LMFDB - Embedded newform 4275.2.a.bf.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.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:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1425)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.36147$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.36147 q^{2} +3.57653 q^{4} -0.784934 q^{7} +3.72294 q^{8} -2.93800 q^{11} -4.00000 q^{13} -1.85360 q^{14} +1.63853 q^{16} -5.15307 q^{17} -1.00000 q^{19} -6.93800 q^{22} -3.78493 q^{23} -9.44588 q^{26} -2.80734 q^{28} +5.00000 q^{29} +1.06200 q^{31} -3.57653 q^{32} -12.1688 q^{34} +0.722938 q^{37} -2.36147 q^{38} -7.93800 q^{41} -4.00000 q^{43} -10.5079 q^{44} -8.93800 q^{46} -3.87601 q^{47} -6.38388 q^{49} -14.3061 q^{52} +6.44588 q^{53} -2.92226 q^{56} +11.8073 q^{58} +4.66094 q^{59} +8.87601 q^{61} +2.50787 q^{62} -11.7229 q^{64} -8.93800 q^{67} -18.4301 q^{68} +2.66094 q^{71} +8.44588 q^{73} +1.70719 q^{74} -3.57653 q^{76} +2.30614 q^{77} +3.35480 q^{79} -18.7453 q^{82} +4.93800 q^{83} -9.44588 q^{86} -10.9380 q^{88} -0.569868 q^{89} +3.13974 q^{91} -13.5369 q^{92} -9.15307 q^{94} -8.59894 q^{97} -15.0753 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} - 3 q^{8} + 3 q^{11} - 12 q^{13} - 15 q^{14} + 12 q^{16} - 6 q^{17} - 3 q^{19} - 9 q^{22} - 9 q^{23} + 27 q^{28} + 15 q^{29} + 15 q^{31} - 6 q^{32} + 6 q^{34} - 12 q^{37} - 12 q^{41} - 12 q^{43}+ \cdots - 57 q^{98}+O(q^{100}) $ 3 * q + 6 * q^4 - 3 * q^8 + 3 * q^11 - 12 * q^13 - 15 * q^14 + 12 * q^16 - 6 * q^17 - 3 * q^19 - 9 * q^22 - 9 * q^23 + 27 * q^28 + 15 * q^29 + 15 * q^31 - 6 * q^32 + 6 * q^34 - 12 * q^37 - 12 * q^41 - 12 * q^43 - 15 * q^44 - 15 * q^46 + 12 * q^47 + 21 * q^49 - 24 * q^52 - 9 * q^53 - 30 * q^56 - 12 * q^59 + 3 * q^61 - 9 * q^62 - 21 * q^64 - 15 * q^67 - 60 * q^68 - 18 * q^71 - 3 * q^73 + 24 * q^74 - 6 * q^76 - 12 * q^77 + 3 * q^79 - 9 * q^82 + 3 * q^83 - 21 * q^88 + 3 * q^89 + 9 * q^92 - 18 * q^94 + 12 * q^97 - 57 * q^98

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 $

$2$	2.36147	1.66981	0.834905	−	0.550394i	$-0.185522\pi$
$2$	2.36147	1.66981	0.834905	+	0.550394i	$0.185522\pi$
$3$	0	0
$3$	0	0
$4$	3.57653	1.78827
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.784934	−0.296677	−0.148339	−	0.988937i	$-0.547393\pi$
$7$	−0.784934	−0.296677	−0.148339	+	0.988937i	$0.547393\pi$
$8$	3.72294	1.31626
$9$	0	0
$10$	0	0
$11$	−2.93800	−0.885841	−0.442921	−	0.896561i	$-0.646058\pi$
$11$	−2.93800	−0.885841	−0.442921	+	0.896561i	$0.646058\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−1.85360	−0.495395
$15$	0	0
$16$	1.63853	0.409633
$17$	−5.15307	−1.24980	−0.624901	−	0.780704i	$-0.714861\pi$
$17$	−5.15307	−1.24980	−0.624901	+	0.780704i	$0.714861\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−6.93800	−1.47919
$23$	−3.78493	−0.789213	−0.394607	−	0.918850i	$-0.629119\pi$
$23$	−3.78493	−0.789213	−0.394607	+	0.918850i	$0.629119\pi$
$24$	0	0
$25$	0	0
$26$	−9.44588	−1.85249
$27$	0	0
$28$	−2.80734	−0.530538
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	1.06200	0.190740	0.0953701	−	0.995442i	$-0.469597\pi$
$31$	1.06200	0.190740	0.0953701	+	0.995442i	$0.469597\pi$
$32$	−3.57653	−0.632248
$33$	0	0
$34$	−12.1688	−2.08693
$35$	0	0
$36$	0	0
$37$	0.722938	0.118850	0.0594251	−	0.998233i	$-0.481073\pi$
$37$	0.722938	0.118850	0.0594251	+	0.998233i	$0.481073\pi$
$38$	−2.36147	−0.383081
$39$	0	0
$40$	0	0
$41$	−7.93800	−1.23971	−0.619854	−	0.784717i	$-0.712808\pi$
$41$	−7.93800	−1.23971	−0.619854	+	0.784717i	$0.712808\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−10.5079	−1.58412
$45$	0	0
$46$	−8.93800	−1.31784
$47$	−3.87601	−0.565374	−0.282687	−	0.959212i	$-0.591226\pi$
$47$	−3.87601	−0.565374	−0.282687	+	0.959212i	$0.591226\pi$
$48$	0	0
$49$	−6.38388	−0.911983
$50$	0	0
$51$	0	0
$52$	−14.3061	−1.98390
$53$	6.44588	0.885409	0.442705	−	0.896668i	$-0.354019\pi$
$53$	6.44588	0.885409	0.442705	+	0.896668i	$0.354019\pi$
$54$	0	0
$55$	0	0
$56$	−2.92226	−0.390503
$57$	0	0
$58$	11.8073	1.55038
$59$	4.66094	0.606803	0.303401	−	0.952863i	$-0.401878\pi$
$59$	4.66094	0.606803	0.303401	+	0.952863i	$0.401878\pi$
$60$	0	0
$61$	8.87601	1.13646	0.568228	−	0.822871i	$-0.307629\pi$
$61$	8.87601	1.13646	0.568228	+	0.822871i	$0.307629\pi$
$62$	2.50787	0.318500
$63$	0	0
$64$	−11.7229	−1.46537
$65$	0	0
$66$	0	0
$67$	−8.93800	−1.09195	−0.545975	−	0.837801i	$-0.683841\pi$
$67$	−8.93800	−1.09195	−0.545975	+	0.837801i	$0.683841\pi$
$68$	−18.4301	−2.23498
$69$	0	0
$70$	0	0
$71$	2.66094	0.315796	0.157898	−	0.987455i	$-0.449528\pi$
$71$	2.66094	0.315796	0.157898	+	0.987455i	$0.449528\pi$
$72$	0	0
$73$	8.44588	0.988515	0.494257	−	0.869316i	$-0.335440\pi$
