LMFDB - Embedded newform 7448.2.a.bo.1.5

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 13x^{4} + 50x^{3} - 53x^{2} - 25x + 28 $ x^7 - x^6 - 14x^5 + 13x^4 + 50x^3 - 53x^2 - 25*x + 28
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			1.5
Root			$-0.779807$ 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)

$f(q)$	$=$	$q+0.779807 q^{3} +1.35094 q^{5} -2.39190 q^{9} +O(q^{10})$	$q+0.779807 q^{3} +1.35094 q^{5} -2.39190 q^{9} +4.41231 q^{11} +3.18423 q^{13} +1.05348 q^{15} -6.98121 q^{17} -1.00000 q^{19} -2.93236 q^{23} -3.17495 q^{25} -4.20464 q^{27} +0.446035 q^{29} -5.22518 q^{31} +3.44075 q^{33} +2.03871 q^{37} +2.48308 q^{39} +1.38401 q^{41} -8.10268 q^{43} -3.23133 q^{45} -6.87510 q^{47} -5.44399 q^{51} -12.2939 q^{53} +5.96079 q^{55} -0.779807 q^{57} -3.86258 q^{59} +2.87510 q^{61} +4.30172 q^{65} -3.83104 q^{67} -2.28667 q^{69} -1.63827 q^{71} +1.47144 q^{73} -2.47585 q^{75} -3.84118 q^{79} +3.89690 q^{81} +0.170087 q^{83} -9.43123 q^{85} +0.347821 q^{87} -1.60106 q^{89} -4.07463 q^{93} -1.35094 q^{95} +4.88299 q^{97} -10.5538 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{3} - q^{5} + 8 q^{9}+O(q^{10}) $ 7 * q - q^3 - q^5 + 8 * q^9	$ 7 q - q^{3} - q^{5} + 8 q^{9} + 3 q^{11} - 6 q^{13} - 4 q^{15} - 10 q^{17} - 7 q^{19} - 2 q^{23} + 8 q^{25} + 2 q^{27} + 3 q^{29} + 6 q^{31} - 21 q^{33} - 7 q^{37} - 2 q^{39} - 9 q^{41} + q^{43} - 24 q^{45} - 15 q^{47} + 6 q^{51} + 5 q^{53} + 6 q^{55} + q^{57} + 9 q^{59} - 13 q^{61} - 6 q^{65} - 2 q^{67} + 20 q^{69} - q^{71} - 42 q^{73} - 40 q^{75} - 3 q^{79} - q^{81} + 12 q^{83} - 16 q^{85} + 2 q^{87} - 41 q^{89} - 2 q^{93} + q^{95} - 5 q^{97} + 9 q^{99}+O(q^{100}) $ 7 * q - q^3 - q^5 + 8 * q^9 + 3 * q^11 - 6 * q^13 - 4 * q^15 - 10 * q^17 - 7 * q^19 - 2 * q^23 + 8 * q^25 + 2 * q^27 + 3 * q^29 + 6 * q^31 - 21 * q^33 - 7 * q^37 - 2 * q^39 - 9 * q^41 + q^43 - 24 * q^45 - 15 * q^47 + 6 * q^51 + 5 * q^53 + 6 * q^55 + q^57 + 9 * q^59 - 13 * q^61 - 6 * q^65 - 2 * q^67 + 20 * q^69 - q^71 - 42 * q^73 - 40 * q^75 - 3 * q^79 - q^81 + 12 * q^83 - 16 * q^85 + 2 * q^87 - 41 * q^89 - 2 * q^93 + q^95 - 5 * q^97 + 9 * 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
$3$	0.779807	0.450222	0.225111	−	0.974333i	$-0.427726\pi$
$3$	0.779807	0.450222	0.225111	+	0.974333i	$0.427726\pi$
$4$	0	0
$5$	1.35094	0.604161	0.302080	−	0.953282i	$-0.402319\pi$
$5$	1.35094	0.604161	0.302080	+	0.953282i	$0.402319\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.39190	−0.797301
$10$	0	0
$11$	4.41231	1.33036	0.665182	−	0.746682i	$-0.268354\pi$
$11$	4.41231	1.33036	0.665182	+	0.746682i	$0.268354\pi$
$12$	0	0
$13$	3.18423	0.883146	0.441573	−	0.897225i	$-0.354421\pi$
$13$	3.18423	0.883146	0.441573	+	0.897225i	$0.354421\pi$
$14$	0	0
$15$	1.05348	0.272006
$16$	0	0
$17$	−6.98121	−1.69319	−0.846596	−	0.532236i	$-0.821352\pi$
$17$	−6.98121	−1.69319	−0.846596	+	0.532236i	$0.821352\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.93236	−0.611439	−0.305719	−	0.952122i	$-0.598897\pi$
$23$	−2.93236	−0.611439	−0.305719	+	0.952122i	$0.598897\pi$
$24$	0	0
$25$	−3.17495	−0.634990
$26$	0	0
$27$	−4.20464	−0.809183
$28$	0	0
$29$	0.446035	0.0828267	0.0414134	−	0.999142i	$-0.486814\pi$
$29$	0.446035	0.0828267	0.0414134	+	0.999142i	$0.486814\pi$
$30$	0	0
$31$	−5.22518	−0.938471	−0.469235	−	0.883073i	$-0.655470\pi$
$31$	−5.22518	−0.938471	−0.469235	+	0.883073i	$0.655470\pi$
$32$	0	0
$33$	3.44075	0.598958
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.03871	0.335162	0.167581	−	0.985858i	$-0.446404\pi$
$37$	2.03871	0.335162	0.167581	+	0.985858i	$0.446404\pi$
$38$	0	0
$39$	2.48308	0.397611
$40$	0	0
$41$	1.38401	0.216146	0.108073	−	0.994143i	$-0.465532\pi$
$41$	1.38401	0.216146	0.108073	+	0.994143i	$0.465532\pi$
$42$	0	0
$43$	−8.10268	−1.23565	−0.617824	−	0.786317i	$-0.711985\pi$
$43$	−8.10268	−1.23565	−0.617824	+	0.786317i	$0.711985\pi$
$44$	0	0
$45$	−3.23133	−0.481698
$46$	0	0
$47$	−6.87510	−1.00284	−0.501418	−	0.865205i	$-0.667188\pi$
$47$	−6.87510	−1.00284	−0.501418	+	0.865205i	$0.667188\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−5.44399	−0.762311
$52$	0	0
$53$	−12.2939	−1.68870	−0.844351	−	0.535790i	$-0.820014\pi$
$53$	−12.2939	−1.68870	−0.844351	+	0.535790i	$0.820014\pi$
$54$	0	0
$55$	5.96079	0.803753
$56$	0	0
$57$	−0.779807	−0.103288
$58$	0	0
$59$	−3.86258	−0.502865	−0.251433	−	0.967875i	$-0.580902\pi$
$59$	−3.86258	−0.502865	−0.251433	+	0.967875i	$0.580902\pi$
$60$	0	0
$61$	2.87510	0.368119	0.184059	−	0.982915i	$-0.441076\pi$
$61$	2.87510	0.368119	0.184059	+	0.982915i	$0.441076\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.30172	0.533562
$66$	0	0
$67$	−3.83104	−0.468036	−0.234018	−	0.972232i	$-0.575187\pi$
