LMFDB - Embedded newform 7448.2.a.bm.1.3

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.98211824.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 15x^{3} + 8x^{2} - 9x + 1 $ x^6 - 3x^5 - 6x^4 + 15x^3 + 8x^2 - 9*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.129411$ 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)

$f(q)$	$=$	$q-0.870589 q^{3} -0.696873 q^{5} -2.24208 q^{9} +O(q^{10})$	$q-0.870589 q^{3} -0.696873 q^{5} -2.24208 q^{9} +2.28239 q^{11} +3.44492 q^{13} +0.606690 q^{15} -1.60669 q^{17} +1.00000 q^{19} -3.13072 q^{23} -4.51437 q^{25} +4.56369 q^{27} +5.41121 q^{29} -6.32957 q^{31} -1.98702 q^{33} +1.04719 q^{37} -2.99911 q^{39} -6.58512 q^{41} +2.10867 q^{43} +1.56244 q^{45} +4.62703 q^{47} +1.39877 q^{51} +7.87418 q^{53} -1.59054 q^{55} -0.870589 q^{57} -2.24584 q^{59} -1.54579 q^{61} -2.40067 q^{65} +0.258278 q^{67} +2.72557 q^{69} -4.57165 q^{71} -7.75142 q^{73} +3.93016 q^{75} +8.00248 q^{79} +2.75313 q^{81} -2.52213 q^{83} +1.11966 q^{85} -4.71094 q^{87} -16.2117 q^{89} +5.51046 q^{93} -0.696873 q^{95} +8.33509 q^{97} -5.11729 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - q^{5} + 3 q^{9}+O(q^{10}) $ 6 * q - 3 * q^3 - q^5 + 3 * q^9	$ 6 q - 3 q^{3} - q^{5} + 3 q^{9} + 3 q^{11} - 6 q^{13} - 8 q^{15} + 2 q^{17} + 6 q^{19} + 2 q^{23} + 5 q^{25} - 6 q^{27} - q^{29} - 14 q^{31} - 19 q^{33} - 7 q^{37} + 22 q^{39} + 21 q^{41} + q^{43} + 4 q^{45} - 23 q^{47} + 2 q^{51} - 15 q^{53} - 2 q^{55} - 3 q^{57} - 25 q^{59} - 21 q^{61} + 6 q^{65} + 2 q^{67} + 20 q^{69} + 13 q^{71} + 2 q^{73} - 24 q^{75} - 17 q^{79} - 2 q^{81} - 4 q^{83} + 40 q^{85} - 30 q^{87} + q^{89} + 6 q^{93} - q^{95} + 13 q^{97} + 41 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 - q^5 + 3 * q^9 + 3 * q^11 - 6 * q^13 - 8 * q^15 + 2 * q^17 + 6 * q^19 + 2 * q^23 + 5 * q^25 - 6 * q^27 - q^29 - 14 * q^31 - 19 * q^33 - 7 * q^37 + 22 * q^39 + 21 * q^41 + q^43 + 4 * q^45 - 23 * q^47 + 2 * q^51 - 15 * q^53 - 2 * q^55 - 3 * q^57 - 25 * q^59 - 21 * q^61 + 6 * q^65 + 2 * q^67 + 20 * q^69 + 13 * q^71 + 2 * q^73 - 24 * q^75 - 17 * q^79 - 2 * q^81 - 4 * q^83 + 40 * q^85 - 30 * q^87 + q^89 + 6 * q^93 - q^95 + 13 * q^97 + 41 * 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
$3$	−0.870589	−0.502635	−0.251317	−	0.967905i	$-0.580864\pi$
$3$	−0.870589	−0.502635	−0.251317	+	0.967905i	$0.580864\pi$
$4$	0	0
$5$	−0.696873	−0.311651	−0.155826	−	0.987785i	$-0.549804\pi$
$5$	−0.696873	−0.311651	−0.155826	+	0.987785i	$0.549804\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.24208	−0.747358
$10$	0	0
$11$	2.28239	0.688166	0.344083	−	0.938939i	$-0.388190\pi$
$11$	2.28239	0.688166	0.344083	+	0.938939i	$0.388190\pi$
$12$	0	0
$13$	3.44492	0.955449	0.477725	−	0.878510i	$-0.341462\pi$
$13$	3.44492	0.955449	0.477725	+	0.878510i	$0.341462\pi$
$14$	0	0
$15$	0.606690	0.156647
$16$	0	0
$17$	−1.60669	−0.389680	−0.194840	−	0.980835i	$-0.562419\pi$
$17$	−1.60669	−0.389680	−0.194840	+	0.980835i	$0.562419\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.13072	−0.652800	−0.326400	−	0.945232i	$-0.605836\pi$
$23$	−3.13072	−0.652800	−0.326400	+	0.945232i	$0.605836\pi$
$24$	0	0
$25$	−4.51437	−0.902874
$26$	0	0
$27$	4.56369	0.878283
$28$	0	0
$29$	5.41121	1.00484	0.502418	−	0.864625i	$-0.332444\pi$
$29$	5.41121	1.00484	0.502418	+	0.864625i	$0.332444\pi$
$30$	0	0
$31$	−6.32957	−1.13683	−0.568413	−	0.822744i	$-0.692442\pi$
$31$	−6.32957	−1.13683	−0.568413	+	0.822744i	$0.692442\pi$
$32$	0	0
$33$	−1.98702	−0.345896
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.04719	0.172156	0.0860781	−	0.996288i	$-0.472567\pi$
$37$	1.04719	0.172156	0.0860781	+	0.996288i	$0.472567\pi$
$38$	0	0
$39$	−2.99911	−0.480242
$40$	0	0
$41$	−6.58512	−1.02842	−0.514211	−	0.857663i	$-0.671915\pi$
$41$	−6.58512	−1.02842	−0.514211	+	0.857663i	$0.671915\pi$
$42$	0	0
$43$	2.10867	0.321570	0.160785	−	0.986989i	$-0.448597\pi$
$43$	2.10867	0.321570	0.160785	+	0.986989i	$0.448597\pi$
$44$	0	0
$45$	1.56244	0.232915
$46$	0	0
$47$	4.62703	0.674922	0.337461	−	0.941340i	$-0.390432\pi$
$47$	4.62703	0.674922	0.337461	+	0.941340i	$0.390432\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.39877	0.195866
$52$	0	0
$53$	7.87418	1.08160	0.540801	−	0.841150i	$-0.318121\pi$
$53$	7.87418	1.08160	0.540801	+	0.841150i	$0.318121\pi$
$54$	0	0
$55$	−1.59054	−0.214468
$56$	0	0
$57$	−0.870589	−0.115312
$58$	0	0
$59$	−2.24584	−0.292384	−0.146192	−	0.989256i	$-0.546702\pi$
$59$	−2.24584	−0.292384	−0.146192	+	0.989256i	$0.546702\pi$
$60$	0	0
$61$	−1.54579	−0.197919	−0.0989594	−	0.995091i	$-0.531551\pi$
$61$	−1.54579	−0.197919	−0.0989594	+	0.995091i	$0.531551\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.40067	−0.297767