$73$	8.44588	0.988515	0.494257	+	0.869316i	$0.335440\pi$
$74$	1.70719	0.198457
$75$	0	0
$76$	−3.57653	−0.410257
$77$	2.30614	0.262809
$78$	0	0
$79$	3.35480	0.377445	0.188722	−	0.982030i	$-0.439565\pi$
$79$	3.35480	0.377445	0.188722	+	0.982030i	$0.439565\pi$
$80$	0	0
$81$	0	0
$82$	−18.7453	−2.07008
$83$	4.93800	0.542016	0.271008	−	0.962577i	$-0.412643\pi$
$83$	4.93800	0.542016	0.271008	+	0.962577i	$0.412643\pi$
$84$	0	0
$85$	0	0
$86$	−9.44588	−1.01857
$87$	0	0
$88$	−10.9380	−1.16600
$89$	−0.569868	−0.0604059	−0.0302029	−	0.999544i	$-0.509615\pi$
$89$	−0.569868	−0.0604059	−0.0302029	+	0.999544i	$0.509615\pi$
$90$	0	0
$91$	3.13974	0.329134
$92$	−13.5369	−1.41132
$93$	0	0
$94$	−9.15307	−0.944067
$95$	0	0
$96$	0	0
$97$	−8.59894	−0.873091	−0.436545	−	0.899682i	$-0.643798\pi$
$97$	−8.59894	−0.873091	−0.436545	+	0.899682i	$0.643798\pi$
$98$	−15.0753	−1.52284
$99$	0	0
$100$	0	0
$101$	1.15307	0.114735	0.0573674	−	0.998353i	$-0.481729\pi$
$101$	1.15307	0.114735	0.0573674	+	0.998353i	$0.481729\pi$
$102$	0	0
$103$	19.9537	1.96610	0.983051	−	0.183335i	$-0.0586892\pi$
$103$	19.9537	1.96610	0.983051	+	0.183335i	$0.0586892\pi$
$104$	−14.8918	−1.46026
$105$	0	0
$106$	15.2217	1.47847
$107$	−12.2308	−1.18240	−0.591198	−	0.806526i	$-0.701345\pi$
$107$	−12.2308	−1.18240	−0.591198	+	0.806526i	$0.701345\pi$
$108$	0	0
$109$	−2.41680	−0.231487	−0.115744	−	0.993279i	$-0.536925\pi$
$109$	−2.41680	−0.231487	−0.115744	+	0.993279i	$0.536925\pi$
$110$	0	0
$111$	0	0
$112$	−1.28614	−0.121529
$113$	−20.6900	−1.94635	−0.973177	−	0.230060i	$-0.926108\pi$
$113$	−20.6900	−1.94635	−0.973177	+	0.230060i	$0.926108\pi$
$114$	0	0
$115$	0	0
$116$	17.8827	1.66036
$117$	0	0
$118$	11.0067	1.01325
$119$	4.04482	0.370788
$120$	0	0
$121$	−2.36814	−0.215285
$122$	20.9604	1.89767
$123$	0	0
$124$	3.79827	0.341094
$125$	0	0
$126$	0	0
$127$	13.5369	1.20121	0.600605	−	0.799546i	$-0.294926\pi$
$127$	13.5369	1.20121	0.600605	+	0.799546i	$0.294926\pi$
$128$	−20.5303	−1.81464
$129$	0	0
$130$	0	0
$131$	4.50787	0.393855	0.196927	−	0.980418i	$-0.436904\pi$
$131$	4.50787	0.393855	0.196927	+	0.980418i	$0.436904\pi$
$132$	0	0
$133$	0.784934	0.0680624
$134$	−21.1068	−1.82335
$135$	0	0
$136$	−19.1846	−1.64506
$137$	−16.2928	−1.39199	−0.695994	−	0.718047i	$-0.745036\pi$
$137$	−16.2928	−1.39199	−0.695994	+	0.718047i	$0.745036\pi$
$138$	0	0
$139$	−2.78493	−0.236215	−0.118108	−	0.993001i	$-0.537683\pi$
$139$	−2.78493	−0.236215	−0.118108	+	0.993001i	$0.537683\pi$
$140$	0	0
$141$	0	0
$142$	6.28373	0.527319
$143$	11.7520	0.982753
$144$	0	0
$145$	0	0
$146$	19.9447	1.65063
$147$	0	0
$148$	2.58561	0.212536
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	22.5989	1.83908	0.919538	−	0.393001i	$-0.128563\pi$
$151$	22.5989	1.83908	0.919538	+	0.393001i	$0.128563\pi$
$152$	−3.72294	−0.301970
$153$	0	0
$154$	5.44588	0.438841
$155$	0	0
$156$	0	0
$157$	14.2441	1.13681	0.568403	−	0.822750i	$-0.307561\pi$
$157$	14.2441	1.13681	0.568403	+	0.822750i	$0.307561\pi$
$158$	7.92226	0.630261
$159$	0	0
$160$	0	0
$161$	2.97092	0.234142
$162$	0	0
$163$	19.9828	1.56518	0.782588	−	0.622540i	$-0.213899\pi$
$163$	19.9828	1.56518	0.782588	+	0.622540i	$0.213899\pi$
$164$	−28.3905	−2.21693
$165$	0	0
$166$	11.6609	0.905065
$167$	−14.7849	−1.14409	−0.572046	−	0.820221i	$-0.693850\pi$
$167$	−14.7849	−1.14409	−0.572046	+	0.820221i	$0.693850\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	−14.3061	−1.09083
$173$	22.7520	1.72980	0.864902	−	0.501941i	$-0.167381\pi$
$173$	22.7520	1.72980	0.864902	+	0.501941i	$0.167381\pi$
$174$	0	0
$175$	0	0
$176$	−4.81401	−0.362870
$177$	0	0
$178$	−1.34573	−0.100866
$179$	−22.1068	−1.65234	−0.826171	−	0.563420i	$-0.809485\pi$
$179$	−22.1068	−1.65234	−0.826171	+	0.563420i	$0.809485\pi$
$180$	0	0
$181$	13.1979	0.980991	0.490496	−	0.871444i	$-0.336816\pi$
$181$	13.1979	0.980991	0.490496	+	0.871444i	$0.336816\pi$
$182$	7.41439	0.549591
$183$	0	0
$184$	−14.0911	−1.03881
$185$	0	0
$186$	0	0
$187$	15.1397	1.10713
$188$	−13.8627	−1.01104
$189$	0	0
$190$	0	0
$191$	−13.6743	−0.989436	−0.494718	−	0.869054i	$-0.664729\pi$
$191$	−13.6743	−0.989436	−0.494718	+	0.869054i	$0.664729\pi$
$192$	0	0
$193$	−24.5989	−1.77067	−0.885335	−	0.464953i	$-0.846071\pi$
$193$	−24.5989	−1.77067	−0.885335	+	0.464953i	$0.846071\pi$
$194$	−20.3061	−1.45790
$195$	0	0
$196$	−22.8322	−1.63087
$197$	−25.6147	−1.82497	−0.912485	−	0.409109i	$-0.865840\pi$
$197$	−25.6147	−1.82497	−0.912485	+	0.409109i	$0.865840\pi$
$198$	0	0
$199$	−0.230809	−0.0163616	−0.00818081	−	0.999967i	$-0.502604\pi$
$199$	−0.230809	−0.0163616	−0.00818081	+	0.999967i	$0.502604\pi$
$200$	0	0
$201$	0	0
$202$	2.72294	0.191585
$203$	−3.92467	−0.275458
$204$	0	0
