$	−16.2943	−1.56071	−0.780355	−	0.625336i	$-0.784962\pi$
$109$	−16.2943	−1.56071	−0.780355	+	0.625336i	$0.784962\pi$
$110$	0	0
$111$	−13.1002	−1.24341
$112$	0	0
$113$	13.9404	1.31141	0.655704	−	0.755018i	$-0.272372\pi$
$113$	13.9404	1.31141	0.655704	+	0.755018i	$0.272372\pi$
$114$	0	0
$115$	−4.05063	−0.377723
$116$	0	0
$117$	0.175591	0.0162334
$118$	0	0
$119$	0	0
$120$	0	0
$121$	21.6637	1.96943
$122$	0	0
$123$	−5.71330	−0.515151
$124$	0	0
$125$	7.71162	0.689749
$126$	0	0
$127$	0.718299	0.0637387	0.0318694	−	0.999492i	$-0.489854\pi$
$127$	0.718299	0.0637387	0.0318694	+	0.999492i	$0.489854\pi$
$128$	0	0
$129$	−27.4569	−2.41744
$130$	0	0
$131$	−14.6245	−1.27775	−0.638874	−	0.769311i	$-0.720599\pi$
$131$	−14.6245	−1.27775	−0.638874	+	0.769311i	$0.720599\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.10955	0.439760
$136$	0	0
$137$	−16.9146	−1.44511	−0.722557	−	0.691312i	$-0.757033\pi$
$137$	−16.9146	−1.44511	−0.722557	+	0.691312i	$0.757033\pi$
$138$	0	0
$139$	13.3401	1.13149	0.565744	−	0.824581i	$-0.308589\pi$
$139$	13.3401	1.13149	0.565744	+	0.824581i	$0.308589\pi$
$140$	0	0
$141$	−32.3441	−2.72386
$142$	0	0
$143$	0.194468	0.0162623
$144$	0	0
$145$	−7.69880	−0.639350
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.5061	−0.942615	−0.471307	−	0.881969i	$-0.656218\pi$
$149$	−11.5061	−0.942615	−0.471307	+	0.881969i	$0.656218\pi$
$150$	0	0
$151$	−5.25790	−0.427882	−0.213941	−	0.976847i	$-0.568630\pi$
$151$	−5.25790	−0.427882	−0.213941	+	0.976847i	$0.568630\pi$
$152$	0	0
$153$	21.8562	1.76697
$154$	0	0
$155$	−2.10177	−0.168818
$156$	0	0
$157$	1.36711	0.109108	0.0545538	−	0.998511i	$-0.482626\pi$
$157$	1.36711	0.109108	0.0545538	+	0.998511i	$0.482626\pi$
$158$	0	0
$159$	−9.80734	−0.777773
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.2099	0.799698	0.399849	−	0.916581i	$-0.369063\pi$
$163$	10.2099	0.799698	0.399849	+	0.916581i	$0.369063\pi$
$164$	0	0
$165$	13.5168	1.05228
$166$	0	0
$167$	−4.13699	−0.320130	−0.160065	−	0.987106i	$-0.551170\pi$
$167$	−4.13699	−0.320130	−0.160065	+	0.987106i	$0.551170\pi$
$168$	0	0
$169$	−12.9988	−0.999911
$170$	0	0
$171$	−5.16044	−0.394629
$172$	0	0
$173$	−18.2842	−1.39012	−0.695062	−	0.718950i	$-0.744623\pi$
$173$	−18.2842	−1.39012	−0.695062	+	0.718950i	$0.744623\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−21.0064	−1.57894
$178$	0	0
$179$	1.21843	0.0910700	0.0455350	−	0.998963i	$-0.485501\pi$
$179$	1.21843	0.0910700	0.0455350	+	0.998963i	$0.485501\pi$
$180$	0	0
$181$	14.0374	1.04339	0.521695	−	0.853132i	$-0.325300\pi$
$181$	14.0374	1.04339	0.521695	+	0.853132i	$0.325300\pi$
$182$	0	0
$183$	7.12019	0.526340
$184$	0	0
$185$	−3.79667	−0.279137
$186$	0	0
$187$	24.2058	1.77011
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.13812	0.588854	0.294427	−	0.955674i	$-0.404871\pi$
$191$	8.13812	0.588854	0.294427	+	0.955674i	$0.404871\pi$
$192$	0	0
$193$	−7.37542	−0.530894	−0.265447	−	0.964125i	$-0.585520\pi$
$193$	−7.37542	−0.530894	−0.265447	+	0.964125i	$0.585520\pi$
$194$	0	0
$195$	0.0804741	0.00576287
$196$	0	0
$197$	27.6022	1.96657	0.983287	−	0.182063i	$-0.0582774\pi$
$197$	27.6022	1.96657	0.983287	+	0.182063i	$0.0582774\pi$
$198$	0	0
$199$	−2.64715	−0.187651	−0.0938257	−	0.995589i	$-0.529910\pi$
$199$	−2.64715	−0.187651	−0.0938257	+	0.995589i	$0.529910\pi$
$200$	0	0
$201$	19.9482	1.40703
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.65582	−0.115648
$206$	0	0
$207$	25.2480	1.75485
$208$	0	0
$209$	−5.71522	−0.395330
$210$	0	0
$211$	−12.7616	−0.878545	−0.439272	−	0.898354i	$-0.644764\pi$
$211$	−12.7616	−0.878545	−0.439272	+	0.898354i	$0.644764\pi$
$212$	0	0
$213$	−34.4442	−2.36008
$214$	0	0
$215$	−7.95751	−0.542698
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.14618	0.145025
$220$	0	0
$221$	0.144113	0.00969409
$222$	0	0
$223$	7.90397	0.529289	0.264645	−	0.964346i	$-0.414745\pi$