$66$	0	0
$67$	0.258278	0.0315537	0.0157768	−	0.999876i	$-0.494978\pi$
$67$	0.258278	0.0315537	0.0157768	+	0.999876i	$0.494978\pi$
$68$	0	0
$69$	2.72557	0.328120
$70$	0	0
$71$	−4.57165	−0.542555	−0.271277	−	0.962501i	$-0.587446\pi$
$71$	−4.57165	−0.542555	−0.271277	+	0.962501i	$0.587446\pi$
$72$	0	0
$73$	−7.75142	−0.907235	−0.453618	−	0.891196i	$-0.649867\pi$
$73$	−7.75142	−0.907235	−0.453618	+	0.891196i	$0.649867\pi$
$74$	0	0
$75$	3.93016	0.453816
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00248	0.900350	0.450175	−	0.892940i	$-0.351362\pi$
$79$	8.00248	0.900350	0.450175	+	0.892940i	$0.351362\pi$
$80$	0	0
$81$	2.75313	0.305903
$82$	0	0
$83$	−2.52213	−0.276839	−0.138420	−	0.990374i	$-0.544202\pi$
$83$	−2.52213	−0.276839	−0.138420	+	0.990374i	$0.544202\pi$
$84$	0	0
$85$	1.11966	0.121444
$86$	0	0
$87$	−4.71094	−0.505065
$88$	0	0
$89$	−16.2117	−1.71843	−0.859216	−	0.511613i	$-0.829048\pi$
$89$	−16.2117	−1.71843	−0.859216	+	0.511613i	$0.829048\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.51046	0.571408
$94$	0	0
$95$	−0.696873	−0.0714977
$96$	0	0
$97$	8.33509	0.846300	0.423150	−	0.906060i	$-0.360924\pi$
$97$	8.33509	0.846300	0.423150	+	0.906060i	$0.360924\pi$
$98$	0	0
$99$	−5.11729	−0.514307
$100$	0	0
$101$	2.31386	0.230237	0.115119	−	0.993352i	$-0.463275\pi$
$101$	2.31386	0.230237	0.115119	+	0.993352i	$0.463275\pi$
$102$	0	0
$103$	−9.51334	−0.937377	−0.468689	−	0.883363i	$-0.655273\pi$
$103$	−9.51334	−0.937377	−0.468689	+	0.883363i	$0.655273\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.52545	0.534165	0.267082	−	0.963674i	$-0.413940\pi$
$107$	5.52545	0.534165	0.267082	+	0.963674i	$0.413940\pi$
$108$	0	0
$109$	6.91908	0.662728	0.331364	−	0.943503i	$-0.392491\pi$
$109$	6.91908	0.662728	0.331364	+	0.943503i	$0.392491\pi$
$110$	0	0
$111$	−0.911668	−0.0865317
$112$	0	0
$113$	16.7207	1.57295	0.786476	−	0.617621i	$-0.211904\pi$
$113$	16.7207	1.57295	0.786476	+	0.617621i	$0.211904\pi$
$114$	0	0
$115$	2.18171	0.203446
$116$	0	0
$117$	−7.72377	−0.714063
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.79070	−0.526427
$122$	0	0
$123$	5.73293	0.516921
$124$	0	0
$125$	6.63031	0.593033
$126$	0	0
$127$	7.39023	0.655777	0.327888	−	0.944716i	$-0.393663\pi$
$127$	7.39023	0.655777	0.327888	+	0.944716i	$0.393663\pi$
$128$	0	0
$129$	−1.83579	−0.161632
$130$	0	0
$131$	−10.4626	−0.914122	−0.457061	−	0.889435i	$-0.651098\pi$
$131$	−10.4626	−0.914122	−0.457061	+	0.889435i	$0.651098\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.18032	−0.273718
$136$	0	0
$137$	−0.904147	−0.0772465	−0.0386233	−	0.999254i	$-0.512297\pi$
$137$	−0.904147	−0.0772465	−0.0386233	+	0.999254i	$0.512297\pi$
$138$	0	0
$139$	9.86463	0.836707	0.418353	−	0.908284i	$-0.362607\pi$
$139$	9.86463	0.836707	0.418353	+	0.908284i	$0.362607\pi$
$140$	0	0
$141$	−4.02824	−0.339239
$142$	0	0
$143$	7.86265	0.657508
$144$	0	0
$145$	−3.77093	−0.313158
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.70086	0.139340	0.0696701	−	0.997570i	$-0.477805\pi$
$149$	1.70086	0.139340	0.0696701	+	0.997570i	$0.477805\pi$
$150$	0	0
$151$	2.82941	0.230254	0.115127	−	0.993351i	$-0.463272\pi$
$151$	2.82941	0.230254	0.115127	+	0.993351i	$0.463272\pi$
$152$	0	0
$153$	3.60232	0.291230
$154$	0	0
$155$	4.41091	0.354293
$156$	0	0
$157$	−5.17054	−0.412654	−0.206327	−	0.978483i	$-0.566151\pi$
$157$	−5.17054	−0.412654	−0.206327	+	0.978483i	$0.566151\pi$
$158$	0	0
$159$	−6.85518	−0.543651
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−7.44542	−0.583170	−0.291585	−	0.956545i	$-0.594183\pi$
$163$	−7.44542	−0.583170	−0.291585	+	0.956545i	$0.594183\pi$
$164$	0	0
$165$	1.38470	0.107799
$166$	0	0
$167$	−11.8553	−0.917390	−0.458695	−	0.888594i	$-0.651683\pi$
$167$	−11.8553	−0.917390	−0.458695	+	0.888594i	$0.651683\pi$
$168$	0	0
$169$	−1.13251	−0.0871163
$170$	0	0
$171$	−2.24208	−0.171456
$172$	0	0
$173$	−9.70507	−0.737863	−0.368931	−	0.929457i	$-0.620276\pi$
$173$	−9.70507	−0.737863	−0.368931	+	0.929457i	$0.620276\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.95521	0.146962
$178$	0	0
$179$	12.9216	0.965806	0.482903	−	0.875674i	$-0.339582\pi$
$179$	12.9216	0.965806	0.482903	+	0.875674i	$0.339582\pi$
$180$	0	0
$181$	22.4115	1.66583	0.832917	−	0.553397i	$-0.186669\pi$
$181$	22.4115	1.66583	0.832917	+	0.553397i	$0.186669\pi$
$182$	0	0
$183$	1.34575	0.0994808
$184$	0	0
$185$	−0.729755	−0.0536527
$186$	0	0
$187$	−3.66709	−0.268164
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−22.6892	−1.64173	−0.820867	−	0.571119i	$-0.806509\pi$
$191$	−22.6892	−1.64173	−0.820867	+	0.571119i	$0.806509\pi$
$192$	0	0
$193$	−19.5706	−1.40872	−0.704361	−	0.709842i	$-0.748767\pi$