$205$	0	0
$206$	47.1201	3.28302
$207$	0	0
$208$	−6.55412	−0.454447
$209$	2.93800	0.203226
$210$	0	0
$211$	5.35480	0.368640	0.184320	−	0.982866i	$-0.440992\pi$
$211$	5.35480	0.368640	0.184320	+	0.982866i	$0.440992\pi$
$212$	23.0539	1.58335
$213$	0	0
$214$	−28.8827	−1.97438
$215$	0	0
$216$	0	0
$217$	−0.833597	−0.0565883
$218$	−5.70719	−0.386540
$219$	0	0
$220$	0	0
$221$	20.6123	1.38653
$222$	0	0
$223$	−1.78493	−0.119528	−0.0597640	−	0.998213i	$-0.519035\pi$
$223$	−1.78493	−0.119528	−0.0597640	+	0.998213i	$0.519035\pi$
$224$	2.80734	0.187574
$225$	0	0
$226$	−48.8588	−3.25004
$227$	8.96708	0.595166	0.297583	−	0.954696i	$-0.403819\pi$
$227$	8.96708	0.595166	0.297583	+	0.954696i	$0.403819\pi$
$228$	0	0
$229$	26.5660	1.75553	0.877766	−	0.479089i	$-0.159033\pi$
$229$	26.5660	1.75553	0.877766	+	0.479089i	$0.159033\pi$
$230$	0	0
$231$	0	0
$232$	18.6147	1.22211
$233$	9.56987	0.626943	0.313471	−	0.949598i	$-0.398508\pi$
$233$	9.56987	0.626943	0.313471	+	0.949598i	$0.398508\pi$
$234$	0	0
$235$	0	0
$236$	16.6700	1.08513
$237$	0	0
$238$	9.55172	0.619146
$239$	−16.9051	−1.09350	−0.546749	−	0.837296i	$-0.684135\pi$
$239$	−16.9051	−1.09350	−0.546749	+	0.837296i	$0.684135\pi$
$240$	0	0
$241$	−9.27706	−0.597588	−0.298794	−	0.954318i	$-0.596584\pi$
$241$	−9.27706	−0.597588	−0.298794	+	0.954318i	$0.596584\pi$
$242$	−5.59228	−0.359485
$243$	0	0
$244$	31.7453	2.03229
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	3.95375	0.251063
$249$	0	0
$250$	0	0
$251$	−25.1531	−1.58765	−0.793824	−	0.608148i	$-0.791913\pi$
$251$	−25.1531	−1.58765	−0.793824	+	0.608148i	$0.791913\pi$
$252$	0	0
$253$	11.1201	0.699118
$254$	31.9671	2.00579
$255$	0	0
$256$	−25.0357	−1.56473
$257$	21.1821	1.32131	0.660653	−	0.750691i	$-0.270279\pi$
$257$	21.1821	1.32131	0.660653	+	0.750691i	$0.270279\pi$
$258$	0	0
$259$	−0.567458	−0.0352601
$260$	0	0
$261$	0	0
$262$	10.6452	0.657663
$263$	−3.78493	−0.233389	−0.116695	−	0.993168i	$-0.537230\pi$
$263$	−3.78493	−0.233389	−0.116695	+	0.993168i	$0.537230\pi$
$264$	0	0
$265$	0	0
$266$	1.85360	0.113651
$267$	0	0
$268$	−31.9671	−1.95270
$269$	−16.3061	−0.994203	−0.497101	−	0.867692i	$-0.665602\pi$
$269$	−16.3061	−0.994203	−0.497101	+	0.867692i	$0.665602\pi$
$270$	0	0
$271$	−2.66094	−0.161641	−0.0808203	−	0.996729i	$-0.525754\pi$
$271$	−2.66094	−0.161641	−0.0808203	+	0.996729i	$0.525754\pi$
$272$	−8.44347	−0.511960
$273$	0	0
$274$	−38.4750	−2.32436
$275$	0	0
$276$	0	0
$277$	−21.1821	−1.27271	−0.636356	−	0.771396i	$-0.719559\pi$
$277$	−21.1821	−1.27271	−0.636356	+	0.771396i	$0.719559\pi$
$278$	−6.57653	−0.394434
$279$	0	0
$280$	0	0
$281$	26.3839	1.57393	0.786965	−	0.616997i	$-0.211651\pi$
$281$	26.3839	1.57393	0.786965	+	0.616997i	$0.211651\pi$
$282$	0	0
$283$	−18.8918	−1.12300	−0.561499	−	0.827477i	$-0.689775\pi$
$283$	−18.8918	−1.12300	−0.561499	+	0.827477i	$0.689775\pi$
$284$	9.51695	0.564727
$285$	0	0
$286$	27.7520	1.64101
$287$	6.23081	0.367793
$288$	0	0
$289$	9.55412	0.562007
$290$	0	0
$291$	0	0
$292$	30.2070	1.76773
$293$	7.49213	0.437695	0.218847	−	0.975759i	$-0.429770\pi$
$293$	7.49213	0.437695	0.218847	+	0.975759i	$0.429770\pi$
$294$	0	0
$295$	0	0
$296$	2.69145	0.156437
$297$	0	0
$298$	0	0
$299$	15.1397	0.875554
$300$	0	0
$301$	3.13974	0.180971
$302$	53.3667	3.07091
$303$	0	0
$304$	−1.63853	−0.0939762
$305$	0	0
$306$	0	0
$307$	−20.8140	−1.18792	−0.593959	−	0.804495i	$-0.702436\pi$
$307$	−20.8140	−1.18792	−0.593959	+	0.804495i	$0.702436\pi$
$308$	8.24799	0.469973
$309$	0	0
$310$	0	0
$311$	−26.8918	−1.52489	−0.762446	−	0.647052i	$-0.776002\pi$
$311$	−26.8918	−1.52489	−0.762446	+	0.647052i	$0.776002\pi$
$312$	0	0
$313$	−24.2599	−1.37125	−0.685625	−	0.727955i	$-0.740471\pi$
$313$	−24.2599	−1.37125	−0.685625	+	0.727955i	$0.740471\pi$
$314$	33.6371	1.89825
$315$	0	0
$316$	11.9986	0.674972
$317$	−23.3061	−1.30900	−0.654502	−	0.756061i	$-0.727121\pi$
$317$	−23.3061	−1.30900	−0.654502	+	0.756061i	$0.727121\pi$
$318$	0	0
$319$	−14.6900	−0.822483
$320$	0	0
$321$	0	0
$322$	7.01574	0.390972
$323$	5.15307	0.286724
$324$	0	0
$325$	0	0
$326$	47.1888	2.61355
$327$	0	0
$328$	−29.5527	−1.63177
$329$	3.04241	0.167733
$330$	0	0
$331$	−23.8298	−1.30980	−0.654901	−	0.755715i	$-0.727290\pi$
$331$	−23.8298	−1.30980	−0.654901	+	0.755715i	$0.727290\pi$
$332$	17.6609	0.969270
$333$	0	0
$334$	−34.9142	−1.91042
$335$	0	0
$336$	0	0
$337$	6.43013	0.350272	0.175136	−	0.984544i	$-0.443964\pi$
$337$	6.43013	0.350272	0.175136	+	0.984544i	$0.443964\pi$
$338$	7.08441	0.385341
$339$	0	0
$340$	0	0
$341$	−3.12015	−0.168966
$342$	0	0
$343$	10.5055	0.567242
$344$	−14.8918	−0.802909
$345$	0	0
$346$	53.7282	2.88844
$347$	9.27706	0.498019	0.249009	−	0.968501i	$-0.419895\pi$