$223$	7.90397	0.529289	0.264645	+	0.964346i	$0.414745\pi$
$224$	0	0
$225$	−22.2651	−1.48434
$226$	0	0
$227$	27.8147	1.84613	0.923064	−	0.384647i	$-0.125677\pi$
$227$	27.8147	1.84613	0.923064	+	0.384647i	$0.125677\pi$
$228$	0	0
$229$	10.5868	0.699599	0.349799	−	0.936825i	$-0.386250\pi$
$229$	10.5868	0.699599	0.349799	+	0.936825i	$0.386250\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.19703	0.340469	0.170234	−	0.985404i	$-0.445548\pi$
$233$	5.19703	0.340469	0.170234	+	0.985404i	$0.445548\pi$
$234$	0	0
$235$	−9.37391	−0.611487
$236$	0	0
$237$	34.1805	2.22026
$238$	0	0
$239$	3.63655	0.235229	0.117614	−	0.993059i	$-0.462475\pi$
$239$	3.63655	0.235229	0.117614	+	0.993059i	$0.462475\pi$
$240$	0	0
$241$	−7.67108	−0.494138	−0.247069	−	0.968998i	$-0.579467\pi$
$241$	−7.67108	−0.494138	−0.247069	+	0.968998i	$0.579467\pi$
$242$	0	0
$243$	12.3764	0.793947
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.0340264	−0.00216505
$248$	0	0
$249$	−39.3437	−2.49331
$250$	0	0
$251$	−16.3670	−1.03308	−0.516539	−	0.856264i	$-0.672780\pi$
$251$	−16.3670	−1.03308	−0.516539	+	0.856264i	$0.672780\pi$
$252$	0	0
$253$	27.9622	1.75797
$254$	0	0
$255$	10.0168	0.627274
$256$	0	0
$257$	15.9176	0.992913	0.496457	−	0.868062i	$-0.334634\pi$
$257$	15.9176	0.992913	0.496457	+	0.868062i	$0.334634\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	47.9873	2.97034
$262$	0	0
$263$	11.7662	0.725537	0.362768	−	0.931879i	$-0.381832\pi$
$263$	11.7662	0.725537	0.362768	+	0.931879i	$0.381832\pi$
$264$	0	0
$265$	−2.84235	−0.174604
$266$	0	0
$267$	15.4182	0.943576
$268$	0	0
$269$	9.96808	0.607765	0.303882	−	0.952710i	$-0.401717\pi$
$269$	9.96808	0.607765	0.303882	+	0.952710i	$0.401717\pi$
$270$	0	0
$271$	−13.9395	−0.846765	−0.423382	−	0.905951i	$-0.639157\pi$
$271$	−13.9395	−0.846765	−0.423382	+	0.905951i	$0.639157\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−24.6587	−1.48697
$276$	0	0
$277$	−4.94578	−0.297163	−0.148582	−	0.988900i	$-0.547471\pi$
$277$	−4.94578	−0.297163	−0.148582	+	0.988900i	$0.547471\pi$
$278$	0	0
$279$	13.1005	0.784309
$280$	0	0
$281$	−10.0464	−0.599318	−0.299659	−	0.954046i	$-0.596873\pi$
$281$	−10.0464	−0.599318	−0.299659	+	0.954046i	$0.596873\pi$
$282$	0	0
$283$	19.8855	1.18207	0.591035	−	0.806646i	$-0.298719\pi$
$283$	19.8855	1.18207	0.591035	+	0.806646i	$0.298719\pi$
$284$	0	0
$285$	−2.36505	−0.140093
$286$	0	0
$287$	0	0
$288$	0	0
$289$	0.938022	0.0551777
$290$	0	0
$291$	24.2658	1.42248
$292$	0	0
$293$	−6.94575	−0.405775	−0.202888	−	0.979202i	$-0.565033\pi$
$293$	−6.94575	−0.405775	−0.202888	+	0.979202i	$0.565033\pi$
$294$	0	0
$295$	−6.08804	−0.354460
$296$	0	0
$297$	−35.2722	−2.04670
$298$	0	0
$299$	0.166477	0.00962764
$300$	0	0
$301$	0	0
$302$	0	0
$303$	35.3896	2.03308
$304$	0	0
$305$	2.06356	0.118159
$306$	0	0
$307$	20.9772	1.19723	0.598617	−	0.801035i	$-0.295717\pi$
$307$	20.9772	1.19723	0.598617	+	0.801035i	$0.295717\pi$
$308$	0	0
$309$	40.6419	2.31204
$310$	0	0
$311$	−7.46043	−0.423042	−0.211521	−	0.977373i	$-0.567842\pi$
$311$	−7.46043	−0.423042	−0.211521	+	0.977373i	$0.567842\pi$
$312$	0	0
$313$	7.22765	0.408531	0.204266	−	0.978916i	$-0.434519\pi$
$313$	7.22765	0.408531	0.204266	+	0.978916i	$0.434519\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.75967	−0.323495	−0.161748	−	0.986832i	$-0.551713\pi$
$317$	−5.75967	−0.323495	−0.161748	+	0.986832i	$0.551713\pi$
$318$	0	0
$319$	53.1462	2.97562
$320$	0	0
$321$	1.16665	0.0651160
$322$	0	0
$323$	−4.23533	−0.235660
$324$	0	0
$325$	−0.146809	−0.00814351
$326$	0	0
$327$	46.5471	2.57406
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12.0852	0.664265	0.332133	−	0.943233i	$-0.392232\pi$
$331$	12.0852	0.664265	0.332133	+	0.943233i	$0.392232\pi$
$332$	0	0
$333$	23.6650	1.29683
$334$	0	0
$335$	5.78135	0.315869
$336$	0	0
$337$	−12.3876	−0.674794	−0.337397	−	0.941363i	$-0.609546\pi$