$67$	−3.83104	−0.468036	−0.234018	+	0.972232i	$0.575187\pi$
$68$	0	0
$69$	−2.28667	−0.275283
$70$	0	0
$71$	−1.63827	−0.194427	−0.0972137	−	0.995264i	$-0.530993\pi$
$71$	−1.63827	−0.194427	−0.0972137	+	0.995264i	$0.530993\pi$
$72$	0	0
$73$	1.47144	0.172219	0.0861095	−	0.996286i	$-0.472557\pi$
$73$	1.47144	0.172219	0.0861095	+	0.996286i	$0.472557\pi$
$74$	0	0
$75$	−2.47585	−0.285886
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−3.84118	−0.432166	−0.216083	−	0.976375i	$-0.569328\pi$
$79$	−3.84118	−0.432166	−0.216083	+	0.976375i	$0.569328\pi$
$80$	0	0
$81$	3.89690	0.432989
$82$	0	0
$83$	0.170087	0.0186695	0.00933475	−	0.999956i	$-0.497029\pi$
$83$	0.170087	0.0186695	0.00933475	+	0.999956i	$0.497029\pi$
$84$	0	0
$85$	−9.43123	−1.02296
$86$	0	0
$87$	0.347821	0.0372904
$88$	0	0
$89$	−1.60106	−0.169712	−0.0848562	−	0.996393i	$-0.527043\pi$
$89$	−1.60106	−0.169712	−0.0848562	+	0.996393i	$0.527043\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.07463	−0.422520
$94$	0	0
$95$	−1.35094	−0.138604
$96$	0	0
$97$	4.88299	0.495793	0.247896	−	0.968787i	$-0.420261\pi$
$97$	4.88299	0.495793	0.247896	+	0.968787i	$0.420261\pi$
$98$	0	0
$99$	−10.5538	−1.06070
$100$	0	0
$101$	6.00096	0.597118	0.298559	−	0.954391i	$-0.403494\pi$
$101$	6.00096	0.597118	0.298559	+	0.954391i	$0.403494\pi$
$102$	0	0
$103$	−14.8601	−1.46421	−0.732105	−	0.681192i	$-0.761462\pi$
$103$	−14.8601	−1.46421	−0.732105	+	0.681192i	$0.761462\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.9911	1.83593	0.917967	−	0.396656i	$-0.129829\pi$
$107$	18.9911	1.83593	0.917967	+	0.396656i	$0.129829\pi$
$108$	0	0
$109$	−7.74246	−0.741593	−0.370796	−	0.928714i	$-0.620915\pi$
$109$	−7.74246	−0.741593	−0.370796	+	0.928714i	$0.620915\pi$
$110$	0	0
$111$	1.58980	0.150897
$112$	0	0
$113$	5.79082	0.544755	0.272377	−	0.962190i	$-0.412190\pi$
$113$	5.79082	0.544755	0.272377	+	0.962190i	$0.412190\pi$
$114$	0	0
$115$	−3.96145	−0.369407
$116$	0	0
$117$	−7.61636	−0.704133
$118$	0	0
$119$	0	0
$120$	0	0
$121$	8.46852	0.769866
$122$	0	0
$123$	1.07926	0.0973134
$124$	0	0
$125$	−11.0439	−0.987797
$126$	0	0
$127$	−9.12309	−0.809544	−0.404772	−	0.914418i	$-0.632649\pi$
$127$	−9.12309	−0.809544	−0.404772	+	0.914418i	$0.632649\pi$
$128$	0	0
$129$	−6.31852	−0.556315
$130$	0	0
$131$	−0.136031	−0.0118850	−0.00594252	−	0.999982i	$-0.501892\pi$
$131$	−0.136031	−0.0118850	−0.00594252	+	0.999982i	$0.501892\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.68024	−0.488877
$136$	0	0
$137$	10.3101	0.880852	0.440426	−	0.897789i	$-0.354827\pi$
$137$	10.3101	0.880852	0.440426	+	0.897789i	$0.354827\pi$
$138$	0	0
$139$	8.77677	0.744436	0.372218	−	0.928145i	$-0.378597\pi$
$139$	8.77677	0.744436	0.372218	+	0.928145i	$0.378597\pi$
$140$	0	0
$141$	−5.36125	−0.451499
$142$	0	0
$143$	14.0498	1.17490
$144$	0	0
$145$	0.602569	0.0500407
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.70209	−0.467134	−0.233567	−	0.972341i	$-0.575040\pi$
$149$	−5.70209	−0.467134	−0.233567	+	0.972341i	$0.575040\pi$
$150$	0	0
$151$	4.93421	0.401540	0.200770	−	0.979638i	$-0.435656\pi$
$151$	4.93421	0.401540	0.200770	+	0.979638i	$0.435656\pi$
$152$	0	0
$153$	16.6984	1.34998
$154$	0	0
$155$	−7.05894	−0.566987
$156$	0	0
$157$	−20.0885	−1.60324	−0.801619	−	0.597835i	$-0.796028\pi$
$157$	−20.0885	−1.60324	−0.801619	+	0.597835i	$0.796028\pi$
$158$	0	0
$159$	−9.58690	−0.760290
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.93130	0.386250	0.193125	−	0.981174i	$-0.438138\pi$
$163$	4.93130	0.386250	0.193125	+	0.981174i	$0.438138\pi$
$164$	0	0
$165$	4.64827	0.361867
$166$	0	0
$167$	14.8508	1.14919	0.574594	−	0.818438i	$-0.305160\pi$
$167$	14.8508	1.14919	0.574594	+	0.818438i	$0.305160\pi$
$168$	0	0
$169$	−2.86070	−0.220054
$170$	0	0
$171$	2.39190	0.182913
$172$	0	0
$173$	−22.3922	−1.70245	−0.851225	−	0.524801i	$-0.824140\pi$
$173$	−22.3922	−1.70245	−0.851225	+	0.524801i	$0.824140\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.01207	−0.226401
$178$	0	0
$179$	−6.46680	−0.483351	−0.241676	−	0.970357i	$-0.577697\pi$
$179$	−6.46680	−0.483351	−0.241676	+	0.970357i	$0.577697\pi$
$180$	0	0
$181$	−7.68136	−0.570951	−0.285475	−	0.958386i	$-0.592152\pi$
$181$	−7.68136	−0.570951	−0.285475	+	0.958386i	$0.592152\pi$
$182$	0	0
$183$	2.24202	0.165735
$184$	0	0
$185$	2.75419	0.202492
$186$	0	0
$187$	−30.8033	−2.25256
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.1056	1.16536	0.582680	−	0.812702i	$-0.302004\pi$
$191$	16.1056	1.16536	0.582680	+	0.812702i	$0.302004\pi$
$192$	0	0
$193$	6.72760	0.484263	0.242132	−	0.970243i	$-0.422153\pi$
$193$	6.72760	0.484263	0.242132	+	0.970243i	$0.422153\pi$
$194$	0	0