$193$	−19.5706	−1.40872	−0.704361	+	0.709842i	$0.748767\pi$
$194$	0	0
$195$	2.09000	0.149668
$196$	0	0
$197$	−0.321941	−0.0229374	−0.0114687	−	0.999934i	$-0.503651\pi$
$197$	−0.321941	−0.0229374	−0.0114687	+	0.999934i	$0.503651\pi$
$198$	0	0
$199$	−25.9064	−1.83646	−0.918229	−	0.396051i	$-0.870380\pi$
$199$	−25.9064	−1.83646	−0.918229	+	0.396051i	$0.870380\pi$
$200$	0	0
$201$	−0.224854	−0.0158600
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.58899	0.320509
$206$	0	0
$207$	7.01930	0.487875
$208$	0	0
$209$	2.28239	0.157876
$210$	0	0
$211$	−17.7189	−1.21982	−0.609910	−	0.792471i	$-0.708794\pi$
$211$	−17.7189	−1.21982	−0.609910	+	0.792471i	$0.708794\pi$
$212$	0	0
$213$	3.98003	0.272707
$214$	0	0
$215$	−1.46948	−0.100218
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.74830	0.456008
$220$	0	0
$221$	−5.53492	−0.372319
$222$	0	0
$223$	−18.4948	−1.23851	−0.619253	−	0.785192i	$-0.712564\pi$
$223$	−18.4948	−1.23851	−0.619253	+	0.785192i	$0.712564\pi$
$224$	0	0
$225$	10.1216	0.674770
$226$	0	0
$227$	−19.8265	−1.31593	−0.657967	−	0.753047i	$-0.728583\pi$
$227$	−19.8265	−1.31593	−0.657967	+	0.753047i	$0.728583\pi$
$228$	0	0
$229$	−10.3092	−0.681253	−0.340627	−	0.940199i	$-0.610639\pi$
$229$	−10.3092	−0.681253	−0.340627	+	0.940199i	$0.610639\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.6430	1.15583	0.577917	−	0.816096i	$-0.303866\pi$
$233$	17.6430	1.15583	0.577917	+	0.816096i	$0.303866\pi$
$234$	0	0
$235$	−3.22445	−0.210340
$236$	0	0
$237$	−6.96687	−0.452547
$238$	0	0
$239$	−14.9210	−0.965159	−0.482579	−	0.875852i	$-0.660300\pi$
$239$	−14.9210	−0.965159	−0.482579	+	0.875852i	$0.660300\pi$
$240$	0	0
$241$	23.0259	1.48323	0.741614	−	0.670827i	$-0.234061\pi$
$241$	23.0259	1.48323	0.741614	+	0.670827i	$0.234061\pi$
$242$	0	0
$243$	−16.0879	−1.03204
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.44492	0.219195
$248$	0	0
$249$	2.19574	0.139149
$250$	0	0
$251$	−2.64676	−0.167062	−0.0835312	−	0.996505i	$-0.526620\pi$
$251$	−2.64676	−0.167062	−0.0835312	+	0.996505i	$0.526620\pi$
$252$	0	0
$253$	−7.14552	−0.449235
$254$	0	0
$255$	−0.974763	−0.0610420
$256$	0	0
$257$	−8.83424	−0.551065	−0.275532	−	0.961292i	$-0.588854\pi$
$257$	−8.83424	−0.551065	−0.275532	+	0.961292i	$0.588854\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−12.1323	−0.750973
$262$	0	0
$263$	18.9766	1.17015	0.585075	−	0.810979i	$-0.301065\pi$
$263$	18.9766	1.17015	0.585075	+	0.810979i	$0.301065\pi$
$264$	0	0
$265$	−5.48731	−0.337083
$266$	0	0
$267$	14.1137	0.863744
$268$	0	0
$269$	−15.4518	−0.942115	−0.471057	−	0.882103i	$-0.656128\pi$
$269$	−15.4518	−0.942115	−0.471057	+	0.882103i	$0.656128\pi$
$270$	0	0
$271$	4.07073	0.247279	0.123640	−	0.992327i	$-0.460543\pi$
$271$	4.07073	0.247279	0.123640	+	0.992327i	$0.460543\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.3035	−0.621327
$276$	0	0
$277$	−10.0450	−0.603543	−0.301772	−	0.953380i	$-0.597578\pi$
$277$	−10.0450	−0.603543	−0.301772	+	0.953380i	$0.597578\pi$
$278$	0	0
$279$	14.1914	0.849616
$280$	0	0
$281$	−11.9719	−0.714185	−0.357092	−	0.934069i	$-0.616232\pi$
$281$	−11.9719	−0.714185	−0.357092	+	0.934069i	$0.616232\pi$
$282$	0	0
$283$	−17.3298	−1.03015	−0.515075	−	0.857145i	$-0.672236\pi$
$283$	−17.3298	−1.03015	−0.515075	+	0.857145i	$0.672236\pi$
$284$	0	0
$285$	0.606690	0.0359372
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.4185	−0.848150
$290$	0	0
$291$	−7.25643	−0.425380
$292$	0	0
$293$	22.4657	1.31246	0.656231	−	0.754560i	$-0.272150\pi$
$293$	22.4657	1.31246	0.656231	+	0.754560i	$0.272150\pi$
$294$	0	0
$295$	1.56507	0.0911219
$296$	0	0
$297$	10.4161	0.604405
$298$	0	0
$299$	−10.7851	−0.623717
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.01442	−0.115725
$304$	0	0
$305$	1.07722	0.0616816
$306$	0	0
$307$	−14.7152	−0.839841	−0.419921	−	0.907561i	$-0.637942\pi$
$307$	−14.7152	−0.839841	−0.419921	+	0.907561i	$0.637942\pi$
$308$	0	0
$309$	8.28221	0.471158
$310$	0	0
$311$	−25.8708	−1.46700	−0.733501	−	0.679689i	$-0.762115\pi$
$311$	−25.8708	−1.46700	−0.733501	+	0.679689i	$0.762115\pi$
$312$	0	0
$313$	−14.5111	−0.820215	−0.410108	−	0.912037i	$-0.634509\pi$
$313$	−14.5111	−0.820215	−0.410108	+	0.912037i	$0.634509\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.1580	1.24452	0.622258	−	0.782812i	$-0.286216\pi$
$317$	22.1580	1.24452	0.622258	+	0.782812i	$0.286216\pi$
$318$	0	0
$319$	12.3505	0.691494
$320$	0	0
$321$	−4.81039	−0.268490
$322$	0	0
$323$	−1.60669	−0.0893986
$324$	0	0
$325$	−15.5516	−0.862650
$326$	0	0
$327$	−6.02367	−0.333110
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.71927	−0.259395	−0.129697	−	0.991554i	$-0.541401\pi$
$331$	−4.71927	−0.259395	−0.129697	+	0.991554i	$0.541401\pi$
$332$	0	0