$337$	−12.3876	−0.674794	−0.337397	+	0.941363i	$0.609546\pi$
$338$	0	0
$339$	−39.8230	−2.16289
$340$	0	0
$341$	14.5089	0.785702
$342$	0	0
$343$	0	0
$344$	0	0
$345$	11.5712	0.622974
$346$	0	0
$347$	21.5682	1.15784	0.578922	−	0.815383i	$-0.303474\pi$
$347$	21.5682	1.15784	0.578922	+	0.815383i	$0.303474\pi$
$348$	0	0
$349$	−28.6816	−1.53529	−0.767645	−	0.640875i	$-0.778572\pi$
$349$	−28.6816	−1.53529	−0.767645	+	0.640875i	$0.778572\pi$
$350$	0	0
$351$	−0.209998	−0.0112089
$352$	0	0
$353$	−14.8765	−0.791796	−0.395898	−	0.918295i	$-0.629567\pi$
$353$	−14.8765	−0.791796	−0.395898	+	0.918295i	$0.629567\pi$
$354$	0	0
$355$	−9.98256	−0.529819
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.08709	0.215708	0.107854	−	0.994167i	$-0.465602\pi$
$359$	4.08709	0.215708	0.107854	+	0.994167i	$0.465602\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−61.8856	−3.24815
$364$	0	0
$365$	0.622002	0.0325571
$366$	0	0
$367$	20.5519	1.07280	0.536399	−	0.843964i	$-0.319784\pi$
$367$	20.5519	1.07280	0.536399	+	0.843964i	$0.319784\pi$
$368$	0	0
$369$	10.3209	0.537284
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−8.40552	−0.435221	−0.217611	−	0.976036i	$-0.569826\pi$
$373$	−8.40552	−0.435221	−0.217611	+	0.976036i	$0.569826\pi$
$374$	0	0
$375$	−22.0294	−1.13759
$376$	0	0
$377$	0.316414	0.0162962
$378$	0	0
$379$	−21.0086	−1.07914	−0.539570	−	0.841941i	$-0.681413\pi$
$379$	−21.0086	−1.07914	−0.539570	+	0.841941i	$0.681413\pi$
$380$	0	0
$381$	−2.05193	−0.105123
$382$	0	0
$383$	−10.6143	−0.542366	−0.271183	−	0.962528i	$-0.587415\pi$
$383$	−10.6143	−0.542366	−0.271183	+	0.962528i	$0.587415\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	49.5999	2.52130
$388$	0	0
$389$	34.0262	1.72520	0.862598	−	0.505890i	$-0.168836\pi$
$389$	34.0262	1.72520	0.862598	+	0.505890i	$0.168836\pi$
$390$	0	0
$391$	20.7218	1.04794
$392$	0	0
$393$	41.7770	2.10737
$394$	0	0
$395$	9.90615	0.498432
$396$	0	0
$397$	−0.0407018	−0.00204277	−0.00102138	−	0.999999i	$-0.500325\pi$
$397$	−0.0407018	−0.00204277	−0.00102138	+	0.999999i	$0.500325\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−25.9584	−1.29630	−0.648149	−	0.761513i	$-0.724457\pi$
$401$	−25.9584	−1.29630	−0.648149	+	0.761513i	$0.724457\pi$
$402$	0	0
$403$	0.0863811	0.00430295
$404$	0	0
$405$	−1.77904	−0.0884013
$406$	0	0
$407$	26.2091	1.29914
$408$	0	0
$409$	−18.8658	−0.932852	−0.466426	−	0.884560i	$-0.654459\pi$
$409$	−18.8658	−0.932852	−0.466426	+	0.884560i	$0.654459\pi$
$410$	0	0
$411$	48.3191	2.38341
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−11.4026	−0.559729
$416$	0	0
$417$	−38.1078	−1.86615
$418$	0	0
$419$	31.3570	1.53189	0.765944	−	0.642907i	$-0.222272\pi$
$419$	31.3570	1.53189	0.765944	+	0.642907i	$0.222272\pi$
$420$	0	0
$421$	9.59485	0.467624	0.233812	−	0.972282i	$-0.424880\pi$
$421$	9.59485	0.467624	0.233812	+	0.972282i	$0.424880\pi$
$422$	0	0
$423$	58.4285	2.84089
$424$	0	0
$425$	−18.2736	−0.886400
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.555528	−0.0268211
$430$	0	0
$431$	32.2382	1.55286	0.776431	−	0.630202i	$-0.217028\pi$
$431$	32.2382	1.55286	0.776431	+	0.630202i	$0.217028\pi$
$432$	0	0
$433$	−40.2685	−1.93518	−0.967590	−	0.252525i	$-0.918739\pi$
$433$	−40.2685	−1.93518	−0.967590	+	0.252525i	$0.918739\pi$
$434$	0	0
$435$	21.9928	1.05447
$436$	0	0
$437$	−4.89259	−0.234045
$438$	0	0
$439$	−30.3094	−1.44659	−0.723295	−	0.690540i	$-0.757373\pi$
$439$	−30.3094	−1.44659	−0.723295	+	0.690540i	$0.757373\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.7267	−0.984755	−0.492378	−	0.870382i	$-0.663872\pi$
$443$	−20.7267	−0.984755	−0.492378	+	0.870382i	$0.663872\pi$
$444$	0	0
$445$	4.46847	0.211826
$446$	0	0
$447$	32.8688	1.55464
$448$	0	0
$449$	−27.0998	−1.27892	−0.639459	−	0.768825i	$-0.720842\pi$
$449$	−27.0998	−1.27892	−0.639459	+	0.768825i	$0.720842\pi$
$450$	0	0
$451$	11.4304	0.538238
$452$	0	0
$453$	15.0200	0.705700