$195$	3.35451	0.240221
$196$	0	0
$197$	16.9465	1.20739	0.603693	−	0.797217i	$-0.293695\pi$
$197$	16.9465	1.20739	0.603693	+	0.797217i	$0.293695\pi$
$198$	0	0
$199$	8.85719	0.627869	0.313935	−	0.949445i	$-0.398353\pi$
$199$	8.85719	0.627869	0.313935	+	0.949445i	$0.398353\pi$
$200$	0	0
$201$	−2.98747	−0.210720
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.86972	0.130587
$206$	0	0
$207$	7.01391	0.487500
$208$	0	0
$209$	−4.41231	−0.305206
$210$	0	0
$211$	24.4661	1.68432	0.842159	−	0.539230i	$-0.181285\pi$
$211$	24.4661	1.68432	0.842159	+	0.539230i	$0.181285\pi$
$212$	0	0
$213$	−1.27754	−0.0875354
$214$	0	0
$215$	−10.9463	−0.746530
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.14744	0.0775367
$220$	0	0
$221$	−22.2298	−1.49533
$222$	0	0
$223$	28.0782	1.88026	0.940128	−	0.340820i	$-0.110705\pi$
$223$	28.0782	1.88026	0.940128	+	0.340820i	$0.110705\pi$
$224$	0	0
$225$	7.59416	0.506278
$226$	0	0
$227$	0.498256	0.0330704	0.0165352	−	0.999863i	$-0.494736\pi$
$227$	0.498256	0.0330704	0.0165352	+	0.999863i	$0.494736\pi$
$228$	0	0
$229$	16.8409	1.11288	0.556439	−	0.830888i	$-0.312167\pi$
$229$	16.8409	1.11288	0.556439	+	0.830888i	$0.312167\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.4097	−1.73016	−0.865080	−	0.501634i	$-0.832733\pi$
$233$	−26.4097	−1.73016	−0.865080	+	0.501634i	$0.832733\pi$
$234$	0	0
$235$	−9.28788	−0.605875
$236$	0	0
$237$	−2.99538	−0.194571
$238$	0	0
$239$	16.0092	1.03555	0.517774	−	0.855517i	$-0.326761\pi$
$239$	16.0092	1.03555	0.517774	+	0.855517i	$0.326761\pi$
$240$	0	0
$241$	−20.5906	−1.32636	−0.663180	−	0.748460i	$-0.730793\pi$
$241$	−20.5906	−1.32636	−0.663180	+	0.748460i	$0.730793\pi$
$242$	0	0
$243$	15.6527	1.00412
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.18423	−0.202608
$248$	0	0
$249$	0.132635	0.00840541
$250$	0	0
$251$	−11.6455	−0.735058	−0.367529	−	0.930012i	$-0.619796\pi$
$251$	−11.6455	−0.735058	−0.367529	+	0.930012i	$0.619796\pi$
$252$	0	0
$253$	−12.9385	−0.813435
$254$	0	0
$255$	−7.35453	−0.460559
$256$	0	0
$257$	−30.5900	−1.90815	−0.954077	−	0.299561i	$-0.903160\pi$
$257$	−30.5900	−1.90815	−0.954077	+	0.299561i	$0.903160\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.06687	−0.0660378
$262$	0	0
$263$	−3.88338	−0.239460	−0.119730	−	0.992807i	$-0.538203\pi$
$263$	−3.88338	−0.239460	−0.119730	+	0.992807i	$0.538203\pi$
$264$	0	0
$265$	−16.6084	−1.02025
$266$	0	0
$267$	−1.24852	−0.0764082
$268$	0	0
$269$	−10.2048	−0.622197	−0.311098	−	0.950378i	$-0.600697\pi$
$269$	−10.2048	−0.622197	−0.311098	+	0.950378i	$0.600697\pi$
$270$	0	0
$271$	18.3602	1.11530	0.557651	−	0.830075i	$-0.311703\pi$
$271$	18.3602	1.11530	0.557651	+	0.830075i	$0.311703\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−14.0089	−0.844767
$276$	0	0
$277$	6.57918	0.395304	0.197652	−	0.980272i	$-0.436668\pi$
$277$	6.57918	0.395304	0.197652	+	0.980272i	$0.436668\pi$
$278$	0	0
$279$	12.4981	0.748243
$280$	0	0
$281$	21.3550	1.27393	0.636966	−	0.770892i	$-0.280189\pi$
$281$	21.3550	1.27393	0.636966	+	0.770892i	$0.280189\pi$
$282$	0	0
$283$	−33.1920	−1.97306	−0.986530	−	0.163578i	$-0.947697\pi$
$283$	−33.1920	−1.97306	−0.986530	+	0.163578i	$0.947697\pi$
$284$	0	0
$285$	−1.05348	−0.0624025
$286$	0	0
$287$	0	0
$288$	0	0
$289$	31.7373	1.86690
$290$	0	0
$291$	3.80779	0.223217
$292$	0	0
$293$	−29.7952	−1.74065	−0.870326	−	0.492476i	$-0.836092\pi$
$293$	−29.7952	−1.74065	−0.870326	+	0.492476i	$0.836092\pi$
$294$	0	0
$295$	−5.21813	−0.303811
$296$	0	0
$297$	−18.5522	−1.07651
$298$	0	0
$299$	−9.33729	−0.539989
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4.67959	0.268835
$304$	0	0
$305$	3.88410	0.222403
$306$	0	0
$307$	−11.0513	−0.630730	−0.315365	−	0.948970i	$-0.602127\pi$
$307$	−11.0513	−0.630730	−0.315365	+	0.948970i	$0.602127\pi$
$308$	0	0
$309$	−11.5880	−0.659219
$310$	0	0
$311$	7.20227	0.408403	0.204202	−	0.978929i	$-0.434540\pi$
$311$	7.20227	0.408403	0.204202	+	0.978929i	$0.434540\pi$
$312$	0	0
$313$	−21.7818	−1.23118	−0.615590	−	0.788066i	$-0.711082\pi$
$313$	−21.7818	−1.23118	−0.615590	+	0.788066i	$0.711082\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.83325	−0.496125	−0.248062	−	0.968744i	$-0.579794\pi$
$317$	−8.83325	−0.496125	−0.248062	+	0.968744i	$0.579794\pi$
$318$	0	0
$319$	1.96805	0.110190
$320$	0	0
$321$	14.8094	0.826578
$322$	0	0
$323$	6.98121	0.388445
$324$	0	0
$325$	−10.1098	−0.560788
$326$	0	0
$327$	−6.03762	−0.333881
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−12.5243	−0.688395	−0.344198	−	0.938897i	$-0.611849\pi$
$331$	−12.5243	−0.688395	−0.344198	+	0.938897i	$0.611849\pi$
$332$	0	0
$333$	−4.87640	−0.267225
$334$	0	0
$335$	−5.17552	−0.282769
$336$	0	0