$333$	−2.34787	−0.128662
$334$	0	0
$335$	−0.179987	−0.00983374
$336$	0	0
$337$	−5.15117	−0.280602	−0.140301	−	0.990109i	$-0.544807\pi$
$337$	−5.15117	−0.280602	−0.140301	+	0.990109i	$0.544807\pi$
$338$	0	0
$339$	−14.5569	−0.790620
$340$	0	0
$341$	−14.4466	−0.782325
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.89938	−0.102259
$346$	0	0
$347$	−20.2945	−1.08947	−0.544734	−	0.838609i	$-0.683369\pi$
$347$	−20.2945	−1.08947	−0.544734	+	0.838609i	$0.683369\pi$
$348$	0	0
$349$	6.90187	0.369448	0.184724	−	0.982790i	$-0.440861\pi$
$349$	6.90187	0.369448	0.184724	+	0.982790i	$0.440861\pi$
$350$	0	0
$351$	15.7216	0.839155
$352$	0	0
$353$	21.0852	1.12225	0.561126	−	0.827730i	$-0.310368\pi$
$353$	21.0852	1.12225	0.561126	+	0.827730i	$0.310368\pi$
$354$	0	0
$355$	3.18586	0.169088
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.15567	−0.272106	−0.136053	−	0.990702i	$-0.543442\pi$
$359$	−5.15567	−0.272106	−0.136053	+	0.990702i	$0.543442\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.04132	0.264601
$364$	0	0
$365$	5.40176	0.282741
$366$	0	0
$367$	−23.3478	−1.21875	−0.609374	−	0.792883i	$-0.708579\pi$
$367$	−23.3478	−1.21875	−0.609374	+	0.792883i	$0.708579\pi$
$368$	0	0
$369$	14.7643	0.768600
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−24.1393	−1.24989	−0.624943	−	0.780671i	$-0.714878\pi$
$373$	−24.1393	−1.24989	−0.624943	+	0.780671i	$0.714878\pi$
$374$	0	0
$375$	−5.77227	−0.298079
$376$	0	0
$377$	18.6412	0.960070
$378$	0	0
$379$	−7.95691	−0.408719	−0.204360	−	0.978896i	$-0.565511\pi$
$379$	−7.95691	−0.408719	−0.204360	+	0.978896i	$0.565511\pi$
$380$	0	0
$381$	−6.43385	−0.329616
$382$	0	0
$383$	−6.04541	−0.308906	−0.154453	−	0.988000i	$-0.549362\pi$
$383$	−6.04541	−0.308906	−0.154453	+	0.988000i	$0.549362\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.72780	−0.240328
$388$	0	0
$389$	−1.36351	−0.0691329	−0.0345665	−	0.999402i	$-0.511005\pi$
$389$	−1.36351	−0.0691329	−0.0345665	+	0.999402i	$0.511005\pi$
$390$	0	0
$391$	5.03009	0.254383
$392$	0	0
$393$	9.10862	0.459469
$394$	0	0
$395$	−5.57672	−0.280595
$396$	0	0
$397$	−13.6305	−0.684096	−0.342048	−	0.939682i	$-0.611121\pi$
$397$	−13.6305	−0.684096	−0.342048	+	0.939682i	$0.611121\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.0342	0.700833	0.350416	−	0.936594i	$-0.386040\pi$
$401$	14.0342	0.700833	0.350416	+	0.936594i	$0.386040\pi$
$402$	0	0
$403$	−21.8049	−1.08618
$404$	0	0
$405$	−1.91858	−0.0953350
$406$	0	0
$407$	2.39008	0.118472
$408$	0	0
$409$	−10.1571	−0.502234	−0.251117	−	0.967957i	$-0.580798\pi$
$409$	−10.1571	−0.502234	−0.251117	+	0.967957i	$0.580798\pi$
$410$	0	0
$411$	0.787140	0.0388268
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.75760	0.0862774
$416$	0	0
$417$	−8.58804	−0.420558
$418$	0	0
$419$	0.655527	0.0320246	0.0160123	−	0.999872i	$-0.494903\pi$
$419$	0.655527	0.0320246	0.0160123	+	0.999872i	$0.494903\pi$
$420$	0	0
$421$	−2.18499	−0.106490	−0.0532450	−	0.998581i	$-0.516956\pi$
$421$	−2.18499	−0.106490	−0.0532450	+	0.998581i	$0.516956\pi$
$422$	0	0
$423$	−10.3742	−0.504409
$424$	0	0
$425$	7.25319	0.351831
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−6.84514	−0.330486
$430$	0	0
$431$	−7.72381	−0.372043	−0.186021	−	0.982546i	$-0.559559\pi$
$431$	−7.72381	−0.372043	−0.186021	+	0.982546i	$0.559559\pi$
$432$	0	0
$433$	8.77876	0.421880	0.210940	−	0.977499i	$-0.432348\pi$
$433$	8.77876	0.421880	0.210940	+	0.977499i	$0.432348\pi$
$434$	0	0
$435$	3.28293	0.157404
$436$	0	0
$437$	−3.13072	−0.149763
$438$	0	0
$439$	−6.06620	−0.289524	−0.144762	−	0.989467i	$-0.546242\pi$
$439$	−6.06620	−0.289524	−0.144762	+	0.989467i	$0.546242\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.4481	0.781471	0.390735	−	0.920503i	$-0.372221\pi$
$443$	16.4481	0.781471	0.390735	+	0.920503i	$0.372221\pi$
$444$	0	0
$445$	11.2975	0.535551
$446$	0	0
$447$	−1.48075	−0.0700372
$448$	0	0
$449$	−11.1856	−0.527879	−0.263940	−	0.964539i	$-0.585022\pi$
$449$	−11.1856	−0.527879	−0.263940	+	0.964539i	$0.585022\pi$
$450$	0	0
$451$	−15.0298	−0.707726
$452$	0	0
$453$	−2.46325	−0.115734
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.32776	0.155666	0.0778330	−	0.996966i	$-0.475200\pi$
$457$	3.32776	0.155666	0.0778330	+	0.996966i	$0.475200\pi$
$458$	0	0
$459$	−7.33244	−0.342249
$460$	0	0
$461$	27.2929	1.27116	0.635579	−	0.772036i	$-0.280762\pi$
$461$	27.2929	1.27116	0.635579	+	0.772036i	$0.280762\pi$
$462$	0	0
$463$	−29.5021	−1.37108	−0.685540	−	0.728035i	$-0.740434\pi$
$463$	−29.5021	−1.37108	−0.685540	+	0.728035i	$0.740434\pi$
$464$	0	0
$465$	−3.84009	−0.178080
$466$	0	0
$467$	−30.1526	−1.39530	−0.697648	−	0.716440i	$-0.745770\pi$