$347$	9.27706	0.498019	0.249009	+	0.968501i	$0.419895\pi$
$348$	0	0
$349$	−8.01574	−0.429073	−0.214536	−	0.976716i	$-0.568824\pi$
$349$	−8.01574	−0.429073	−0.214536	+	0.976716i	$0.568824\pi$
$350$	0	0
$351$	0	0
$352$	10.5079	0.560071
$353$	−22.3061	−1.18724	−0.593618	−	0.804747i	$-0.702301\pi$
$353$	−22.3061	−1.18724	−0.593618	+	0.804747i	$0.702301\pi$
$354$	0	0
$355$	0	0
$356$	−2.03815	−0.108022
$357$	0	0
$358$	−52.2046	−2.75910
$359$	9.40106	0.496169	0.248084	−	0.968738i	$-0.420199\pi$
$359$	9.40106	0.496169	0.248084	+	0.968738i	$0.420199\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	31.1664	1.63807
$363$	0	0
$364$	11.2294	0.588579
$365$	0	0
$366$	0	0
$367$	−7.32188	−0.382199	−0.191100	−	0.981571i	$-0.561205\pi$
$367$	−7.32188	−0.382199	−0.191100	+	0.981571i	$0.561205\pi$
$368$	−6.20173	−0.323288
$369$	0	0
$370$	0	0
$371$	−5.05959	−0.262681
$372$	0	0
$373$	−9.02908	−0.467508	−0.233754	−	0.972296i	$-0.575101\pi$
$373$	−9.02908	−0.467508	−0.233754	+	0.972296i	$0.575101\pi$
$374$	35.7520	1.84869
$375$	0	0
$376$	−14.4301	−0.744177
$377$	−20.0000	−1.03005
$378$	0	0
$379$	28.0448	1.44057	0.720283	−	0.693681i	$-0.244012\pi$
$379$	28.0448	1.44057	0.720283	+	0.693681i	$0.244012\pi$
$380$	0	0
$381$	0	0
$382$	−32.2914	−1.65217
$383$	−33.6767	−1.72080	−0.860399	−	0.509621i	$-0.829786\pi$
$383$	−33.6767	−1.72080	−0.860399	+	0.509621i	$0.829786\pi$
$384$	0	0
$385$	0	0
$386$	−58.0896	−2.95668
$387$	0	0
$388$	−30.7544	−1.56132
$389$	−0.678118	−0.0343819	−0.0171910	−	0.999852i	$-0.505472\pi$
$389$	−0.678118	−0.0343819	−0.0171910	+	0.999852i	$0.505472\pi$
$390$	0	0
$391$	19.5040	0.986361
$392$	−23.7668	−1.20040
$393$	0	0
$394$	−60.4883	−3.04736
$395$	0	0
$396$	0	0
$397$	−10.8140	−0.542740	−0.271370	−	0.962475i	$-0.587477\pi$
$397$	−10.8140	−0.542740	−0.271370	+	0.962475i	$0.587477\pi$
$398$	−0.545048	−0.0273208
$399$	0	0
$400$	0	0
$401$	28.2599	1.41123	0.705616	−	0.708595i	$-0.250671\pi$
$401$	28.2599	1.41123	0.705616	+	0.708595i	$0.250671\pi$
$402$	0	0
$403$	−4.24799	−0.211607
$404$	4.12399	0.205176
$405$	0	0
$406$	−9.26799	−0.459962
$407$	−2.12399	−0.105282
$408$	0	0
$409$	29.1846	1.44308	0.721542	−	0.692371i	$-0.243434\pi$
$409$	29.1846	1.44308	0.721542	+	0.692371i	$0.243434\pi$
$410$	0	0
$411$	0	0
$412$	71.3653	3.51591
$413$	−3.65853	−0.180025
$414$	0	0
$415$	0	0
$416$	14.3061	0.701416
$417$	0	0
$418$	6.93800	0.339349
$419$	26.7678	1.30769	0.653845	−	0.756628i	$-0.273155\pi$
$419$	26.7678	1.30769	0.653845	+	0.756628i	$0.273155\pi$
$420$	0	0
$421$	29.5040	1.43794	0.718969	−	0.695042i	$-0.244614\pi$
$421$	29.5040	1.43794	0.718969	+	0.695042i	$0.244614\pi$
$422$	12.6452	0.615559
$423$	0	0
$424$	23.9976	1.16543
$425$	0	0
$426$	0	0
$427$	−6.96708	−0.337161
$428$	−43.7439	−2.11444
$429$	0	0
$430$	0	0
$431$	−2.90893	−0.140118	−0.0700590	−	0.997543i	$-0.522319\pi$
$431$	−2.90893	−0.140118	−0.0700590	+	0.997543i	$0.522319\pi$
$432$	0	0
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−1.96851	−0.0944917
$435$	0	0
$436$	−8.64376	−0.413961
$437$	3.78493	0.181058
$438$	0	0
$439$	33.4130	1.59471	0.797357	−	0.603508i	$-0.206231\pi$
$439$	33.4130	1.59471	0.797357	+	0.603508i	$0.206231\pi$
$440$	0	0
$441$	0	0
$442$	48.6753	2.31525
$443$	23.5818	1.12040	0.560202	−	0.828356i	$-0.310724\pi$
$443$	23.5818	1.12040	0.560202	+	0.828356i	$0.310724\pi$
$444$	0	0
$445$	0	0
$446$	−4.21507	−0.199589
$447$	0	0
$448$	9.20173	0.434741
$449$	6.87601	0.324499	0.162249	−	0.986750i	$-0.448125\pi$
$449$	6.87601	0.324499	0.162249	+	0.986750i	$0.448125\pi$
$450$	0	0
$451$	23.3219	1.09818
$452$	−73.9986	−3.48060
$453$	0	0
$454$	21.1755	0.993814
$455$	0	0
$456$	0	0
$457$	−33.2599	−1.55583	−0.777916	−	0.628368i	$-0.783723\pi$
$457$	−33.2599	−1.55583	−0.777916	+	0.628368i	$0.783723\pi$
$458$	62.7348	2.93141
$459$	0	0
$460$	0	0
$461$	−16.2928	−0.758832	−0.379416	−	0.925226i	$-0.623875\pi$
$461$	−16.2928	−0.758832	−0.379416	+	0.925226i	$0.623875\pi$
$462$	0	0
$463$	33.4459	1.55436	0.777181	−	0.629277i	$-0.216649\pi$
$463$	33.4459	1.55436	0.777181	+	0.629277i	$0.216649\pi$
$464$	8.19266	0.380335
$465$	0	0
$466$	22.5989	1.04688
$467$	−20.5212	−0.949608	−0.474804	−	0.880092i	$-0.657481\pi$
$467$	−20.5212	−0.949608	−0.474804	+	0.880092i	$0.657481\pi$
$468$	0	0
$469$	7.01574	0.323957
$470$	0	0
$471$	0	0
$472$	17.3524	0.798709
$473$	11.7520	0.540358
$474$	0	0
$475$	0	0
$476$	14.4664	0.663068
$477$	0	0
$478$	−39.9208	−1.82594
$479$	7.84309	0.358360	0.179180	−	0.983816i	$-0.442656\pi$
$479$	7.84309	0.358360	0.179180	+	0.983816i	$0.442656\pi$
$480$	0	0
$481$	−2.89175	−0.131852
$482$	−21.9075	−0.997859
$483$	0	0
$484$	−8.46972	−0.384987
$485$	0	0
$486$	0	0