$337$	19.9056	1.08433	0.542164	−	0.840273i	$-0.317605\pi$
$337$	19.9056	1.08433	0.542164	+	0.840273i	$0.317605\pi$
$338$	0	0
$339$	4.51572	0.245260
$340$	0	0
$341$	−23.0552	−1.24851
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.08917	−0.166315
$346$	0	0
$347$	−2.49887	−0.134146	−0.0670732	−	0.997748i	$-0.521366\pi$
$347$	−2.49887	−0.134146	−0.0670732	+	0.997748i	$0.521366\pi$
$348$	0	0
$349$	1.21989	0.0652993	0.0326497	−	0.999467i	$-0.489605\pi$
$349$	1.21989	0.0652993	0.0326497	+	0.999467i	$0.489605\pi$
$350$	0	0
$351$	−13.3885	−0.714627
$352$	0	0
$353$	−31.1955	−1.66037	−0.830186	−	0.557487i	$-0.811766\pi$
$353$	−31.1955	−1.66037	−0.830186	+	0.557487i	$0.811766\pi$
$354$	0	0
$355$	−2.21322	−0.117465
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.3343	−0.598199	−0.299100	−	0.954222i	$-0.596686\pi$
$359$	−11.3343	−0.598199	−0.299100	+	0.954222i	$0.596686\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	6.60381	0.346610
$364$	0	0
$365$	1.98783	0.104048
$366$	0	0
$367$	1.22126	0.0637493	0.0318746	−	0.999492i	$-0.489852\pi$
$367$	1.22126	0.0637493	0.0318746	+	0.999492i	$0.489852\pi$
$368$	0	0
$369$	−3.31041	−0.172333
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−18.3376	−0.949486	−0.474743	−	0.880124i	$-0.657459\pi$
$373$	−18.3376	−0.949486	−0.474743	+	0.880124i	$0.657459\pi$
$374$	0	0
$375$	−8.61211	−0.444727
$376$	0	0
$377$	1.42028	0.0731481
$378$	0	0
$379$	−24.3363	−1.25007	−0.625035	−	0.780597i	$-0.714915\pi$
$379$	−24.3363	−1.25007	−0.625035	+	0.780597i	$0.714915\pi$
$380$	0	0
$381$	−7.11425	−0.364474
$382$	0	0
$383$	−11.0978	−0.567072	−0.283536	−	0.958962i	$-0.591508\pi$
$383$	−11.0978	−0.567072	−0.283536	+	0.958962i	$0.591508\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	19.3808	0.985182
$388$	0	0
$389$	−0.260670	−0.0132165	−0.00660825	−	0.999978i	$-0.502103\pi$
$389$	−0.260670	−0.0132165	−0.00660825	+	0.999978i	$0.502103\pi$
$390$	0	0
$391$	20.4714	1.03528
$392$	0	0
$393$	−0.106078	−0.00535090
$394$	0	0
$395$	−5.18922	−0.261098
$396$	0	0
$397$	13.8153	0.693370	0.346685	−	0.937982i	$-0.387307\pi$
$397$	13.8153	0.693370	0.346685	+	0.937982i	$0.387307\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.7826	1.28752	0.643761	−	0.765226i	$-0.277373\pi$
$401$	25.7826	1.28752	0.643761	+	0.765226i	$0.277373\pi$
$402$	0	0
$403$	−16.6382	−0.828806
$404$	0	0
$405$	5.26449	0.261595
$406$	0	0
$407$	8.99544	0.445887
$408$	0	0
$409$	−31.9070	−1.57770	−0.788850	−	0.614585i	$-0.789323\pi$
$409$	−31.9070	−1.57770	−0.788850	+	0.614585i	$0.789323\pi$
$410$	0	0
$411$	8.03989	0.396578
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0.229778	0.0112794
$416$	0	0
$417$	6.84419	0.335161
$418$	0	0
$419$	38.6127	1.88635	0.943177	−	0.332292i	$-0.107822\pi$
$419$	38.6127	1.88635	0.943177	+	0.332292i	$0.107822\pi$
$420$	0	0
$421$	−6.44899	−0.314305	−0.157152	−	0.987574i	$-0.550231\pi$
$421$	−6.44899	−0.314305	−0.157152	+	0.987574i	$0.550231\pi$
$422$	0	0
$423$	16.4446	0.799562
$424$	0	0
$425$	22.1650	1.07516
$426$	0	0
$427$	0	0
$428$	0	0
$429$	10.9561	0.528967
$430$	0	0
$431$	10.7138	0.516067	0.258033	−	0.966136i	$-0.416926\pi$
$431$	10.7138	0.516067	0.258033	+	0.966136i	$0.416926\pi$
$432$	0	0
$433$	4.99389	0.239991	0.119995	−	0.992774i	$-0.461712\pi$
$433$	4.99389	0.239991	0.119995	+	0.992774i	$0.461712\pi$
$434$	0	0
$435$	0.469888	0.0225294
$436$	0	0
$437$	2.93236	0.140274
$438$	0	0
$439$	−4.51302	−0.215395	−0.107697	−	0.994184i	$-0.534348\pi$
$439$	−4.51302	−0.215395	−0.107697	+	0.994184i	$0.534348\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.08651	−0.289179	−0.144589	−	0.989492i	$-0.546186\pi$
$443$	−6.08651	−0.289179	−0.144589	+	0.989492i	$0.546186\pi$
$444$	0	0
$445$	−2.16295	−0.102534
$446$	0	0
$447$	−4.44653	−0.210314
$448$	0	0
$449$	6.20512	0.292838	0.146419	−	0.989223i	$-0.453225\pi$
$449$	6.20512	0.292838	0.146419	+	0.989223i	$0.453225\pi$
$450$	0	0
$451$	6.10668	0.287552
$452$	0	0
$453$	3.84773	0.180782
$454$	0	0
$455$	0	0
$456$	0	0
$457$	38.3435	1.79364	0.896818	−	0.442400i	$-0.145873\pi$
$457$	38.3435	1.79364	0.896818	+	0.442400i	$0.145873\pi$
$458$	0	0
$459$	29.3535	1.37010
$460$	0	0
$461$	26.5778	1.23785	0.618925	−	0.785450i	$-0.287569\pi$
$461$	26.5778	1.23785	0.618925	+	0.785450i	$0.287569\pi$
$462$	0	0
$463$	−13.5240	−0.628515	−0.314257	−	0.949338i	$-0.601755\pi$
$463$	−13.5240	−0.628515	−0.314257	+	0.949338i	$0.601755\pi$
$464$	0	0
$465$	−5.50461	−0.255270
$466$	0	0
$467$	−17.8354	−0.825322	−0.412661	−	0.910885i	$-0.635401\pi$
$467$	−17.8354	−0.825322	−0.412661	+	0.910885i	$0.635401\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−15.6652	−0.721812
$472$	0	0
$473$	−35.7516	−1.64386