$467$	−30.1526	−1.39530	−0.697648	+	0.716440i	$0.745770\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4.50141	0.207414
$472$	0	0
$473$	4.81281	0.221293
$474$	0	0
$475$	−4.51437	−0.207133
$476$	0	0
$477$	−17.6545	−0.808344
$478$	0	0
$479$	−1.31884	−0.0602592	−0.0301296	−	0.999546i	$-0.509592\pi$
$479$	−1.31884	−0.0602592	−0.0301296	+	0.999546i	$0.509592\pi$
$480$	0	0
$481$	3.60747	0.164487
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.80850	−0.263750
$486$	0	0
$487$	18.7561	0.849922	0.424961	−	0.905212i	$-0.360288\pi$
$487$	18.7561	0.849922	0.424961	+	0.905212i	$0.360288\pi$
$488$	0	0
$489$	6.48190	0.293121
$490$	0	0
$491$	−24.8943	−1.12346	−0.561732	−	0.827319i	$-0.689865\pi$
$491$	−24.8943	−1.12346	−0.561732	+	0.827319i	$0.689865\pi$
$492$	0	0
$493$	−8.69413	−0.391564
$494$	0	0
$495$	3.56610	0.160284
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9.14698	0.409475	0.204738	−	0.978817i	$-0.434366\pi$
$499$	9.14698	0.409475	0.204738	+	0.978817i	$0.434366\pi$
$500$	0	0
$501$	10.3211	0.461112
$502$	0	0
$503$	7.55793	0.336991	0.168496	−	0.985702i	$-0.446109\pi$
$503$	7.55793	0.336991	0.168496	+	0.985702i	$0.446109\pi$
$504$	0	0
$505$	−1.61247	−0.0717538
$506$	0	0
$507$	0.985953	0.0437877
$508$	0	0
$509$	5.12398	0.227116	0.113558	−	0.993531i	$-0.463775\pi$
$509$	5.12398	0.227116	0.113558	+	0.993531i	$0.463775\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.56369	0.201492
$514$	0	0
$515$	6.62959	0.292135
$516$	0	0
$517$	10.5607	0.464458
$518$	0	0
$519$	8.44913	0.370875
$520$	0	0
$521$	30.0466	1.31636	0.658182	−	0.752859i	$-0.271326\pi$
$521$	30.0466	1.31636	0.658182	+	0.752859i	$0.271326\pi$
$522$	0	0
$523$	12.6132	0.551538	0.275769	−	0.961224i	$-0.411067\pi$
$523$	12.6132	0.551538	0.275769	+	0.961224i	$0.411067\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.1697	0.442998
$528$	0	0
$529$	−13.1986	−0.573852
$530$	0	0
$531$	5.03535	0.218516
$532$	0	0
$533$	−22.6852	−0.982606
$534$	0	0
$535$	−3.85053	−0.166473
$536$	0	0
$537$	−11.2494	−0.485448
$538$	0	0
$539$	0	0
$540$	0	0
$541$	20.6312	0.887005	0.443502	−	0.896273i	$-0.353736\pi$
$541$	20.6312	0.887005	0.443502	+	0.896273i	$0.353736\pi$
$542$	0	0
$543$	−19.5112	−0.837306
$544$	0	0
$545$	−4.82172	−0.206540
$546$	0	0
$547$	−12.9380	−0.553188	−0.276594	−	0.960987i	$-0.589206\pi$
$547$	−12.9380	−0.553188	−0.276594	+	0.960987i	$0.589206\pi$
$548$	0	0
$549$	3.46579	0.147916
$550$	0	0
$551$	5.41121	0.230525
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.635317	0.0269677
$556$	0	0
$557$	−1.80817	−0.0766145	−0.0383073	−	0.999266i	$-0.512197\pi$
$557$	−1.80817	−0.0766145	−0.0383073	+	0.999266i	$0.512197\pi$
$558$	0	0
$559$	7.26422	0.307244
$560$	0	0
$561$	3.19253	0.134789
$562$	0	0
$563$	28.6205	1.20621	0.603105	−	0.797662i	$-0.293930\pi$
$563$	28.6205	1.20621	0.603105	+	0.797662i	$0.293930\pi$
$564$	0	0
$565$	−11.6522	−0.490212
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.1092	−0.759177	−0.379588	−	0.925155i	$-0.623934\pi$
$569$	−18.1092	−0.759177	−0.379588	+	0.925155i	$0.623934\pi$
$570$	0	0
$571$	−25.0463	−1.04815	−0.524077	−	0.851671i	$-0.675589\pi$
$571$	−25.0463	−1.04815	−0.524077	+	0.851671i	$0.675589\pi$
$572$	0	0
$573$	19.7530	0.825192
$574$	0	0
$575$	14.1332	0.589396
$576$	0	0
$577$	−1.83623	−0.0764433	−0.0382217	−	0.999269i	$-0.512169\pi$
$577$	−1.83623	−0.0764433	−0.0382217	+	0.999269i	$0.512169\pi$
$578$	0	0
$579$	17.0380	0.708073
$580$	0	0
$581$	0	0
$582$	0	0
$583$	17.9720	0.744322
$584$	0	0
$585$	5.38249	0.222539
$586$	0	0
$587$	3.43073	0.141601	0.0708006	−	0.997490i	$-0.477445\pi$
$587$	3.43073	0.141601	0.0708006	+	0.997490i	$0.477445\pi$
$588$	0	0
$589$	−6.32957	−0.260806
$590$	0	0
$591$	0.280278	0.0115291
$592$	0	0
$593$	−11.4706	−0.471042	−0.235521	−	0.971869i	$-0.575680\pi$
$593$	−11.4706	−0.471042	−0.235521	+	0.971869i	$0.575680\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.5538	0.923067
$598$	0	0
$599$	21.5266	0.879555	0.439778	−	0.898107i	$-0.355057\pi$
$599$	21.5266	0.879555	0.439778	+	0.898107i	$0.355057\pi$
$600$	0	0
$601$	−9.86358	−0.402344	−0.201172	−	0.979556i	$-0.564475\pi$
$601$	−9.86358	−0.402344	−0.201172	+	0.979556i	$0.564475\pi$
$602$	0	0
$603$	−0.579078	−0.0235819
$604$	0	0
$605$	4.03538	0.164062
$606$	0	0
$607$	3.21655	0.130556	0.0652778	−	0.997867i	$-0.479207\pi$
$607$	3.21655	0.130556	0.0652778	+	0.997867i	$0.479207\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	15.9398	0.644854
$612$	0	0
$613$	9.74470	0.393585	0.196792	−	0.980445i	$-0.436948\pi$
$613$	9.74470	0.393585	0.196792	+	0.980445i	$0.436948\pi$
$614$	0	0
$615$	−3.99513	−0.161099
$616$	0	0
$617$	25.1002	1.01049	0.505247	−	0.862975i	$-0.331401\pi$
$617$	25.1002	1.01049	0.505247	+	0.862975i	$0.331401\pi$