$487$	7.16640	0.324741	0.162370	−	0.986730i	$-0.448086\pi$
$487$	7.16640	0.324741	0.162370	+	0.986730i	$0.448086\pi$
$488$	33.0448	1.49587
$489$	0	0
$490$	0	0
$491$	−16.2928	−0.735284	−0.367642	−	0.929967i	$-0.619835\pi$
$491$	−16.2928	−0.735284	−0.367642	+	0.929967i	$0.619835\pi$
$492$	0	0
$493$	−25.7653	−1.16041
$494$	9.44588	0.424990
$495$	0	0
$496$	1.74011	0.0781334
$497$	−2.08866	−0.0936893
$498$	0	0
$499$	40.2890	1.80358	0.901791	−	0.432173i	$-0.142253\pi$
$499$	40.2890	1.80358	0.901791	+	0.432173i	$0.142253\pi$
$500$	0	0
$501$	0	0
$502$	−59.3982	−2.65107
$503$	15.1397	0.675047	0.337524	−	0.941317i	$-0.390411\pi$
$503$	15.1397	0.675047	0.337524	+	0.941317i	$0.390411\pi$
$504$	0	0
$505$	0	0
$506$	26.2599	1.16739
$507$	0	0
$508$	48.4154	2.14808
$509$	−7.55412	−0.334831	−0.167415	−	0.985886i	$-0.553542\pi$
$509$	−7.55412	−0.334831	−0.167415	+	0.985886i	$0.553542\pi$
$510$	0	0
$511$	−6.62945	−0.293270
$512$	−18.0606	−0.798172
$513$	0	0
$514$	50.0210	2.20633
$515$	0	0
$516$	0	0
$517$	11.3877	0.500831
$518$	−1.34003	−0.0588778
$519$	0	0
$520$	0	0
$521$	33.4616	1.46598	0.732990	−	0.680239i	$-0.238124\pi$
$521$	33.4616	1.46598	0.732990	+	0.680239i	$0.238124\pi$
$522$	0	0
$523$	35.9208	1.57071	0.785354	−	0.619047i	$-0.212481\pi$
$523$	35.9208	1.57071	0.785354	+	0.619047i	$0.212481\pi$
$524$	16.1226	0.704317
$525$	0	0
$526$	−8.93800	−0.389715
$527$	−5.47254	−0.238388
$528$	0	0
$529$	−8.67427	−0.377142
$530$	0	0
$531$	0	0
$532$	2.80734	0.121714
$533$	31.7520	1.37533
$534$	0	0
$535$	0	0
$536$	−33.2756	−1.43729
$537$	0	0
$538$	−38.5064	−1.66013
$539$	18.7559	0.807872
$540$	0	0
$541$	−43.1516	−1.85523	−0.927617	−	0.373533i	$-0.878146\pi$
$541$	−43.1516	−1.85523	−0.927617	+	0.373533i	$0.878146\pi$
$542$	−6.28373	−0.269909
$543$	0	0
$544$	18.4301	0.790185
$545$	0	0
$546$	0	0
$547$	−23.2308	−0.993278	−0.496639	−	0.867957i	$-0.665433\pi$
$547$	−23.2308	−0.993278	−0.496639	+	0.867957i	$0.665433\pi$
$548$	−58.2718	−2.48925
$549$	0	0
$550$	0	0
$551$	−5.00000	−0.213007
$552$	0	0
$553$	−2.63330	−0.111979
$554$	−50.0210	−2.12519
$555$	0	0
$556$	−9.96041	−0.422416
$557$	−8.24799	−0.349478	−0.174739	−	0.984615i	$-0.555908\pi$
$557$	−8.24799	−0.349478	−0.174739	+	0.984615i	$0.555908\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	62.3047	2.62817
$563$	−25.8273	−1.08849	−0.544246	−	0.838925i	$-0.683184\pi$
$563$	−25.8273	−1.08849	−0.544246	+	0.838925i	$0.683184\pi$
$564$	0	0
$565$	0	0
$566$	−44.6123	−1.87519
$567$	0	0
$568$	9.90652	0.415668
$569$	12.1860	0.510863	0.255432	−	0.966827i	$-0.417782\pi$
$569$	12.1860	0.510863	0.255432	+	0.966827i	$0.417782\pi$
$570$	0	0
$571$	12.2308	0.511843	0.255922	−	0.966698i	$-0.417621\pi$
$571$	12.2308	0.511843	0.255922	+	0.966698i	$0.417621\pi$
$572$	42.0315	1.75742
$573$	0	0
$574$	14.7139	0.614145
$575$	0	0
$576$	0	0
$577$	−21.9537	−0.913946	−0.456973	−	0.889480i	$-0.651066\pi$
$577$	−21.9537	−0.913946	−0.456973	+	0.889480i	$0.651066\pi$
$578$	22.5618	0.938446
$579$	0	0
$580$	0	0
$581$	−3.87601	−0.160804
$582$	0	0
$583$	−18.9380	−0.784332
$584$	31.4435	1.30114
$585$	0	0
$586$	17.6924	0.730867
$587$	22.5527	0.930849	0.465425	−	0.885088i	$-0.345902\pi$
$587$	22.5527	0.930849	0.465425	+	0.885088i	$0.345902\pi$
$588$	0	0
$589$	−1.06200	−0.0437588
$590$	0	0
$591$	0	0
$592$	1.18456	0.0486849
$593$	−28.3376	−1.16369	−0.581843	−	0.813301i	$-0.697668\pi$
$593$	−28.3376	−1.16369	−0.581843	+	0.813301i	$0.697668\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	35.7520	1.46201
$599$	−2.27947	−0.0931367	−0.0465684	−	0.998915i	$-0.514829\pi$
$599$	−2.27947	−0.0931367	−0.0465684	+	0.998915i	$0.514829\pi$
$600$	0	0
$601$	−3.56987	−0.145618	−0.0728090	−	0.997346i	$-0.523196\pi$
$601$	−3.56987	−0.145618	−0.0728090	+	0.997346i	$0.523196\pi$
$602$	7.41439	0.302188
$603$	0	0
$604$	80.8259	3.28876
$605$	0	0
$606$	0	0
$607$	−27.6609	−1.12272	−0.561361	−	0.827571i	$-0.689722\pi$
$607$	−27.6609	−1.12272	−0.561361	+	0.827571i	$0.689722\pi$
$608$	3.57653	0.145048
$609$	0	0
$610$	0	0
$611$	15.5040	0.627226
$612$	0	0
$613$	−5.50787	−0.222461	−0.111230	−	0.993795i	$-0.535479\pi$
$613$	−5.50787	−0.222461	−0.111230	+	0.993795i	$0.535479\pi$
$614$	−49.1516	−1.98360
$615$	0	0
$616$	8.58561	0.345924
$617$	−19.4459	−0.782861	−0.391431	−	0.920208i	$-0.628020\pi$
$617$	−19.4459	−0.782861	−0.391431	+	0.920208i	$0.628020\pi$
$618$	0	0
$619$	16.5369	0.664676	0.332338	−	0.943160i	$-0.392163\pi$
$619$	16.5369	0.664676	0.332338	+	0.943160i	$0.392163\pi$
$620$	0	0
$621$	0	0
$622$	−63.5040	−2.54628
$623$	0.447309	0.0179211
$624$	0	0
$625$	0	0
$626$	−57.2890	−2.28973
$627$	0	0
$628$	50.9447	2.03291
$629$	−3.72535	−0.148539
$630$	0	0