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.8071	1.20721	0.603603	−	0.797285i	$-0.293731\pi$
$457$	25.8071	1.20721	0.603603	+	0.797285i	$0.293731\pi$
$458$	0	0
$459$	−26.1389	−1.22006
$460$	0	0
$461$	39.9222	1.85936	0.929681	−	0.368366i	$-0.120083\pi$
$461$	39.9222	1.85936	0.929681	+	0.368366i	$0.120083\pi$
$462$	0	0
$463$	−24.7160	−1.14865	−0.574324	−	0.818628i	$-0.694735\pi$
$463$	−24.7160	−1.14865	−0.574324	+	0.818628i	$0.694735\pi$
$464$	0	0
$465$	6.00403	0.278430
$466$	0	0
$467$	−0.671659	−0.0310807	−0.0155403	−	0.999879i	$-0.504947\pi$
$467$	−0.671659	−0.0310807	−0.0155403	+	0.999879i	$0.504947\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.90536	−0.179950
$472$	0	0
$473$	54.9322	2.52578
$474$	0	0
$475$	4.31456	0.197966
$476$	0	0
$477$	17.7166	0.811189
$478$	0	0
$479$	17.9591	0.820573	0.410286	−	0.911957i	$-0.365429\pi$
$479$	17.9591	0.820573	0.410286	+	0.911957i	$0.365429\pi$
$480$	0	0
$481$	0.156040	0.00711481
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.03267	0.319337
$486$	0	0
$487$	−29.7951	−1.35014	−0.675072	−	0.737752i	$-0.735887\pi$
$487$	−29.7951	−1.35014	−0.675072	+	0.737752i	$0.735887\pi$
$488$	0	0
$489$	−29.1660	−1.31893
$490$	0	0
$491$	11.8193	0.533397	0.266698	−	0.963780i	$-0.414067\pi$
$491$	11.8193	0.533397	0.266698	+	0.963780i	$0.414067\pi$
$492$	0	0
$493$	39.3846	1.77380
$494$	0	0
$495$	−24.4176	−1.09749
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−17.4426	−0.780839	−0.390419	−	0.920637i	$-0.627670\pi$
$499$	−17.4426	−0.780839	−0.390419	+	0.920637i	$0.627670\pi$
$500$	0	0
$501$	11.8179	0.527986
$502$	0	0
$503$	−28.8112	−1.28463	−0.642313	−	0.766442i	$-0.722025\pi$
$503$	−28.8112	−1.28463	−0.642313	+	0.766442i	$0.722025\pi$
$504$	0	0
$505$	10.2566	0.456412
$506$	0	0
$507$	37.1331	1.64914
$508$	0	0
$509$	20.6433	0.914996	0.457498	−	0.889211i	$-0.348746\pi$
$509$	20.6433	0.914996	0.457498	+	0.889211i	$0.348746\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	6.17162	0.272484
$514$	0	0
$515$	11.7788	0.519035
$516$	0	0
$517$	64.7098	2.84594
$518$	0	0
$519$	52.2316	2.29271
$520$	0	0
$521$	11.4233	0.500463	0.250232	−	0.968186i	$-0.419493\pi$
$521$	11.4233	0.500463	0.250232	+	0.968186i	$0.419493\pi$
$522$	0	0
$523$	1.56114	0.0682641	0.0341320	−	0.999417i	$-0.489133\pi$
$523$	1.56114	0.0682641	0.0341320	+	0.999417i	$0.489133\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.7520	0.468365
$528$	0	0
$529$	0.937483	0.0407601
$530$	0	0
$531$	37.9473	1.64677
$532$	0	0
$533$	0.0680528	0.00294769
$534$	0	0
$535$	0.338117	0.0146181
$536$	0	0
$537$	−3.48064	−0.150201
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.3691	0.531791	0.265896	−	0.964002i	$-0.414332\pi$
$541$	12.3691	0.531791	0.265896	+	0.964002i	$0.414332\pi$
$542$	0	0
$543$	−40.0998	−1.72085
$544$	0	0
$545$	13.4902	0.577857
$546$	0	0
$547$	9.11300	0.389644	0.194822	−	0.980839i	$-0.437587\pi$
$547$	9.11300	0.389644	0.194822	+	0.980839i	$0.437587\pi$
$548$	0	0
$549$	−12.8624	−0.548953
$550$	0	0
$551$	−9.29907	−0.396154
$552$	0	0
$553$	0	0
$554$	0	0
$555$	10.8458	0.460377
$556$	0	0
$557$	−19.8032	−0.839087	−0.419543	−	0.907735i	$-0.637810\pi$
$557$	−19.8032	−0.839087	−0.419543	+	0.907735i	$0.637810\pi$
$558$	0	0
$559$	0.327047	0.0138326
$560$	0	0
$561$	−69.1476	−2.91941
$562$	0	0
$563$	−14.1184	−0.595018	−0.297509	−	0.954719i	$-0.596156\pi$
$563$	−14.1184	−0.595018	−0.297509	+	0.954719i	$0.596156\pi$
$564$	0	0
$565$	−11.5414	−0.485552
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−43.5823	−1.82706	−0.913532	−	0.406766i	$-0.866657\pi$
$569$	−43.5823	−1.82706	−0.913532	+	0.406766i	$0.866657\pi$
$570$	0	0
$571$	42.4524	1.77658	0.888289	−	0.459286i	$-0.151895\pi$
$571$	42.4524	1.77658	0.888289	+	0.459286i	$0.151895\pi$
$572$	0	0
$573$	−23.2478	−0.971189
$574$	0	0
$575$	−21.1094	−0.880324
$576$	0	0