$618$	0	0
$619$	−14.7814	−0.594114	−0.297057	−	0.954860i	$-0.596005\pi$
$619$	−14.7814	−0.594114	−0.297057	+	0.954860i	$0.596005\pi$
$620$	0	0
$621$	−14.2876	−0.573343
$622$	0	0
$623$	0	0
$624$	0	0
$625$	17.9514	0.718054
$626$	0	0
$627$	−1.98702	−0.0793540
$628$	0	0
$629$	−1.68250	−0.0670858
$630$	0	0
$631$	31.6274	1.25907	0.629533	−	0.776974i	$-0.283246\pi$
$631$	31.6274	1.25907	0.629533	+	0.776974i	$0.283246\pi$
$632$	0	0
$633$	15.4259	0.613124
$634$	0	0
$635$	−5.15005	−0.204374
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10.2500	0.405483
$640$	0	0
$641$	20.1388	0.795434	0.397717	−	0.917508i	$-0.369803\pi$
$641$	20.1388	0.795434	0.397717	+	0.917508i	$0.369803\pi$
$642$	0	0
$643$	−45.2873	−1.78596	−0.892978	−	0.450100i	$-0.851388\pi$
$643$	−45.2873	−1.78596	−0.892978	+	0.450100i	$0.851388\pi$
$644$	0	0
$645$	1.27931	0.0503728
$646$	0	0
$647$	−36.9656	−1.45327	−0.726633	−	0.687025i	$-0.758916\pi$
$647$	−36.9656	−1.45327	−0.726633	+	0.687025i	$0.758916\pi$
$648$	0	0
$649$	−5.12589	−0.201209
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.38211	−0.367150	−0.183575	−	0.983006i	$-0.558767\pi$
$653$	−9.38211	−0.367150	−0.183575	+	0.983006i	$0.558767\pi$
$654$	0	0
$655$	7.29111	0.284887
$656$	0	0
$657$	17.3793	0.678030
$658$	0	0
$659$	−31.7917	−1.23843	−0.619214	−	0.785222i	$-0.712549\pi$
$659$	−31.7917	−1.23843	−0.619214	+	0.785222i	$0.712549\pi$
$660$	0	0
$661$	−11.0330	−0.429135	−0.214567	−	0.976709i	$-0.568834\pi$
$661$	−11.0330	−0.429135	−0.214567	+	0.976709i	$0.568834\pi$
$662$	0	0
$663$	4.81864	0.187141
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−16.9410	−0.655957
$668$	0	0
$669$	16.1014	0.622516
$670$	0	0
$671$	−3.52811	−0.136201
$672$	0	0
$673$	6.14913	0.237031	0.118516	−	0.992952i	$-0.462186\pi$
$673$	6.14913	0.237031	0.118516	+	0.992952i	$0.462186\pi$
$674$	0	0
$675$	−20.6022	−0.792978
$676$	0	0
$677$	−47.8516	−1.83909	−0.919544	−	0.392988i	$-0.871441\pi$
$677$	−47.8516	−1.83909	−0.919544	+	0.392988i	$0.871441\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	17.2608	0.661434
$682$	0	0
$683$	−33.9242	−1.29807	−0.649036	−	0.760758i	$-0.724827\pi$
$683$	−33.9242	−1.29807	−0.649036	+	0.760758i	$0.724827\pi$
$684$	0	0
$685$	0.630076	0.0240740
$686$	0	0
$687$	8.97510	0.342422
$688$	0	0
$689$	27.1259	1.03342
$690$	0	0
$691$	18.9068	0.719250	0.359625	−	0.933097i	$-0.382905\pi$
$691$	18.9068	0.719250	0.359625	+	0.933097i	$0.382905\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.87440	−0.260761
$696$	0	0
$697$	10.5802	0.400755
$698$	0	0
$699$	−15.3598	−0.580962
$700$	0	0
$701$	9.98818	0.377248	0.188624	−	0.982049i	$-0.439597\pi$
$701$	9.98818	0.377248	0.188624	+	0.982049i	$0.439597\pi$
$702$	0	0
$703$	1.04719	0.0394953
$704$	0	0
$705$	2.80717	0.105724
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−19.4169	−0.729218	−0.364609	−	0.931161i	$-0.618797\pi$
$709$	−19.4169	−0.729218	−0.364609	+	0.931161i	$0.618797\pi$
$710$	0	0
$711$	−17.9422	−0.672884
$712$	0	0
$713$	19.8161	0.742119
$714$	0	0
$715$	−5.47927	−0.204913
$716$	0	0
$717$	12.9900	0.485122
$718$	0	0
$719$	−25.2132	−0.940293	−0.470146	−	0.882588i	$-0.655799\pi$
$719$	−25.2132	−0.940293	−0.470146	+	0.882588i	$0.655799\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−20.0461	−0.745522
$724$	0	0
$725$	−24.4282	−0.907240
$726$	0	0
$727$	−44.8631	−1.66388	−0.831940	−	0.554866i	$-0.812770\pi$
$727$	−44.8631	−1.66388	−0.831940	+	0.554866i	$0.812770\pi$
$728$	0	0
$729$	5.74659	0.212837
$730$	0	0
$731$	−3.38799	−0.125309
$732$	0	0
$733$	−38.2598	−1.41316	−0.706579	−	0.707634i	$-0.749763\pi$
$733$	−38.2598	−1.41316	−0.706579	+	0.707634i	$0.749763\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.589491	0.0217142
$738$	0	0
$739$	25.8460	0.950760	0.475380	−	0.879781i	$-0.342311\pi$
$739$	25.8460	0.950760	0.475380	+	0.879781i	$0.342311\pi$
$740$	0	0
$741$	−2.99911	−0.110175
$742$	0	0
$743$	10.7225	0.393369	0.196685	−	0.980467i	$-0.436983\pi$
$743$	10.7225	0.393369	0.196685	+	0.980467i	$0.436983\pi$
$744$	0	0
$745$	−1.18529	−0.0434255
$746$	0	0
$747$	5.65480	0.206898
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−42.2059	−1.54011	−0.770057	−	0.637975i	$-0.779772\pi$
$751$	−42.2059	−1.54011	−0.770057	+	0.637975i	$0.779772\pi$
$752$	0	0
$753$	2.30424	0.0839713
$754$	0	0
$755$	−1.97174	−0.0717590
$756$	0	0
$757$	−2.54838	−0.0926226	−0.0463113	−	0.998927i	$-0.514747\pi$
$757$	−2.54838	−0.0926226	−0.0463113	+	0.998927i	$0.514747\pi$
$758$	0	0
$759$	6.22081	0.225801
$760$	0	0
$761$	22.5154	0.816183	0.408091	−	0.912941i	$-0.366194\pi$
$761$	22.5154	0.816183	0.408091	+	0.912941i	$0.366194\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.51036	−0.0907623