$631$	3.41439	0.135925	0.0679623	−	0.997688i	$-0.478350\pi$
$631$	3.41439	0.135925	0.0679623	+	0.997688i	$0.478350\pi$
$632$	12.4897	0.496814
$633$	0	0
$634$	−55.0367	−2.18579
$635$	0	0
$636$	0	0
$637$	25.5355	1.01175
$638$	−34.6900	−1.37339
$639$	0	0
$640$	0	0
$641$	30.6123	1.20911	0.604556	−	0.796563i	$-0.293351\pi$
$641$	30.6123	1.20911	0.604556	+	0.796563i	$0.293351\pi$
$642$	0	0
$643$	−10.5369	−0.415537	−0.207768	−	0.978178i	$-0.566620\pi$
$643$	−10.5369	−0.415537	−0.207768	+	0.978178i	$0.566620\pi$
$644$	10.6256	0.418708
$645$	0	0
$646$	12.1688	0.478776
$647$	23.0935	0.907898	0.453949	−	0.891028i	$-0.350015\pi$
$647$	23.0935	0.907898	0.453949	+	0.891028i	$0.350015\pi$
$648$	0	0
$649$	−13.6939	−0.537531
$650$	0	0
$651$	0	0
$652$	71.4693	2.79895
$653$	−0.833597	−0.0326212	−0.0163106	−	0.999867i	$-0.505192\pi$
$653$	−0.833597	−0.0326212	−0.0163106	+	0.999867i	$0.505192\pi$
$654$	0	0
$655$	0	0
$656$	−13.0067	−0.507825
$657$	0	0
$658$	7.18456	0.280083
$659$	37.1712	1.44799	0.723993	−	0.689808i	$-0.242305\pi$
$659$	37.1712	1.44799	0.723993	+	0.689808i	$0.242305\pi$
$660$	0	0
$661$	9.98667	0.388436	0.194218	−	0.980958i	$-0.437783\pi$
$661$	9.98667	0.388436	0.194218	+	0.980958i	$0.437783\pi$
$662$	−56.2732	−2.18712
$663$	0	0
$664$	18.3839	0.713433
$665$	0	0
$666$	0	0
$667$	−18.9247	−0.732766
$668$	−52.8788	−2.04594
$669$	0	0
$670$	0	0
$671$	−26.0777	−1.00672
$672$	0	0
$673$	7.13974	0.275217	0.137608	−	0.990487i	$-0.456058\pi$
$673$	7.13974	0.275217	0.137608	+	0.990487i	$0.456058\pi$
$674$	15.1846	0.584887
$675$	0	0
$676$	10.7296	0.412677
$677$	−15.5856	−0.599004	−0.299502	−	0.954096i	$-0.596820\pi$
$677$	−15.5856	−0.599004	−0.299502	+	0.954096i	$0.596820\pi$
$678$	0	0
$679$	6.74960	0.259026
$680$	0	0
$681$	0	0
$682$	−7.36814	−0.282140
$683$	3.49454	0.133715	0.0668574	−	0.997763i	$-0.478703\pi$
$683$	3.49454	0.133715	0.0668574	+	0.997763i	$0.478703\pi$
$684$	0	0
$685$	0	0
$686$	24.8083	0.947186
$687$	0	0
$688$	−6.55412	−0.249874
$689$	−25.7835	−0.982273
$690$	0	0
$691$	37.6280	1.43144	0.715719	−	0.698389i	$-0.246099\pi$
$691$	37.6280	1.43144	0.715719	+	0.698389i	$0.246099\pi$
$692$	81.3734	3.09335
$693$	0	0
$694$	21.9075	0.831597
$695$	0	0
$696$	0	0
$697$	40.9051	1.54939
$698$	−18.9289	−0.716470
$699$	0	0
$700$	0	0
$701$	7.92083	0.299165	0.149583	−	0.988749i	$-0.452207\pi$
$701$	7.92083	0.299165	0.149583	+	0.988749i	$0.452207\pi$
$702$	0	0
$703$	−0.722938	−0.0272661
$704$	34.4420	1.29808
$705$	0	0
$706$	−52.6753	−1.98246
$707$	−0.905083	−0.0340392
$708$	0	0
$709$	−28.0739	−1.05434	−0.527169	−	0.849761i	$-0.676746\pi$
$709$	−28.0739	−1.05434	−0.527169	+	0.849761i	$0.676746\pi$
$710$	0	0
$711$	0	0
$712$	−2.12158	−0.0795097
$713$	−4.01959	−0.150535
$714$	0	0
$715$	0	0
$716$	−79.0658	−2.95483
$717$	0	0
$718$	22.2003	0.828508
$719$	−20.2017	−0.753397	−0.376699	−	0.926336i	$-0.622941\pi$
$719$	−20.2017	−0.753397	−0.376699	+	0.926336i	$0.622941\pi$
$720$	0	0
$721$	−15.6624	−0.583297
$722$	2.36147	0.0878848
$723$	0	0
$724$	47.2027	1.75427
$725$	0	0
$726$	0	0
$727$	17.0644	0.632884	0.316442	−	0.948612i	$-0.397512\pi$
$727$	17.0644	0.632884	0.316442	+	0.948612i	$0.397512\pi$
$728$	11.6890	0.433225
$729$	0	0
$730$	0	0
$731$	20.6123	0.762373
$732$	0	0
$733$	46.3534	1.71210	0.856050	−	0.516892i	$-0.172911\pi$
$733$	46.3534	1.71210	0.856050	+	0.516892i	$0.172911\pi$
$734$	−17.2904	−0.638200
$735$	0	0
$736$	13.5369	0.498979
$737$	26.2599	0.967295
$738$	0	0
$739$	3.46305	0.127390	0.0636952	−	0.997969i	$-0.479711\pi$
$739$	3.46305	0.127390	0.0636952	+	0.997969i	$0.479711\pi$
$740$	0	0
$741$	0	0
$742$	−11.9481	−0.438627
$743$	−21.3972	−0.784988	−0.392494	−	0.919755i	$-0.628388\pi$
$743$	−21.3972	−0.784988	−0.392494	+	0.919755i	$0.628388\pi$
$744$	0	0
$745$	0	0
$746$	−21.3219	−0.780650
$747$	0	0
$748$	54.1478	1.97984
$749$	9.60038	0.350790
$750$	0	0
$751$	−38.5198	−1.40561	−0.702803	−	0.711384i	$-0.748069\pi$
$751$	−38.5198	−1.40561	−0.702803	+	0.711384i	$0.748069\pi$
$752$	−6.35096	−0.231596
$753$	0	0
$754$	−47.2294	−1.71999
$755$	0	0
$756$	0	0
$757$	−44.8760	−1.63105	−0.815523	−	0.578725i	$-0.803551\pi$
$757$	−44.8760	−1.63105	−0.815523	+	0.578725i	$0.803551\pi$
$758$	66.2270	2.40547
$759$	0	0
$760$	0	0
$761$	−51.2427	−1.85755	−0.928773	−	0.370648i	$-0.879136\pi$
$761$	−51.2427	−1.85755	−0.928773	+	0.370648i	$0.879136\pi$
$762$	0	0
$763$	1.89703	0.0686770
$764$	−48.9065	−1.76938
$765$	0	0
$766$	−79.5264	−2.87341
$767$	−18.6438	−0.673187
$768$	0	0
$769$	−31.3061	−1.12893	−0.564464	−	0.825458i	$-0.690917\pi$
$769$	−31.3061	−1.12893	−0.564464	+	0.825458i	$0.690917\pi$
$770$	0	0
$771$	0	0
$772$	−87.9790	−3.16643
$773$	−37.9695	−1.36567	−0.682834	−	0.730574i	$-0.739253\pi$