$474$	0	0
$475$	3.17495	0.145677
$476$	0	0
$477$	29.4059	1.34640
$478$	0	0
$479$	−12.6357	−0.577340	−0.288670	−	0.957429i	$-0.593213\pi$
$479$	−12.6357	−0.577340	−0.288670	+	0.957429i	$0.593213\pi$
$480$	0	0
$481$	6.49172	0.295997
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.59666	0.299539
$486$	0	0
$487$	17.9014	0.811188	0.405594	−	0.914053i	$-0.367065\pi$
$487$	17.9014	0.811188	0.405594	+	0.914053i	$0.367065\pi$
$488$	0	0
$489$	3.84546	0.173898
$490$	0	0
$491$	4.98195	0.224832	0.112416	−	0.993661i	$-0.464141\pi$
$491$	4.98195	0.224832	0.112416	+	0.993661i	$0.464141\pi$
$492$	0	0
$493$	−3.11387	−0.140241
$494$	0	0
$495$	−14.2576	−0.640833
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13.3109	−0.595879	−0.297940	−	0.954585i	$-0.596299\pi$
$499$	−13.3109	−0.595879	−0.297940	+	0.954585i	$0.596299\pi$
$500$	0	0
$501$	11.5807	0.517390
$502$	0	0
$503$	8.99468	0.401053	0.200527	−	0.979688i	$-0.435735\pi$
$503$	8.99468	0.401053	0.200527	+	0.979688i	$0.435735\pi$
$504$	0	0
$505$	8.10697	0.360755
$506$	0	0
$507$	−2.23079	−0.0990729
$508$	0	0
$509$	−15.3199	−0.679043	−0.339522	−	0.940598i	$-0.610265\pi$
$509$	−15.3199	−0.679043	−0.339522	+	0.940598i	$0.610265\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.20464	0.185639
$514$	0	0
$515$	−20.0752	−0.884618
$516$	0	0
$517$	−30.3351	−1.33414
$518$	0	0
$519$	−17.4616	−0.766480
$520$	0	0
$521$	−6.44799	−0.282492	−0.141246	−	0.989975i	$-0.545111\pi$
$521$	−6.44799	−0.282492	−0.141246	+	0.989975i	$0.545111\pi$
$522$	0	0
$523$	12.8087	0.560086	0.280043	−	0.959987i	$-0.409651\pi$
$523$	12.8087	0.560086	0.280043	+	0.959987i	$0.409651\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36.4781	1.58901
$528$	0	0
$529$	−14.4013	−0.626143
$530$	0	0
$531$	9.23891	0.400935
$532$	0	0
$533$	4.40699	0.190888
$534$	0	0
$535$	25.6559	1.10920
$536$	0	0
$537$	−5.04286	−0.217615
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.0025	0.730995	0.365497	−	0.930812i	$-0.380899\pi$
$541$	17.0025	0.730995	0.365497	+	0.930812i	$0.380899\pi$
$542$	0	0
$543$	−5.98997	−0.257054
$544$	0	0
$545$	−10.4596	−0.448041
$546$	0	0
$547$	1.66785	0.0713121	0.0356561	−	0.999364i	$-0.488648\pi$
$547$	1.66785	0.0713121	0.0356561	+	0.999364i	$0.488648\pi$
$548$	0	0
$549$	−6.87695	−0.293501
$550$	0	0
$551$	−0.446035	−0.0190017
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.14773	0.0911662
$556$	0	0
$557$	35.2596	1.49400	0.746998	−	0.664827i	$-0.231495\pi$
$557$	35.2596	1.49400	0.746998	+	0.664827i	$0.231495\pi$
$558$	0	0
$559$	−25.8008	−1.09126
$560$	0	0
$561$	−24.0206	−1.01415
$562$	0	0
$563$	20.2697	0.854266	0.427133	−	0.904189i	$-0.359524\pi$
$563$	20.2697	0.854266	0.427133	+	0.904189i	$0.359524\pi$
$564$	0	0
$565$	7.82308	0.329120
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.3485	−0.978821	−0.489410	−	0.872054i	$-0.662788\pi$
$569$	−23.3485	−0.978821	−0.489410	+	0.872054i	$0.662788\pi$
$570$	0	0
$571$	20.6203	0.862934	0.431467	−	0.902129i	$-0.357996\pi$
$571$	20.6203	0.862934	0.431467	+	0.902129i	$0.357996\pi$
$572$	0	0
$573$	12.5592	0.524670
$574$	0	0
$575$	9.31008	0.388257
$576$	0	0
$577$	−32.0949	−1.33613	−0.668064	−	0.744103i	$-0.732877\pi$
$577$	−32.0949	−1.33613	−0.668064	+	0.744103i	$0.732877\pi$
$578$	0	0
$579$	5.24623	0.218026
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−54.2447	−2.24659
$584$	0	0
$585$	−10.2893	−0.425409
$586$	0	0
$587$	46.2494	1.90892	0.954459	−	0.298342i	$-0.0964335\pi$
$587$	46.2494	1.90892	0.954459	+	0.298342i	$0.0964335\pi$
$588$	0	0
$589$	5.22518	0.215300
$590$	0	0
$591$	13.2150	0.543591
$592$	0	0
$593$	−17.6407	−0.724418	−0.362209	−	0.932097i	$-0.617977\pi$
$593$	−17.6407	−0.724418	−0.362209	+	0.932097i	$0.617977\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.90689	0.282680
$598$	0	0
$599$	9.26515	0.378564	0.189282	−	0.981923i	$-0.439384\pi$
$599$	9.26515	0.378564	0.189282	+	0.981923i	$0.439384\pi$
$600$	0	0
$601$	−15.8442	−0.646299	−0.323150	−	0.946348i	$-0.604742\pi$
$601$	−15.8442	−0.646299	−0.323150	+	0.946348i	$0.604742\pi$
$602$	0	0
$603$	9.16346	0.373165
$604$	0	0
$605$	11.4405	0.465123
$606$	0	0
$607$	−30.7851	−1.24953	−0.624765	−	0.780813i	$-0.714805\pi$
$607$	−30.7851	−1.24953	−0.624765	+	0.780813i	$0.714805\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.8919	−0.885651
$612$	0	0
$613$	42.5516	1.71864	0.859322	−	0.511436i	$-0.170886\pi$
$613$	42.5516	1.71864	0.859322	+	0.511436i	$0.170886\pi$
$614$	0	0
$615$	1.45802	0.0587930
$616$	0	0
$617$	−22.9716	−0.924800	−0.462400	−	0.886671i	$-0.653012\pi$
$617$	−22.9716	−0.924800	−0.462400	+	0.886671i	$0.653012\pi$
$618$	0	0
$619$	−14.7166	−0.591509	−0.295755	−	0.955264i	$-0.595571\pi$