$766$	0	0
$767$	−7.73676	−0.279358
$768$	0	0
$769$	−48.9303	−1.76447	−0.882235	−	0.470809i	$-0.843962\pi$
$769$	−48.9303	−1.76447	−0.882235	+	0.470809i	$0.843962\pi$
$770$	0	0
$771$	7.69099	0.276984
$772$	0	0
$773$	−16.3072	−0.586530	−0.293265	−	0.956031i	$-0.594742\pi$
$773$	−16.3072	−0.586530	−0.293265	+	0.956031i	$0.594742\pi$
$774$	0	0
$775$	28.5740	1.02641
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.58512	−0.235936
$780$	0	0
$781$	−10.4343	−0.373368
$782$	0	0
$783$	24.6951	0.882530
$784$	0	0
$785$	3.60321	0.128604
$786$	0	0
$787$	−22.6080	−0.805890	−0.402945	−	0.915224i	$-0.632013\pi$
$787$	−22.6080	−0.805890	−0.402945	+	0.915224i	$0.632013\pi$
$788$	0	0
$789$	−16.5209	−0.588158
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5.32514	−0.189101
$794$	0	0
$795$	4.77719	0.169429
$796$	0	0
$797$	2.27087	0.0804384	0.0402192	−	0.999191i	$-0.487194\pi$
$797$	2.27087	0.0804384	0.0402192	+	0.999191i	$0.487194\pi$
$798$	0	0
$799$	−7.43421	−0.263003
$800$	0	0
$801$	36.3478	1.28428
$802$	0	0
$803$	−17.6918	−0.624329
$804$	0	0
$805$	0	0
$806$	0	0
$807$	13.4522	0.473540
$808$	0	0
$809$	−20.6368	−0.725553	−0.362776	−	0.931876i	$-0.618171\pi$
$809$	−20.6368	−0.725553	−0.362776	+	0.931876i	$0.618171\pi$
$810$	0	0
$811$	34.1345	1.19863	0.599313	−	0.800515i	$-0.295441\pi$
$811$	34.1345	1.19863	0.599313	+	0.800515i	$0.295441\pi$
$812$	0	0
$813$	−3.54393	−0.124291
$814$	0	0
$815$	5.18851	0.181746
$816$	0	0
$817$	2.10867	0.0737732
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16.8551	−0.588246	−0.294123	−	0.955768i	$-0.595028\pi$
$821$	−16.8551	−0.588246	−0.294123	+	0.955768i	$0.595028\pi$
$822$	0	0
$823$	−12.6602	−0.441308	−0.220654	−	0.975352i	$-0.570819\pi$
$823$	−12.6602	−0.441308	−0.220654	+	0.975352i	$0.570819\pi$
$824$	0	0
$825$	8.97015	0.312301
$826$	0	0
$827$	21.4523	0.745970	0.372985	−	0.927837i	$-0.378334\pi$
$827$	21.4523	0.745970	0.372985	+	0.927837i	$0.378334\pi$
$828$	0	0
$829$	30.8325	1.07086	0.535429	−	0.844580i	$-0.320150\pi$
$829$	30.8325	1.07086	0.535429	+	0.844580i	$0.320150\pi$
$830$	0	0
$831$	8.74503	0.303362
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8.26163	0.285906
$836$	0	0
$837$	−28.8862	−0.998454
$838$	0	0
$839$	−21.1597	−0.730515	−0.365258	−	0.930906i	$-0.619019\pi$
$839$	−21.1597	−0.730515	−0.365258	+	0.930906i	$0.619019\pi$
$840$	0	0
$841$	0.281163	0.00969529
$842$	0	0
$843$	10.4226	0.358974
$844$	0	0
$845$	0.789217	0.0271499
$846$	0	0
$847$	0	0
$848$	0	0
$849$	15.0871	0.517789
$850$	0	0
$851$	−3.27844	−0.112384
$852$	0	0
$853$	31.8237	1.08962	0.544812	−	0.838558i	$-0.316601\pi$
$853$	31.8237	1.08962	0.544812	+	0.838558i	$0.316601\pi$
$854$	0	0
$855$	1.56244	0.0534344
$856$	0	0
$857$	8.16700	0.278980	0.139490	−	0.990224i	$-0.455454\pi$
$857$	8.16700	0.278980	0.139490	+	0.990224i	$0.455454\pi$
$858$	0	0
$859$	14.7826	0.504376	0.252188	−	0.967678i	$-0.418850\pi$
$859$	14.7826	0.504376	0.252188	+	0.967678i	$0.418850\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13.9084	−0.473446	−0.236723	−	0.971577i	$-0.576073\pi$
$863$	−13.9084	−0.473446	−0.236723	+	0.971577i	$0.576073\pi$
$864$	0	0
$865$	6.76320	0.229956
$866$	0	0
$867$	12.5526	0.426310
$868$	0	0
$869$	18.2648	0.619590
$870$	0	0
$871$	0.889747	0.0301479
$872$	0	0
$873$	−18.6879	−0.632489
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.42594	−0.115686	−0.0578429	−	0.998326i	$-0.518422\pi$
$877$	−3.42594	−0.115686	−0.0578429	+	0.998326i	$0.518422\pi$
$878$	0	0
$879$	−19.5584	−0.659689
$880$	0	0
$881$	29.1521	0.982158	0.491079	−	0.871115i	$-0.336603\pi$
$881$	29.1521	0.982158	0.491079	+	0.871115i	$0.336603\pi$
$882$	0	0
$883$	13.1808	0.443568	0.221784	−	0.975096i	$-0.428812\pi$
$883$	13.1808	0.443568	0.221784	+	0.975096i	$0.428812\pi$
$884$	0	0
$885$	−1.36253	−0.0458010
$886$	0	0
$887$	2.28567	0.0767453	0.0383726	−	0.999263i	$-0.487783\pi$
$887$	2.28567	0.0767453	0.0383726	+	0.999263i	$0.487783\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.28370	0.210512
$892$	0	0
$893$	4.62703	0.154838
$894$	0	0
$895$	−9.00472	−0.300995
$896$	0	0
$897$	9.38937	0.313502
$898$	0	0
$899$	−34.2506	−1.14232
$900$	0	0
$901$	−12.6514	−0.421478
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−15.6180	−0.519159
$906$	0	0
$907$	11.5657	0.384032	0.192016	−	0.981392i	$-0.438497\pi$
$907$	11.5657	0.384032	0.192016	+	0.981392i	$0.438497\pi$
$908$	0	0
$909$	−5.18784	−0.172070
$910$	0	0
$911$	−49.0967	−1.62665	−0.813323	−	0.581813i	$-0.802344\pi$
$911$	−49.0967	−1.62665	−0.813323	+	0.581813i	$0.802344\pi$
$912$	0	0
$913$	−5.75648	−0.190512
$914$	0	0
$915$	−0.937818	−0.0310033
$916$	0	0
$917$	0	0
$918$	0	0
$919$	27.9695	0.922628	0.461314	−	0.887237i	$-0.347378\pi$