$773$	−37.9695	−1.36567	−0.682834	+	0.730574i	$0.739253\pi$
$774$	0	0
$775$	0	0
$776$	−32.0133	−1.14921
$777$	0	0
$778$	−1.60135	−0.0574113
$779$	7.93800	0.284408
$780$	0	0
$781$	−7.81785	−0.279745
$782$	46.0582	1.64704
$783$	0	0
$784$	−10.4602	−0.373578
$785$	0	0
$786$	0	0
$787$	−29.5369	−1.05288	−0.526439	−	0.850213i	$-0.676473\pi$
$787$	−29.5369	−1.05288	−0.526439	+	0.850213i	$0.676473\pi$
$788$	−91.6118	−3.26354
$789$	0	0
$790$	0	0
$791$	16.2403	0.577439
$792$	0	0
$793$	−35.5040	−1.26079
$794$	−25.5369	−0.906272
$795$	0	0
$796$	−0.825497	−0.0292590
$797$	−12.7716	−0.452393	−0.226197	−	0.974082i	$-0.572629\pi$
$797$	−12.7716	−0.452393	−0.226197	+	0.974082i	$0.572629\pi$
$798$	0	0
$799$	19.9733	0.706606
$800$	0	0
$801$	0	0
$802$	66.7348	2.35649
$803$	−24.8140	−0.875667
$804$	0	0
$805$	0	0
$806$	−10.0315	−0.353344
$807$	0	0
$808$	4.29281	0.151020
$809$	−5.56987	−0.195826	−0.0979131	−	0.995195i	$-0.531217\pi$
$809$	−5.56987	−0.195826	−0.0979131	+	0.995195i	$0.531217\pi$
$810$	0	0
$811$	−1.55028	−0.0544377	−0.0272189	−	0.999629i	$-0.508665\pi$
$811$	−1.55028	−0.0544377	−0.0272189	+	0.999629i	$0.508665\pi$
$812$	−14.0367	−0.492592
$813$	0	0
$814$	−5.01574	−0.175802
$815$	0	0
$816$	0	0
$817$	4.00000	0.139942
$818$	68.9184	2.40968
$819$	0	0
$820$	0	0
$821$	49.7968	1.73792	0.868961	−	0.494881i	$-0.164788\pi$
$821$	49.7968	1.73792	0.868961	+	0.494881i	$0.164788\pi$
$822$	0	0
$823$	−6.78493	−0.236508	−0.118254	−	0.992983i	$-0.537730\pi$
$823$	−6.78493	−0.236508	−0.118254	+	0.992983i	$0.537730\pi$
$824$	74.2866	2.58789
$825$	0	0
$826$	−8.63951	−0.300607
$827$	−26.8918	−0.935118	−0.467559	−	0.883962i	$-0.654866\pi$
$827$	−26.8918	−0.935118	−0.467559	+	0.883962i	$0.654866\pi$
$828$	0	0
$829$	33.4774	1.16272	0.581358	−	0.813648i	$-0.302521\pi$
$829$	33.4774	1.16272	0.581358	+	0.813648i	$0.302521\pi$
$830$	0	0
$831$	0	0
$832$	46.8918	1.62568
$833$	32.8966	1.13980
$834$	0	0
$835$	0	0
$836$	10.5079	0.363422
$837$	0	0
$838$	63.2112	2.18360
$839$	−26.1335	−0.902228	−0.451114	−	0.892466i	$-0.648973\pi$
$839$	−26.1335	−0.902228	−0.451114	+	0.892466i	$0.648973\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	69.6728	2.40108
$843$	0	0
$844$	19.1516	0.659226
$845$	0	0
$846$	0	0
$847$	1.85883	0.0638702
$848$	10.5618	0.362693
$849$	0	0
$850$	0	0
$851$	−2.73627	−0.0937982
$852$	0	0
$853$	−4.77160	−0.163376	−0.0816882	−	0.996658i	$-0.526031\pi$
$853$	−4.77160	−0.163376	−0.0816882	+	0.996658i	$0.526031\pi$
$854$	−16.4525	−0.562994
$855$	0	0
$856$	−45.5345	−1.55634
$857$	26.9537	0.920722	0.460361	−	0.887732i	$-0.347720\pi$
$857$	26.9537	0.920722	0.460361	+	0.887732i	$0.347720\pi$
$858$	0	0
$859$	−41.7034	−1.42290	−0.711450	−	0.702737i	$-0.751961\pi$
$859$	−41.7034	−1.42290	−0.711450	+	0.702737i	$0.751961\pi$
$860$	0	0
$861$	0	0
$862$	−6.86934	−0.233971
$863$	33.1492	1.12841	0.564206	−	0.825634i	$-0.309182\pi$
$863$	33.1492	1.12841	0.564206	+	0.825634i	$0.309182\pi$
$864$	0	0
$865$	0	0
$866$	14.1688	0.481476
$867$	0	0
$868$	−2.98139	−0.101195
$869$	−9.85642	−0.334356
$870$	0	0
$871$	35.7520	1.21141
$872$	−8.99759	−0.304697
$873$	0	0
$874$	8.93800	0.302332
$875$	0	0
$876$	0	0
$877$	−31.0739	−1.04929	−0.524645	−	0.851321i	$-0.675802\pi$
$877$	−31.0739	−1.04929	−0.524645	+	0.851321i	$0.675802\pi$
$878$	78.9036	2.66287
$879$	0	0
$880$	0	0
$881$	5.44588	0.183476	0.0917381	−	0.995783i	$-0.470758\pi$
$881$	5.44588	0.183476	0.0917381	+	0.995783i	$0.470758\pi$
$882$	0	0
$883$	−23.1492	−0.779033	−0.389517	−	0.921019i	$-0.627358\pi$
$883$	−23.1492	−0.779033	−0.389517	+	0.921019i	$0.627358\pi$
$884$	73.7205	2.47949
$885$	0	0
$886$	55.6876	1.87086
$887$	44.3643	1.48961	0.744804	−	0.667284i	$-0.232543\pi$
$887$	44.3643	1.48961	0.744804	+	0.667284i	$0.232543\pi$
$888$	0	0
$889$	−10.6256	−0.356372
$890$	0	0
$891$	0	0
$892$	−6.38388	−0.213748
$893$	3.87601	0.129706
$894$	0	0
$895$	0	0
$896$	16.1149	0.538362
$897$	0	0
$898$	16.2375	0.541852
$899$	5.30998	0.177098
$900$	0	0
$901$	−33.2160	−1.10659
$902$	55.0739	1.83376
$903$	0	0
$904$	−77.0276	−2.56190
$905$	0	0
$906$	0	0
$907$	0.319472	0.0106079	0.00530395	−	0.999986i	$-0.498312\pi$
$907$	0.319472	0.0106079	0.00530395	+	0.999986i	$0.498312\pi$
$908$	32.0711	1.06432
$909$	0	0
$910$	0	0
$911$	−14.0487	−0.465453	−0.232726	−	0.972542i	$-0.574765\pi$
$911$	−14.0487	−0.465453	−0.232726	+	0.972542i	$0.574765\pi$
$912$	0	0
$913$	−14.5079	−0.480140
$914$	−78.5422	−2.59794
$915$	0	0
$916$	95.0143	3.13936
$917$	−3.53838	−0.116848
$918$	0	0
$919$	−25.8007	−0.851086	−0.425543	−	0.904938i	$-0.639917\pi$
$919$	−25.8007	−0.851086	−0.425543	+	0.904938i	$0.639917\pi$
$920$	0	0
$921$	0	0
$922$	−38.4750	−1.26711