$619$	−14.7166	−0.591509	−0.295755	+	0.955264i	$0.595571\pi$
$620$	0	0
$621$	12.3295	0.494766
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0.955034	0.0382013
$626$	0	0
$627$	−3.44075	−0.137410
$628$	0	0
$629$	−14.2327	−0.567494
$630$	0	0
$631$	35.3003	1.40528	0.702641	−	0.711545i	$-0.252004\pi$
$631$	35.3003	1.40528	0.702641	+	0.711545i	$0.252004\pi$
$632$	0	0
$633$	19.0788	0.758316
$634$	0	0
$635$	−12.3248	−0.489095
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.91859	0.155017
$640$	0	0
$641$	16.8638	0.666080	0.333040	−	0.942913i	$-0.391926\pi$
$641$	16.8638	0.666080	0.333040	+	0.942913i	$0.391926\pi$
$642$	0	0
$643$	−0.146033	−0.00575899	−0.00287949	−	0.999996i	$-0.500917\pi$
$643$	−0.146033	−0.00575899	−0.00287949	+	0.999996i	$0.500917\pi$
$644$	0	0
$645$	−8.53598	−0.336104
$646$	0	0
$647$	−6.54001	−0.257114	−0.128557	−	0.991702i	$-0.541035\pi$
$647$	−6.54001	−0.257114	−0.128557	+	0.991702i	$0.541035\pi$
$648$	0	0
$649$	−17.0429	−0.668993
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−43.1081	−1.68695	−0.843476	−	0.537167i	$-0.819494\pi$
$653$	−43.1081	−1.68695	−0.843476	+	0.537167i	$0.819494\pi$
$654$	0	0
$655$	−0.183770	−0.00718048
$656$	0	0
$657$	−3.51954	−0.137310
$658$	0	0
$659$	0.232585	0.00906024	0.00453012	−	0.999990i	$-0.498558\pi$
$659$	0.232585	0.00906024	0.00453012	+	0.999990i	$0.498558\pi$
$660$	0	0
$661$	14.4225	0.560969	0.280485	−	0.959859i	$-0.409505\pi$
$661$	14.4225	0.560969	0.280485	+	0.959859i	$0.409505\pi$
$662$	0	0
$663$	−17.3349	−0.673232
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.30794	−0.0506434
$668$	0	0
$669$	21.8956	0.846532
$670$	0	0
$671$	12.6858	0.489732
$672$	0	0
$673$	16.4629	0.634598	0.317299	−	0.948326i	$-0.397224\pi$
$673$	16.4629	0.634598	0.317299	+	0.948326i	$0.397224\pi$
$674$	0	0
$675$	13.3495	0.513823
$676$	0	0
$677$	35.7726	1.37485	0.687426	−	0.726255i	$-0.258741\pi$
$677$	35.7726	1.37485	0.687426	+	0.726255i	$0.258741\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0.388543	0.0148890
$682$	0	0
$683$	33.5272	1.28288	0.641441	−	0.767173i	$-0.278337\pi$
$683$	33.5272	1.28288	0.641441	+	0.767173i	$0.278337\pi$
$684$	0	0
$685$	13.9284	0.532176
$686$	0	0
$687$	13.1326	0.501042
$688$	0	0
$689$	−39.1467	−1.49137
$690$	0	0
$691$	−51.2178	−1.94842	−0.974209	−	0.225650i	$-0.927550\pi$
$691$	−51.2178	−1.94842	−0.974209	+	0.225650i	$0.927550\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.8569	0.449759
$696$	0	0
$697$	−9.66204	−0.365976
$698$	0	0
$699$	−20.5945	−0.778955
$700$	0	0
$701$	−35.6479	−1.34640	−0.673202	−	0.739459i	$-0.735082\pi$
$701$	−35.6479	−1.34640	−0.673202	+	0.739459i	$0.735082\pi$
$702$	0	0
$703$	−2.03871	−0.0768914
$704$	0	0
$705$	−7.24275	−0.272778
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−36.7911	−1.38172	−0.690859	−	0.722990i	$-0.742767\pi$
$709$	−36.7911	−1.38172	−0.690859	+	0.722990i	$0.742767\pi$
$710$	0	0
$711$	9.18772	0.344566
$712$	0	0
$713$	15.3221	0.573817
$714$	0	0
$715$	18.9805	0.709831
$716$	0	0
$717$	12.4841	0.466226
$718$	0	0
$719$	−43.0778	−1.60653	−0.803265	−	0.595622i	$-0.796906\pi$
$719$	−43.0778	−1.60653	−0.803265	+	0.595622i	$0.796906\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−16.0567	−0.597156
$724$	0	0
$725$	−1.41614	−0.0525941
$726$	0	0
$727$	18.9257	0.701914	0.350957	−	0.936391i	$-0.385856\pi$
$727$	18.9257	0.701914	0.350957	+	0.936391i	$0.385856\pi$
$728$	0	0
$729$	0.515424	0.0190898
$730$	0	0
$731$	56.5665	2.09219
$732$	0	0
$733$	−10.3469	−0.382171	−0.191085	−	0.981573i	$-0.561201\pi$
$733$	−10.3469	−0.382171	−0.191085	+	0.981573i	$0.561201\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.9037	−0.622657
$738$	0	0
$739$	−26.8138	−0.986363	−0.493181	−	0.869926i	$-0.664166\pi$
$739$	−26.8138	−0.986363	−0.493181	+	0.869926i	$0.664166\pi$
$740$	0	0
$741$	−2.48308	−0.0912183
$742$	0	0
$743$	32.5436	1.19391	0.596954	−	0.802275i	$-0.296377\pi$
$743$	32.5436	1.19391	0.596954	+	0.802275i	$0.296377\pi$
$744$	0	0
$745$	−7.70321	−0.282224
$746$	0	0
$747$	−0.406832	−0.0148852
$748$	0	0
$749$	0	0
$750$	0	0
$751$	37.8559	1.38138	0.690692	−	0.723150i	$-0.257306\pi$
$751$	37.8559	1.38138	0.690692	+	0.723150i	$0.257306\pi$
$752$	0	0
$753$	−9.08125	−0.330939
$754$	0	0
$755$	6.66584	0.242595
$756$	0	0
$757$	−14.0618	−0.511085	−0.255543	−	0.966798i	$-0.582254\pi$
$757$	−14.0618	−0.511085	−0.255543	+	0.966798i	$0.582254\pi$
$758$	0	0
$759$	−10.0895	−0.366226
$760$	0	0
$761$	41.9854	1.52197	0.760984	−	0.648771i	$-0.224716\pi$
$761$	41.9854	1.52197	0.760984	+	0.648771i	$0.224716\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	22.5586	0.815607
$766$	0	0
$767$	−12.2993	−0.444103
$768$	0	0