$919$	27.9695	0.922628	0.461314	+	0.887237i	$0.347378\pi$
$920$	0	0
$921$	12.8109	0.422133
$922$	0	0
$923$	−15.7490	−0.518384
$924$	0	0
$925$	−4.72738	−0.155435
$926$	0	0
$927$	21.3296	0.700557
$928$	0	0
$929$	14.7519	0.483996	0.241998	−	0.970277i	$-0.422197\pi$
$929$	14.7519	0.483996	0.241998	+	0.970277i	$0.422197\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	22.5229	0.737366
$934$	0	0
$935$	2.55550	0.0835737
$936$	0	0
$937$	−29.7752	−0.972715	−0.486357	−	0.873760i	$-0.661675\pi$
$937$	−29.7752	−0.972715	−0.486357	+	0.873760i	$0.661675\pi$
$938$	0	0
$939$	12.6332	0.412269
$940$	0	0
$941$	−6.35846	−0.207280	−0.103640	−	0.994615i	$-0.533049\pi$
$941$	−6.35846	−0.207280	−0.103640	+	0.994615i	$0.533049\pi$
$942$	0	0
$943$	20.6162	0.671354
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.7306	0.381193	0.190596	−	0.981668i	$-0.438958\pi$
$947$	11.7306	0.381193	0.190596	+	0.981668i	$0.438958\pi$
$948$	0	0
$949$	−26.7030	−0.866818
$950$	0	0
$951$	−19.2905	−0.625537
$952$	0	0
$953$	−42.9533	−1.39139	−0.695697	−	0.718336i	$-0.744904\pi$
$953$	−42.9533	−1.39139	−0.695697	+	0.718336i	$0.744904\pi$
$954$	0	0
$955$	15.8115	0.511648
$956$	0	0
$957$	−10.7522	−0.347569
$958$	0	0
$959$	0	0
$960$	0	0
$961$	9.06351	0.292371
$962$	0	0
$963$	−12.3885	−0.399213
$964$	0	0
$965$	13.6382	0.439030
$966$	0	0
$967$	43.8989	1.41169	0.705846	−	0.708365i	$-0.250567\pi$
$967$	43.8989	1.41169	0.705846	+	0.708365i	$0.250567\pi$
$968$	0	0
$969$	1.39877	0.0449349
$970$	0	0
$971$	47.2077	1.51497	0.757483	−	0.652854i	$-0.226429\pi$
$971$	47.2077	1.51497	0.757483	+	0.652854i	$0.226429\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	13.5391	0.433598
$976$	0	0
$977$	−26.2307	−0.839196	−0.419598	−	0.907710i	$-0.637829\pi$
$977$	−26.2307	−0.839196	−0.419598	+	0.907710i	$0.637829\pi$
$978$	0	0
$979$	−37.0013	−1.18257
$980$	0	0
$981$	−15.5131	−0.495295
$982$	0	0
$983$	8.93102	0.284855	0.142428	−	0.989805i	$-0.454509\pi$
$983$	8.93102	0.284855	0.142428	+	0.989805i	$0.454509\pi$
$984$	0	0
$985$	0.224352	0.00714846
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.60166	−0.209921
$990$	0	0
$991$	14.3138	0.454693	0.227347	−	0.973814i	$-0.426995\pi$
$991$	14.3138	0.454693	0.227347	+	0.973814i	$0.426995\pi$
$992$	0	0
$993$	4.10855	0.130381
$994$	0	0
$995$	18.0535	0.572334
$996$	0	0
$997$	25.9196	0.820881	0.410441	−	0.911887i	$-0.365375\pi$
$997$	25.9196	0.820881	0.410441	+	0.911887i	$0.365375\pi$
$998$	0	0
$999$	4.77903	0.151202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bm.1.3	✓	6
7.6	odd	2		7448.2.a.bn.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bm.1.3	✓	6	1.1	even	1	trivial
7448.2.a.bn.1.4	yes	6	7.6	odd	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-0.870589 q^{3} -0.696873 q^{5} -2.24208 q^{9} +O(q^{10})\)	\(q-0.870589 q^{3} -0.696873 q^{5} -2.24208 q^{9} +2.28239 q^{11} +3.44492 q^{13} +0.606690 q^{15} -1.60669 q^{17} +1.00000 q^{19} -3.13072 q^{23} -4.51437 q^{25} +4.56369 q^{27} +5.41121 q^{29} -6.32957 q^{31} -1.98702 q^{33} +1.04719 q^{37} -2.99911 q^{39} -6.58512 q^{41} +2.10867 q^{43} +1.56244 q^{45} +4.62703 q^{47} +1.39877 q^{51} +7.87418 q^{53} -1.59054 q^{55} -0.870589 q^{57} -2.24584 q^{59} -1.54579 q^{61} -2.40067 q^{65} +0.258278 q^{67} +2.72557 q^{69} -4.57165 q^{71} -7.75142 q^{73} +3.93016 q^{75} +8.00248 q^{79} +2.75313 q^{81} -2.52213 q^{83} +1.11966 q^{85} -4.71094 q^{87} -16.2117 q^{89} +5.51046 q^{93} -0.696873 q^{95} +8.33509 q^{97} -5.11729 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - q^{5} + 3 q^{9}+O(q^{10}) \) 6 * q - 3 * q^3 - q^5 + 3 * q^9	\( 6 q - 3 q^{3} - q^{5} + 3 q^{9} + 3 q^{11} - 6 q^{13} - 8 q^{15} + 2 q^{17} + 6 q^{19} + 2 q^{23} + 5 q^{25} - 6 q^{27} - q^{29} - 14 q^{31} - 19 q^{33} - 7 q^{37} + 22 q^{39} + 21 q^{41} + q^{43} + 4 q^{45} - 23 q^{47} + 2 q^{51} - 15 q^{53} - 2 q^{55} - 3 q^{57} - 25 q^{59} - 21 q^{61} + 6 q^{65} + 2 q^{67} + 20 q^{69} + 13 q^{71} + 2 q^{73} - 24 q^{75} - 17 q^{79} - 2 q^{81} - 4 q^{83} + 40 q^{85} - 30 q^{87} + q^{89} + 6 q^{93} - q^{95} + 13 q^{97} + 41 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 - q^5 + 3 * q^9 + 3 * q^11 - 6 * q^13 - 8 * q^15 + 2 * q^17 + 6 * q^19 + 2 * q^23 + 5 * q^25 - 6 * q^27 - q^29 - 14 * q^31 - 19 * q^33 - 7 * q^37 + 22 * q^39 + 21 * q^41 + q^43 + 4 * q^45 - 23 * q^47 + 2 * q^51 - 15 * q^53 - 2 * q^55 - 3 * q^57 - 25 * q^59 - 21 * q^61 + 6 * q^65 + 2 * q^67 + 20 * q^69 + 13 * q^71 + 2 * q^73 - 24 * q^75 - 17 * q^79 - 2 * q^81 - 4 * q^83 + 40 * q^85 - 30 * q^87 + q^89 + 6 * q^93 - q^95 + 13 * q^97 + 41 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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