$577$	24.5373	1.02150	0.510750	−	0.859729i	$-0.329368\pi$
$577$	24.5373	1.02150	0.510750	+	0.859729i	$0.329368\pi$
$578$	0	0
$579$	21.0690	0.875597
$580$	0	0
$581$	0	0
$582$	0	0
$583$	19.6213	0.812630
$584$	0	0
$585$	−0.145374	−0.00601047
$586$	0	0
$587$	−17.6330	−0.727792	−0.363896	−	0.931440i	$-0.618554\pi$
$587$	−17.6330	−0.727792	−0.363896	+	0.931440i	$0.618554\pi$
$588$	0	0
$589$	−2.53865	−0.104603
$590$	0	0
$591$	−78.8497	−3.24344
$592$	0	0
$593$	36.1387	1.48404	0.742019	−	0.670378i	$-0.233868\pi$
$593$	36.1387	1.48404	0.742019	+	0.670378i	$0.233868\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.56197	0.309491
$598$	0	0
$599$	−9.58546	−0.391651	−0.195826	−	0.980639i	$-0.562739\pi$
$599$	−9.58546	−0.391651	−0.195826	+	0.980639i	$0.562739\pi$
$600$	0	0
$601$	−11.6277	−0.474303	−0.237151	−	0.971473i	$-0.576214\pi$
$601$	−11.6277	−0.474303	−0.237151	+	0.971473i	$0.576214\pi$
$602$	0	0
$603$	−36.0357	−1.46749
$604$	0	0
$605$	−17.9356	−0.729186
$606$	0	0
$607$	23.2195	0.942451	0.471225	−	0.882013i	$-0.343812\pi$
$607$	23.2195	0.942451	0.471225	+	0.882013i	$0.343812\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.385260	0.0155859
$612$	0	0
$613$	−2.90464	−0.117317	−0.0586586	−	0.998278i	$-0.518682\pi$
$613$	−2.90464	−0.117317	−0.0586586	+	0.998278i	$0.518682\pi$
$614$	0	0
$615$	4.73010	0.190736
$616$	0	0
$617$	4.32789	0.174234	0.0871171	−	0.996198i	$-0.472235\pi$
$617$	4.32789	0.174234	0.0871171	+	0.996198i	$0.472235\pi$
$618$	0	0
$619$	26.9522	1.08330	0.541651	−	0.840603i	$-0.317799\pi$
$619$	26.9522	1.08330	0.541651	+	0.840603i	$0.317799\pi$
$620$	0	0
$621$	−30.1953	−1.21169
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15.1883	0.607532
$626$	0	0
$627$	16.3264	0.652012
$628$	0	0
$629$	19.4226	0.774429
$630$	0	0
$631$	−8.31113	−0.330861	−0.165430	−	0.986221i	$-0.552901\pi$
$631$	−8.31113	−0.330861	−0.165430	+	0.986221i	$0.552901\pi$
$632$	0	0
$633$	36.4554	1.44897
$634$	0	0
$635$	−0.594687	−0.0235994
$636$	0	0
$637$	0	0
$638$	0	0
$639$	62.2222	2.46147
$640$	0	0
$641$	−31.0052	−1.22463	−0.612317	−	0.790613i	$-0.709762\pi$
$641$	−31.0052	−1.22463	−0.612317	+	0.790613i	$0.709762\pi$
$642$	0	0
$643$	−38.6660	−1.52484	−0.762420	−	0.647083i	$-0.775989\pi$
$643$	−38.6660	−1.52484	−0.762420	+	0.647083i	$0.775989\pi$
$644$	0	0
$645$	22.7318	0.895064
$646$	0	0
$647$	−33.6717	−1.32377	−0.661886	−	0.749605i	$-0.730244\pi$
$647$	−33.6717	−1.32377	−0.661886	+	0.749605i	$0.730244\pi$
$648$	0	0
$649$	42.0269	1.64970
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.3378	−0.756747	−0.378373	−	0.925653i	$-0.623516\pi$
$653$	−19.3378	−0.756747	−0.378373	+	0.925653i	$0.623516\pi$
$654$	0	0
$655$	12.1078	0.473090
$656$	0	0
$657$	−3.87700	−0.151256
$658$	0	0
$659$	−33.8064	−1.31691	−0.658456	−	0.752619i	$-0.728790\pi$
$659$	−33.8064	−1.31691	−0.658456	+	0.752619i	$0.728790\pi$
$660$	0	0
$661$	−20.1948	−0.785487	−0.392744	−	0.919648i	$-0.628474\pi$
$661$	−20.1948	−0.785487	−0.392744	+	0.919648i	$0.628474\pi$
$662$	0	0
$663$	−0.411680	−0.0159883
$664$	0	0
$665$	0	0
$666$	0	0
$667$	45.4966	1.76164
$668$	0	0
$669$	−22.5789	−0.872950
$670$	0	0
$671$	−14.2452	−0.549929
$672$	0	0
$673$	22.7317	0.876242	0.438121	−	0.898916i	$-0.355644\pi$
$673$	22.7317	0.876242	0.438121	+	0.898916i	$0.355644\pi$
$674$	0	0
$675$	26.6279	1.02491
$676$	0	0
$677$	31.9095	1.22638	0.613190	−	0.789935i	$-0.289886\pi$
$677$	31.9095	1.22638	0.613190	+	0.789935i	$0.289886\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−79.4569	−3.04479
$682$	0	0
$683$	−26.7826	−1.02481	−0.512404	−	0.858745i	$-0.671245\pi$
$683$	−26.7826	−1.02481	−0.512404	+	0.858745i	$0.671245\pi$
$684$	0	0
$685$	14.0038	0.535057
$686$	0	0
$687$	−30.2429	−1.15384
$688$	0	0
$689$	0.116818	0.00445042
$690$	0	0
$691$	41.6964	1.58621	0.793104	−	0.609087i	$-0.208464\pi$
$691$	41.6964	1.58621	0.793104	+	0.609087i	$0.208464\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.0444	−0.418937