$769$	4.88600	0.176194	0.0880969	−	0.996112i	$-0.471921\pi$
$769$	4.88600	0.176194	0.0880969	+	0.996112i	$0.471921\pi$
$770$	0	0
$771$	−23.8543	−0.859092
$772$	0	0
$773$	7.43938	0.267576	0.133788	−	0.991010i	$-0.457286\pi$
$773$	7.43938	0.267576	0.133788	+	0.991010i	$0.457286\pi$
$774$	0	0
$775$	16.5897	0.595919
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.38401	−0.0495872
$780$	0	0
$781$	−7.22858	−0.258659
$782$	0	0
$783$	−1.87542	−0.0670220
$784$	0	0
$785$	−27.1385	−0.968614
$786$	0	0
$787$	−4.25930	−0.151828	−0.0759139	−	0.997114i	$-0.524187\pi$
$787$	−4.25930	−0.151828	−0.0759139	+	0.997114i	$0.524187\pi$
$788$	0	0
$789$	−3.02829	−0.107810
$790$	0	0
$791$	0	0
$792$	0	0
$793$	9.15497	0.325103
$794$	0	0
$795$	−12.9514	−0.459338
$796$	0	0
$797$	−14.6435	−0.518699	−0.259350	−	0.965783i	$-0.583508\pi$
$797$	−14.6435	−0.518699	−0.259350	+	0.965783i	$0.583508\pi$
$798$	0	0
$799$	47.9965	1.69799
$800$	0	0
$801$	3.82959	0.135312
$802$	0	0
$803$	6.49245	0.229114
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.95776	−0.280126
$808$	0	0
$809$	36.4606	1.28189	0.640943	−	0.767589i	$-0.278544\pi$
$809$	36.4606	1.28189	0.640943	+	0.767589i	$0.278544\pi$
$810$	0	0
$811$	−25.6188	−0.899597	−0.449799	−	0.893130i	$-0.648504\pi$
$811$	−25.6188	−0.899597	−0.449799	+	0.893130i	$0.648504\pi$
$812$	0	0
$813$	14.3174	0.502133
$814$	0	0
$815$	6.66192	0.233357
$816$	0	0
$817$	8.10268	0.283477
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.2946	0.394183	0.197092	−	0.980385i	$-0.436850\pi$
$821$	11.2946	0.394183	0.197092	+	0.980385i	$0.436850\pi$
$822$	0	0
$823$	27.1764	0.947310	0.473655	−	0.880710i	$-0.342934\pi$
$823$	27.1764	0.947310	0.473655	+	0.880710i	$0.342934\pi$
$824$	0	0
$825$	−10.9242	−0.380332
$826$	0	0
$827$	−22.0446	−0.766565	−0.383282	−	0.923631i	$-0.625206\pi$
$827$	−22.0446	−0.766565	−0.383282	+	0.923631i	$0.625206\pi$
$828$	0	0
$829$	14.1868	0.492728	0.246364	−	0.969177i	$-0.420764\pi$
$829$	14.1868	0.492728	0.246364	+	0.969177i	$0.420764\pi$
$830$	0	0
$831$	5.13048	0.177975
$832$	0	0
$833$	0	0
$834$	0	0
$835$	20.0626	0.694295
$836$	0	0
$837$	21.9700	0.759395
$838$	0	0
$839$	−45.2566	−1.56243	−0.781216	−	0.624261i	$-0.785400\pi$
$839$	−45.2566	−1.56243	−0.781216	+	0.624261i	$0.785400\pi$
$840$	0	0
$841$	−28.8011	−0.993140
$842$	0	0
$843$	16.6528	0.573552
$844$	0	0
$845$	−3.86464	−0.132948
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−25.8834	−0.888315
$850$	0	0
$851$	−5.97823	−0.204931
$852$	0	0
$853$	−27.4083	−0.938442	−0.469221	−	0.883081i	$-0.655465\pi$
$853$	−27.4083	−0.938442	−0.469221	+	0.883081i	$0.655465\pi$
$854$	0	0
$855$	3.23133	0.110509
$856$	0	0
$857$	−29.5873	−1.01068	−0.505342	−	0.862919i	$-0.668633\pi$
$857$	−29.5873	−1.01068	−0.505342	+	0.862919i	$0.668633\pi$
$858$	0	0
$859$	−27.2164	−0.928611	−0.464306	−	0.885675i	$-0.653696\pi$
$859$	−27.2164	−0.928611	−0.464306	+	0.885675i	$0.653696\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35.4232	−1.20582	−0.602910	−	0.797809i	$-0.705992\pi$
$863$	−35.4232	−1.20582	−0.602910	+	0.797809i	$0.705992\pi$
$864$	0	0
$865$	−30.2507	−1.02855
$866$	0	0
$867$	24.7489	0.840518
$868$	0	0
$869$	−16.9485	−0.574938
$870$	0	0
$871$	−12.1989	−0.413344
$872$	0	0
$873$	−11.6796	−0.395296
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.2273	−0.412886	−0.206443	−	0.978459i	$-0.566189\pi$
$877$	−12.2273	−0.412886	−0.206443	+	0.978459i	$0.566189\pi$
$878$	0	0
$879$	−23.2345	−0.783679
$880$	0	0
$881$	27.7388	0.934544	0.467272	−	0.884114i	$-0.345237\pi$
$881$	27.7388	0.934544	0.467272	+	0.884114i	$0.345237\pi$
$882$	0	0
$883$	−9.61931	−0.323716	−0.161858	−	0.986814i	$-0.551749\pi$
$883$	−9.61931	−0.323716	−0.161858	+	0.986814i	$0.551749\pi$
$884$	0	0
$885$	−4.06913	−0.136782
$886$	0	0
$887$	26.0571	0.874913	0.437457	−	0.899240i	$-0.355879\pi$
$887$	26.0571	0.874913	0.437457	+	0.899240i	$0.355879\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	17.1943	0.576032
$892$	0	0
$893$	6.87510	0.230066
$894$	0	0
$895$	−8.73629	−0.292022
$896$	0	0
$897$	−7.28128	−0.243115
$898$	0	0
$899$	−2.33062	−0.0777304
$900$	0	0
$901$	85.8266	2.85930
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.3771	−0.344946
$906$	0	0
$907$	−43.9931	−1.46077	−0.730383	−	0.683038i	$-0.760658\pi$
$907$	−43.9931	−1.46077	−0.730383	+	0.683038i	$0.760658\pi$
$908$	0	0
$909$	−14.3537	−0.476083
$910$	0	0
$911$	−23.0419	−0.763413	−0.381707	−	0.924284i	$-0.624664\pi$
$911$	−23.0419	−0.763413	−0.381707	+	0.924284i	$0.624664\pi$
$912$	0	0
$913$	0.750478	0.0248372
$914$	0	0
$915$	3.02885	0.100131
$916$	0	0
$917$	0	0
$918$	0	0
$919$	12.0853	0.398657	0.199328	−	0.979933i	$-0.436124\pi$