$923$	−10.6438	−0.350344
$924$	0	0
$925$	0	0
$926$	78.9814	2.59549
$927$	0	0
$928$	−17.8827	−0.587028
$929$	9.86267	0.323584	0.161792	−	0.986825i	$-0.448273\pi$
$929$	9.86267	0.323584	0.161792	+	0.986825i	$0.448273\pi$
$930$	0	0
$931$	6.38388	0.209223
$932$	34.2270	1.12114
$933$	0	0
$934$	−48.4602	−1.58567
$935$	0	0
$936$	0	0
$937$	30.4921	0.996134	0.498067	−	0.867138i	$-0.334043\pi$
$937$	30.4921	0.996134	0.498067	+	0.867138i	$0.334043\pi$
$938$	16.5675	0.540947
$939$	0	0
$940$	0	0
$941$	−2.20173	−0.0717744	−0.0358872	−	0.999356i	$-0.511426\pi$
$941$	−2.20173	−0.0717744	−0.0358872	+	0.999356i	$0.511426\pi$
$942$	0	0
$943$	30.0448	0.978394
$944$	7.63710	0.248566
$945$	0	0
$946$	27.7520	0.902296
$947$	11.7520	0.381889	0.190945	−	0.981601i	$-0.438845\pi$
$947$	11.7520	0.381889	0.190945	+	0.981601i	$0.438845\pi$
$948$	0	0
$949$	−33.7835	−1.09666
$950$	0	0
$951$	0	0
$952$	15.0586	0.488052
$953$	−21.7363	−0.704107	−0.352053	−	0.935980i	$-0.614516\pi$
$953$	−21.7363	−0.704107	−0.352053	+	0.935980i	$0.614516\pi$
$954$	0	0
$955$	0	0
$956$	−60.4616	−1.95547
$957$	0	0
$958$	18.5212	0.598393
$959$	12.7888	0.412971
$960$	0	0
$961$	−29.8722	−0.963618
$962$	−6.82878	−0.220169
$963$	0	0
$964$	−33.1797	−1.06865
$965$	0	0
$966$	0	0
$967$	15.2466	0.490296	0.245148	−	0.969486i	$-0.421163\pi$
$967$	15.2466	0.490296	0.245148	+	0.969486i	$0.421163\pi$
$968$	−8.81642	−0.283370
$969$	0	0
$970$	0	0
$971$	−38.5951	−1.23858	−0.619288	−	0.785164i	$-0.712579\pi$
$971$	−38.5951	−1.23858	−0.619288	+	0.785164i	$0.712579\pi$
$972$	0	0
$973$	2.18599	0.0700796
$974$	16.9232	0.542255
$975$	0	0
$976$	14.5436	0.465530
$977$	3.10825	0.0994417	0.0497209	−	0.998763i	$-0.484167\pi$
$977$	3.10825	0.0994417	0.0497209	+	0.998763i	$0.484167\pi$
$978$	0	0
$979$	1.67427	0.0535100
$980$	0	0
$981$	0	0
$982$	−38.4750	−1.22779
$983$	14.4616	0.461254	0.230627	−	0.973042i	$-0.425922\pi$
$983$	14.4616	0.461254	0.230627	+	0.973042i	$0.425922\pi$
$984$	0	0
$985$	0	0
$986$	−60.8441	−1.93767
$987$	0	0
$988$	14.3061	0.455139
$989$	15.1397	0.481416
$990$	0	0
$991$	−23.8298	−0.756977	−0.378489	−	0.925606i	$-0.623556\pi$
$991$	−23.8298	−0.756977	−0.378489	+	0.925606i	$0.623556\pi$
$992$	−3.79827	−0.120595
$993$	0	0
$994$	−4.93231	−0.156443
$995$	0	0
$996$	0	0
$997$	1.79827	0.0569517	0.0284758	−	0.999594i	$-0.490935\pi$
$997$	1.79827	0.0569517	0.0284758	+	0.999594i	$0.490935\pi$
$998$	95.1411	3.01164
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bf.1.3		3
3.2	odd	2		1425.2.a.t.1.1	✓	3
5.4	even	2		4275.2.a.bg.1.1		3
15.2	even	4		1425.2.c.o.799.2		6
15.8	even	4		1425.2.c.o.799.5		6
15.14	odd	2		1425.2.a.w.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.t.1.1	✓	3	3.2	odd	2
1425.2.a.w.1.3	yes	3	15.14	odd	2
1425.2.c.o.799.2		6	15.2	even	4
1425.2.c.o.799.5		6	15.8	even	4
4275.2.a.bf.1.3		3	1.1	even	1	trivial
4275.2.a.bg.1.1		3	5.4	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:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+2.36147 q^{2} +3.57653 q^{4} -0.784934 q^{7} +3.72294 q^{8} -2.93800 q^{11} -4.00000 q^{13} -1.85360 q^{14} +1.63853 q^{16} -5.15307 q^{17} -1.00000 q^{19} -6.93800 q^{22} -3.78493 q^{23} -9.44588 q^{26} -2.80734 q^{28} +5.00000 q^{29} +1.06200 q^{31} -3.57653 q^{32} -12.1688 q^{34} +0.722938 q^{37} -2.36147 q^{38} -7.93800 q^{41} -4.00000 q^{43} -10.5079 q^{44} -8.93800 q^{46} -3.87601 q^{47} -6.38388 q^{49} -14.3061 q^{52} +6.44588 q^{53} -2.92226 q^{56} +11.8073 q^{58} +4.66094 q^{59} +8.87601 q^{61} +2.50787 q^{62} -11.7229 q^{64} -8.93800 q^{67} -18.4301 q^{68} +2.66094 q^{71} +8.44588 q^{73} +1.70719 q^{74} -3.57653 q^{76} +2.30614 q^{77} +3.35480 q^{79} -18.7453 q^{82} +4.93800 q^{83} -9.44588 q^{86} -10.9380 q^{88} -0.569868 q^{89} +3.13974 q^{91} -13.5369 q^{92} -9.15307 q^{94} -8.59894 q^{97} -15.0753 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} - 3 q^{8} + 3 q^{11} - 12 q^{13} - 15 q^{14} + 12 q^{16} - 6 q^{17} - 3 q^{19} - 9 q^{22} - 9 q^{23} + 27 q^{28} + 15 q^{29} + 15 q^{31} - 6 q^{32} + 6 q^{34} - 12 q^{37} - 12 q^{41} - 12 q^{43}+ \cdots - 57 q^{98}+O(q^{100}) \) 3 * q + 6 * q^4 - 3 * q^8 + 3 * q^11 - 12 * q^13 - 15 * q^14 + 12 * q^16 - 6 * q^17 - 3 * q^19 - 9 * q^22 - 9 * q^23 + 27 * q^28 + 15 * q^29 + 15 * q^31 - 6 * q^32 + 6 * q^34 - 12 * q^37 - 12 * q^41 - 12 * q^43 - 15 * q^44 - 15 * q^46 + 12 * q^47 + 21 * q^49 - 24 * q^52 - 9 * q^53 - 30 * q^56 - 12 * q^59 + 3 * q^61 - 9 * q^62 - 21 * q^64 - 15 * q^67 - 60 * q^68 - 18 * q^71 - 3 * q^73 + 24 * q^74 - 6 * q^76 - 12 * q^77 + 3 * q^79 - 9 * q^82 + 3 * q^83 - 21 * q^88 + 3 * q^89 + 9 * q^92 - 18 * q^94 + 12 * q^97 - 57 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more