$696$	0	0
$697$	8.47066	0.320849
$698$	0	0
$699$	−14.8461	−0.561530
$700$	0	0
$701$	−23.5645	−0.890017	−0.445009	−	0.895526i	$-0.646799\pi$
$701$	−23.5645	−0.890017	−0.445009	+	0.895526i	$0.646799\pi$
$702$	0	0
$703$	−4.58585	−0.172959
$704$	0	0
$705$	26.7780	1.00852
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8.24561	0.309670	0.154835	−	0.987940i	$-0.450515\pi$
$709$	8.24561	0.309670	0.154835	+	0.987940i	$0.450515\pi$
$710$	0	0
$711$	−61.7459	−2.31565
$712$	0	0
$713$	12.4206	0.465154
$714$	0	0
$715$	−0.161002	−0.00602114
$716$	0	0
$717$	−10.3883	−0.387960
$718$	0	0
$719$	17.6824	0.659443	0.329721	−	0.944078i	$-0.393045\pi$
$719$	17.6824	0.659443	0.329721	+	0.944078i	$0.393045\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	21.9136	0.814975
$724$	0	0
$725$	−40.1215	−1.49007
$726$	0	0
$727$	−25.0671	−0.929689	−0.464844	−	0.885392i	$-0.653890\pi$
$727$	−25.0671	−0.929689	−0.464844	+	0.885392i	$0.653890\pi$
$728$	0	0
$729$	−41.8015	−1.54821
$730$	0	0
$731$	40.7081	1.50564
$732$	0	0
$733$	0.479157	0.0176981	0.00884903	−	0.999961i	$-0.497183\pi$
$733$	0.479157	0.0176981	0.00884903	+	0.999961i	$0.497183\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−39.9097	−1.47009
$738$	0	0
$739$	−34.6171	−1.27341	−0.636706	−	0.771107i	$-0.719704\pi$
$739$	−34.6171	−1.27341	−0.636706	+	0.771107i	$0.719704\pi$
$740$	0	0
$741$	0.0972015	0.00357079
$742$	0	0
$743$	30.2246	1.10883	0.554417	−	0.832239i	$-0.312941\pi$
$743$	30.2246	1.10883	0.554417	+	0.832239i	$0.312941\pi$
$744$	0	0
$745$	9.52600	0.349006
$746$	0	0
$747$	71.0732	2.60043
$748$	0	0
$749$	0	0
$750$	0	0
$751$	27.6805	1.01008	0.505038	−	0.863097i	$-0.331478\pi$
$751$	27.6805	1.01008	0.505038	+	0.863097i	$0.331478\pi$
$752$	0	0
$753$	46.7549	1.70384
$754$	0	0
$755$	4.35307	0.158424
$756$	0	0
$757$	3.82783	0.139125	0.0695625	−	0.997578i	$-0.477840\pi$
$757$	3.82783	0.139125	0.0695625	+	0.997578i	$0.477840\pi$
$758$	0	0
$759$	−79.8783	−2.89940
$760$	0	0
$761$	−8.54790	−0.309861	−0.154931	−	0.987925i	$-0.549515\pi$
$761$	−8.54790	−0.309861	−0.154931	+	0.987925i	$0.549515\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−18.0950	−0.654224
$766$	0	0
$767$	0.250213	0.00903468
$768$	0	0
$769$	15.0750	0.543619	0.271810	−	0.962351i	$-0.412378\pi$
$769$	15.0750	0.543619	0.271810	+	0.962351i	$0.412378\pi$
$770$	0	0
$771$	−45.4710	−1.63760
$772$	0	0
$773$	0.163207	0.00587016	0.00293508	−	0.999996i	$-0.499066\pi$
$773$	0.163207	0.00587016	0.00293508	+	0.999996i	$0.499066\pi$
$774$	0	0
$775$	−10.9532	−0.393449
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	68.9115	2.46585
$782$	0	0
$783$	−57.3904	−2.05096
$784$	0	0
$785$	−1.13185	−0.0403974
$786$	0	0
$787$	−40.9903	−1.46115	−0.730573	−	0.682834i	$-0.760747\pi$
$787$	−40.9903	−1.46115	−0.730573	+	0.682834i	$0.760747\pi$
$788$	0	0
$789$	−33.6120	−1.19662
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.0848108	−0.00301172
$794$	0	0
$795$	8.11960	0.287972
$796$	0	0
$797$	−36.3930	−1.28911	−0.644554	−	0.764559i	$-0.722957\pi$
$797$	−36.3930	−1.28911	−0.644554	+	0.764559i	$0.722957\pi$
$798$	0	0
$799$	47.9540	1.69649
$800$	0	0
$801$	−27.8524	−0.984116
$802$	0	0
$803$	−4.29379	−0.151525
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−28.4753	−1.00238
$808$	0	0
$809$	−28.0557	−0.986387	−0.493193	−	0.869920i	$-0.664171\pi$
$809$	−28.0557	−0.986387	−0.493193	+	0.869920i	$0.664171\pi$
$810$	0	0
$811$	−53.7818	−1.88853	−0.944267	−	0.329179i	$-0.893228\pi$
$811$	−53.7818	−1.88853	−0.944267	+	0.329179i	$0.893228\pi$
$812$	0	0
$813$	39.8203	1.39656
$814$	0	0
$815$	−8.45285	−0.296090
$816$	0	0
$817$	−9.61156	−0.336266
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−35.1669	−1.22733	−0.613667	−	0.789565i	$-0.710306\pi$
$821$	−35.1669	−1.22733	−0.613667	+	0.789565i	$0.710306\pi$
$822$	0	0