$919$	12.0853	0.398657	0.199328	+	0.979933i	$0.436124\pi$
$920$	0	0
$921$	−8.61786	−0.283968
$922$	0	0
$923$	−5.21664	−0.171708
$924$	0	0
$925$	−6.47280	−0.212824
$926$	0	0
$927$	35.5439	1.16741
$928$	0	0
$929$	−38.6396	−1.26773	−0.633863	−	0.773446i	$-0.718532\pi$
$929$	−38.6396	−1.26773	−0.633863	+	0.773446i	$0.718532\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	5.61638	0.183872
$934$	0	0
$935$	−41.6135	−1.36091
$936$	0	0
$937$	−36.8739	−1.20462	−0.602309	−	0.798263i	$-0.705752\pi$
$937$	−36.8739	−1.20462	−0.602309	+	0.798263i	$0.705752\pi$
$938$	0	0
$939$	−16.9856	−0.554304
$940$	0	0
$941$	−35.2692	−1.14974	−0.574871	−	0.818244i	$-0.694948\pi$
$941$	−35.2692	−1.14974	−0.574871	+	0.818244i	$0.694948\pi$
$942$	0	0
$943$	−4.05840	−0.132160
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.02607	0.0983342	0.0491671	−	0.998791i	$-0.484343\pi$
$947$	3.02607	0.0983342	0.0491671	+	0.998791i	$0.484343\pi$
$948$	0	0
$949$	4.68540	0.152094
$950$	0	0
$951$	−6.88823	−0.223366
$952$	0	0
$953$	−4.87768	−0.158004	−0.0790019	−	0.996874i	$-0.525173\pi$
$953$	−4.87768	−0.158004	−0.0790019	+	0.996874i	$0.525173\pi$
$954$	0	0
$955$	21.7578	0.704064
$956$	0	0
$957$	1.53470	0.0496097
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−3.69745	−0.119273
$962$	0	0
$963$	−45.4247	−1.46379
$964$	0	0
$965$	9.08862	0.292573
$966$	0	0
$967$	−19.4864	−0.626639	−0.313319	−	0.949648i	$-0.601441\pi$
$967$	−19.4864	−0.626639	−0.313319	+	0.949648i	$0.601441\pi$
$968$	0	0
$969$	5.44399	0.174886
$970$	0	0
$971$	23.3448	0.749170	0.374585	−	0.927193i	$-0.377785\pi$
$971$	23.3448	0.749170	0.374585	+	0.927193i	$0.377785\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−7.88365	−0.252479
$976$	0	0
$977$	−37.1069	−1.18715	−0.593577	−	0.804777i	$-0.702285\pi$
$977$	−37.1069	−1.18715	−0.593577	+	0.804777i	$0.702285\pi$
$978$	0	0
$979$	−7.06440	−0.225779
$980$	0	0
$981$	18.5192	0.591272
$982$	0	0
$983$	−0.331062	−0.0105592	−0.00527962	−	0.999986i	$-0.501681\pi$
$983$	−0.331062	−0.0105592	−0.00527962	+	0.999986i	$0.501681\pi$
$984$	0	0
$985$	22.8937	0.729455
$986$	0	0
$987$	0	0
$988$	0	0
$989$	23.7599	0.755522
$990$	0	0
$991$	53.0404	1.68488	0.842442	−	0.538787i	$-0.181117\pi$
$991$	53.0404	1.68488	0.842442	+	0.538787i	$0.181117\pi$
$992$	0	0
$993$	−9.76650	−0.309930
$994$	0	0
$995$	11.9656	0.379334
$996$	0	0
$997$	−40.4169	−1.28002	−0.640009	−	0.768368i	$-0.721069\pi$
$997$	−40.4169	−1.28002	−0.640009	+	0.768368i	$0.721069\pi$
$998$	0	0
$999$	−8.57205	−0.271208

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bo.1.5	✓	7
7.6	odd	2		7448.2.a.bp.1.3	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bo.1.5	✓	7	1.1	even	1	trivial
7448.2.a.bp.1.3	yes	7	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)

\(f(q)\)	\(=\)	\(q+0.779807 q^{3} +1.35094 q^{5} -2.39190 q^{9} +O(q^{10})\)	\(q+0.779807 q^{3} +1.35094 q^{5} -2.39190 q^{9} +4.41231 q^{11} +3.18423 q^{13} +1.05348 q^{15} -6.98121 q^{17} -1.00000 q^{19} -2.93236 q^{23} -3.17495 q^{25} -4.20464 q^{27} +0.446035 q^{29} -5.22518 q^{31} +3.44075 q^{33} +2.03871 q^{37} +2.48308 q^{39} +1.38401 q^{41} -8.10268 q^{43} -3.23133 q^{45} -6.87510 q^{47} -5.44399 q^{51} -12.2939 q^{53} +5.96079 q^{55} -0.779807 q^{57} -3.86258 q^{59} +2.87510 q^{61} +4.30172 q^{65} -3.83104 q^{67} -2.28667 q^{69} -1.63827 q^{71} +1.47144 q^{73} -2.47585 q^{75} -3.84118 q^{79} +3.89690 q^{81} +0.170087 q^{83} -9.43123 q^{85} +0.347821 q^{87} -1.60106 q^{89} -4.07463 q^{93} -1.35094 q^{95} +4.88299 q^{97} -10.5538 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{3} - q^{5} + 8 q^{9}+O(q^{10}) \) 7 * q - q^3 - q^5 + 8 * q^9	\( 7 q - q^{3} - q^{5} + 8 q^{9} + 3 q^{11} - 6 q^{13} - 4 q^{15} - 10 q^{17} - 7 q^{19} - 2 q^{23} + 8 q^{25} + 2 q^{27} + 3 q^{29} + 6 q^{31} - 21 q^{33} - 7 q^{37} - 2 q^{39} - 9 q^{41} + q^{43} - 24 q^{45} - 15 q^{47} + 6 q^{51} + 5 q^{53} + 6 q^{55} + q^{57} + 9 q^{59} - 13 q^{61} - 6 q^{65} - 2 q^{67} + 20 q^{69} - q^{71} - 42 q^{73} - 40 q^{75} - 3 q^{79} - q^{81} + 12 q^{83} - 16 q^{85} + 2 q^{87} - 41 q^{89} - 2 q^{93} + q^{95} - 5 q^{97} + 9 q^{99}+O(q^{100}) \) 7 * q - q^3 - q^5 + 8 * q^9 + 3 * q^11 - 6 * q^13 - 4 * q^15 - 10 * q^17 - 7 * q^19 - 2 * q^23 + 8 * q^25 + 2 * q^27 + 3 * q^29 + 6 * q^31 - 21 * q^33 - 7 * q^37 - 2 * q^39 - 9 * q^41 + q^43 - 24 * q^45 - 15 * q^47 + 6 * q^51 + 5 * q^53 + 6 * q^55 + q^57 + 9 * q^59 - 13 * q^61 - 6 * q^65 - 2 * q^67 + 20 * q^69 - q^71 - 42 * q^73 - 40 * q^75 - 3 * q^79 - q^81 + 12 * q^83 - 16 * q^85 + 2 * q^87 - 41 * q^89 - 2 * q^93 + q^95 - 5 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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