$823$	40.3676	1.40713	0.703563	−	0.710633i	$-0.251591\pi$
$823$	40.3676	1.40713	0.703563	+	0.710633i	$0.251591\pi$
$824$	0	0
$825$	70.4412	2.45245
$826$	0	0
$827$	52.9629	1.84170	0.920851	−	0.389914i	$-0.127495\pi$
$827$	52.9629	1.84170	0.920851	+	0.389914i	$0.127495\pi$
$828$	0	0
$829$	−21.8071	−0.757392	−0.378696	−	0.925521i	$-0.623627\pi$
$829$	−21.8071	−0.757392	−0.378696	+	0.925521i	$0.623627\pi$
$830$	0	0
$831$	14.1284	0.490108
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.42506	0.118529
$836$	0	0
$837$	−15.6676	−0.541551
$838$	0	0
$839$	33.8346	1.16810	0.584051	−	0.811717i	$-0.301467\pi$
$839$	33.8346	1.16810	0.584051	+	0.811717i	$0.301467\pi$
$840$	0	0
$841$	57.4728	1.98182
$842$	0	0
$843$	28.6990	0.988447
$844$	0	0
$845$	10.7619	0.370220
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−56.8059	−1.94957
$850$	0	0
$851$	22.4367	0.769120
$852$	0	0
$853$	58.1329	1.99043	0.995217	−	0.0976929i	$-0.0311463\pi$
$853$	58.1329	1.99043	0.995217	+	0.0976929i	$0.0311463\pi$
$854$	0	0
$855$	4.27238	0.146112
$856$	0	0
$857$	−18.7326	−0.639893	−0.319947	−	0.947436i	$-0.603665\pi$
$857$	−18.7326	−0.639893	−0.319947	+	0.947436i	$0.603665\pi$
$858$	0	0
$859$	23.4300	0.799422	0.399711	−	0.916641i	$-0.369110\pi$
$859$	23.4300	0.799422	0.399711	+	0.916641i	$0.369110\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.7253	1.11398	0.556991	−	0.830519i	$-0.311956\pi$
$863$	32.7253	1.11398	0.556991	+	0.830519i	$0.311956\pi$
$864$	0	0
$865$	15.1377	0.514697
$866$	0	0
$867$	−2.67960	−0.0910039
$868$	0	0
$869$	−68.3840	−2.31977
$870$	0	0
$871$	−0.237609	−0.00805106
$872$	0	0
$873$	−43.8353	−1.48360
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.126372	0.00426727	0.00213363	−	0.999998i	$-0.499321\pi$
$877$	0.126372	0.00426727	0.00213363	+	0.999998i	$0.499321\pi$
$878$	0	0
$879$	19.8416	0.669240
$880$	0	0
$881$	24.2852	0.818190	0.409095	−	0.912492i	$-0.365844\pi$
$881$	24.2852	0.818190	0.409095	+	0.912492i	$0.365844\pi$
$882$	0	0
$883$	−53.0218	−1.78432	−0.892162	−	0.451715i	$-0.850812\pi$
$883$	−53.0218	−1.78432	−0.892162	+	0.451715i	$0.850812\pi$
$884$	0	0
$885$	17.3914	0.584606
$886$	0	0
$887$	−0.526089	−0.0176644	−0.00883218	−	0.999961i	$-0.502811\pi$
$887$	−0.526089	−0.0176644	−0.00883218	+	0.999961i	$0.502811\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	12.2811	0.411431
$892$	0	0
$893$	−11.3224	−0.378889
$894$	0	0
$895$	−1.00875	−0.0337189
$896$	0	0
$897$	−0.475568	−0.0158787
$898$	0	0
$899$	23.6071	0.787340
$900$	0	0
$901$	14.5406	0.484417
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.6217	−0.386318
$906$	0	0
$907$	11.6188	0.385795	0.192898	−	0.981219i	$-0.438211\pi$
$907$	11.6188	0.385795	0.192898	+	0.981219i	$0.438211\pi$
$908$	0	0
$909$	−63.9302	−2.12043
$910$	0	0
$911$	2.23613	0.0740862	0.0370431	−	0.999314i	$-0.488206\pi$
$911$	2.23613	0.0740862	0.0370431	+	0.999314i	$0.488206\pi$
$912$	0	0
$913$	78.7139	2.60505
$914$	0	0
$915$	−5.89488	−0.194879
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−37.9884	−1.25312	−0.626561	−	0.779373i	$-0.715538\pi$
$919$	−37.9884	−1.25312	−0.626561	+	0.779373i	$0.715538\pi$
$920$	0	0
$921$	−59.9246	−1.97458
$922$	0	0
$923$	0.410275	0.0135044
$924$	0	0
$925$	−19.7859	−0.650558
$926$	0	0
$927$	−73.4183	−2.41137
$928$	0	0
$929$	−20.7798	−0.681762	−0.340881	−	0.940106i	$-0.610725\pi$
$929$	−20.7798	−0.681762	−0.340881	+	0.940106i	$0.610725\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	21.3118	0.697718
$934$	0	0
$935$	−20.0403	−0.655386
$936$	0	0
$937$	39.6999	1.29694	0.648470	−	0.761240i	$-0.275409\pi$
$937$	39.6999	1.29694	0.648470	+	0.761240i	$0.275409\pi$
$938$	0	0
$939$	−20.6469	−0.673785
$940$	0	0
$941$	43.5893	1.42097	0.710485	−	0.703713i	$-0.248476\pi$
$941$	43.5893	1.42097	0.710485	+	0.703713i	$0.248476\pi$
$942$	0	0
$943$	9.78519	0.318650
$944$	0	0
$945$	0	0
$946$	0	0
$947$	56.2382	1.82749	0.913747	−	0.406283i	$-0.133175\pi$
$947$	56.2382	1.82749	0.913747	+	0.406283i	$0.133175\pi$
$948$	0	0
$949$	−0.0255638	−0.000829835 0
$950$	0	0
$951$	16.4534	0.533537
$952$	0	0
$953$	−8.77476	−0.284242	−0.142121	−	0.989849i	$-0.545392\pi$
$953$	−8.77476	−0.284242	−0.142121	+	0.989849i	$0.545392\pi$
$954$	0	0
$955$	−6.73764	−0.218025
$956$	0	0
$957$	−151.820	−4.90765
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24.5553	−0.792105
$962$	0	0
$963$	−2.10751	−0.0679137
$964$	0	0
$965$	6.10619	0.196565
$966$	0	0
$967$	56.7355	1.82449	0.912245	−	0.409644i	$-0.134347\pi$
$967$	56.7355	1.82449	0.912245	+	0.409644i	$0.134347\pi$
$968$	0	0
$969$	12.0989	0.388671
$970$	0	0
$971$	−34.9737	−1.12236	−0.561179	−	0.827694i	$-0.689652\pi$
$971$	−34.9737	−1.12236	−0.561179	+	0.827694i	$0.689652\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0.419382	0.0134310
$976$	0	0
$977$	37.2572	1.19196	0.595982	−	0.802998i	$-0.296763\pi$
$977$	37.2572	1.19196	0.595982	+	0.802998i	$0.296763\pi$
$978$	0	0
$979$	−30.8467	−0.985864
$980$	0	0
$981$	−84.0858	−2.68465
$982$	0	0
$983$	−20.8075	−0.663657	−0.331829	−	0.943340i	$-0.607666\pi$
$983$	−20.8075	−0.663657	−0.331829	+	0.943340i	$0.607666\pi$
$984$	0	0
$985$	−22.8521	−0.728129
$986$	0	0
$987$	0	0
$988$	0	0
$989$	47.0255	1.49532
$990$	0	0
$991$	−16.2219	−0.515305	−0.257653	−	0.966238i	$-0.582949\pi$
$991$	−16.2219	−0.515305	−0.257653	+	0.966238i	$0.582949\pi$
$992$	0	0
$993$	−34.5233	−1.09556
$994$	0	0
$995$	2.19160	0.0694784
$996$	0	0
$997$	5.65999	0.179254	0.0896268	−	0.995975i	$-0.471433\pi$
$997$	5.65999	0.179254	0.0896268	+	0.995975i	$0.471433\pi$
$998$	0	0
$999$	−28.3021	−0.895440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.br.1.1		8
7.3	odd	6		1064.2.q.n.457.1	yes	16
7.5	odd	6		1064.2.q.n.305.1	✓	16
7.6	odd	2		7448.2.a.bq.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.n.305.1	✓	16	7.5	odd	6
1064.2.q.n.457.1	yes	16	7.3	odd	6
7448.2.a.bq.1.8		8	7.6	odd	2
7448.2.a.br.1.1		8	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-2.85665 q^{3} -0.827910 q^{5} +5.16044 q^{9} +O(q^{10})\)	\(q-2.85665 q^{3} -0.827910 q^{5} +5.16044 q^{9} +5.71522 q^{11} +0.0340264 q^{13} +2.36505 q^{15} +4.23533 q^{17} -1.00000 q^{19} +4.89259 q^{23} -4.31456 q^{25} -6.17162 q^{27} +9.29907 q^{29} +2.53865 q^{31} -16.3264 q^{33} +4.58585 q^{37} -0.0972015 q^{39} +2.00000 q^{41} +9.61156 q^{43} -4.27238 q^{45} +11.3224 q^{47} -12.0989 q^{51} +3.43316 q^{53} -4.73169 q^{55} +2.85665 q^{57} +7.35351 q^{59} -2.49250 q^{61} -0.0281708 q^{65} -6.98306 q^{67} -13.9764 q^{69} +12.0575 q^{71} -0.751292 q^{73} +12.3252 q^{75} -11.9652 q^{79} +2.14883 q^{81} +13.7727 q^{83} -3.50647 q^{85} -26.5642 q^{87} -5.39729 q^{89} -7.25203 q^{93} +0.827910 q^{95} -8.49448 q^{97} +29.4930 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{5} + 6 q^{9}+O(q^{10}) \) 8 * q + q^5 + 6 * q^9	\( 8 q + q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} - 4 q^{17} - 8 q^{19} + 25 q^{23} + 15 q^{25} - 3 q^{27} + 6 q^{29} - 8 q^{33} + 13 q^{37} - 11 q^{39} + 16 q^{41} + 17 q^{43} - 17 q^{45} + 24 q^{47} + 5 q^{51} + 2 q^{53} - 5 q^{55} - 2 q^{59} + 13 q^{61} - 26 q^{65} + 2 q^{67} + 11 q^{69} + 10 q^{71} - 5 q^{73} + 20 q^{75} + 16 q^{79} - 12 q^{81} + 43 q^{83} + 24 q^{85} - 20 q^{87} - 8 q^{89} + 2 q^{93} - q^{95} + 12 q^{97} + 37 q^{99}+O(q^{100}) \) 8 * q + q^5 + 6 * q^9 + 9 * q^11 + 8 * q^15 - 4 * q^17 - 8 * q^19 + 25 * q^23 + 15 * q^25 - 3 * q^27 + 6 * q^29 - 8 * q^33 + 13 * q^37 - 11 * q^39 + 16 * q^41 + 17 * q^43 - 17 * q^45 + 24 * q^47 + 5 * q^51 + 2 * q^53 - 5 * q^55 - 2 * q^59 + 13 * q^61 - 26 * q^65 + 2 * q^67 + 11 * q^69 + 10 * q^71 - 5 * q^73 + 20 * q^75 + 16 * q^79 - 12 * q^81 + 43 * q^83 + 24 * q^85 - 20 * q^87 - 8 * q^89 + 2 * q^93 - q^95 + 12 * q^97 + 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more