LMFDB - Embedded newform 1300.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,Mod(1,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, 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("1300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.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:	$10.3805522628$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	1300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.539189 q^{3} +3.70928 q^{7} -2.70928 q^{9} +O(q^{10})$	$q-0.539189 q^{3} +3.70928 q^{7} -2.70928 q^{9} +5.51026 q^{11} +1.00000 q^{13} -0.340173 q^{17} -2.24846 q^{19} -2.00000 q^{21} +1.46081 q^{23} +3.07838 q^{27} +0.630898 q^{29} -6.58864 q^{31} -2.97107 q^{33} +5.55252 q^{37} -0.539189 q^{39} -4.68035 q^{41} +10.6381 q^{43} +4.29072 q^{47} +6.75872 q^{49} +0.183417 q^{51} +9.07838 q^{53} +1.21235 q^{57} +0.986669 q^{59} -12.2329 q^{61} -10.0494 q^{63} +4.97107 q^{67} -0.787653 q^{69} +7.32684 q^{71} +11.1278 q^{73} +20.4391 q^{77} +9.75872 q^{79} +6.46800 q^{81} -16.8865 q^{83} -0.340173 q^{87} +14.9939 q^{89} +3.70928 q^{91} +3.55252 q^{93} -12.4391 q^{97} -14.9288 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{7} - q^{9}+O(q^{10}) $ 3 * q + 4 * q^7 - q^9	$ 3 q + 4 q^{7} - q^{9} + 3 q^{13} + 10 q^{17} + 2 q^{19} - 6 q^{21} + 6 q^{23} + 6 q^{27} - 2 q^{29} + 6 q^{33} + 16 q^{37} + 8 q^{41} - 6 q^{43} + 20 q^{47} - 5 q^{49} - 4 q^{51} + 24 q^{53} + 14 q^{57} + 2 q^{59} - 14 q^{61} - 12 q^{63} + 8 q^{69} + 10 q^{71} + 12 q^{73} + 14 q^{77} + 4 q^{79} - 13 q^{81} - 4 q^{83} + 10 q^{87} + 10 q^{89} + 4 q^{91} + 10 q^{93} + 10 q^{97} - 14 q^{99}+O(q^{100}) $ 3 * q + 4 * q^7 - q^9 + 3 * q^13 + 10 * q^17 + 2 * q^19 - 6 * q^21 + 6 * q^23 + 6 * q^27 - 2 * q^29 + 6 * q^33 + 16 * q^37 + 8 * q^41 - 6 * q^43 + 20 * q^47 - 5 * q^49 - 4 * q^51 + 24 * q^53 + 14 * q^57 + 2 * q^59 - 14 * q^61 - 12 * q^63 + 8 * q^69 + 10 * q^71 + 12 * q^73 + 14 * q^77 + 4 * q^79 - 13 * q^81 - 4 * q^83 + 10 * q^87 + 10 * q^89 + 4 * q^91 + 10 * q^93 + 10 * q^97 - 14 * 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
$3$	−0.539189	−0.311301	−0.155650	−	0.987812i	$-0.549747\pi$
$3$	−0.539189	−0.311301	−0.155650	+	0.987812i	$0.549747\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.70928	1.40197	0.700987	−	0.713174i	$-0.252743\pi$
$7$	3.70928	1.40197	0.700987	+	0.713174i	$0.252743\pi$
$8$	0	0
$9$	−2.70928	−0.903092
$10$	0	0
$11$	5.51026	1.66141	0.830703	−	0.556716i	$-0.187939\pi$
$11$	5.51026	1.66141	0.830703	+	0.556716i	$0.187939\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.340173	−0.0825041	−0.0412520	−	0.999149i	$-0.513135\pi$
$17$	−0.340173	−0.0825041	−0.0412520	+	0.999149i	$0.513135\pi$
$18$	0	0
$19$	−2.24846	−0.515833	−0.257917	−	0.966167i	$-0.583036\pi$
$19$	−2.24846	−0.515833	−0.257917	+	0.966167i	$0.583036\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	1.46081	0.304600	0.152300	−	0.988334i	$-0.451332\pi$
$23$	1.46081	0.304600	0.152300	+	0.988334i	$0.451332\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.07838	0.592434
$28$	0	0
$29$	0.630898	0.117155	0.0585774	−	0.998283i	$-0.481344\pi$
$29$	0.630898	0.117155	0.0585774	+	0.998283i	$0.481344\pi$
$30$	0	0
$31$	−6.58864	−1.18335	−0.591677	−	0.806175i	$-0.701534\pi$
$31$	−6.58864	−1.18335	−0.591677	+	0.806175i	$0.701534\pi$
$32$	0	0
$33$	−2.97107	−0.517197
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.55252	0.912829	0.456414	−	0.889767i	$-0.349133\pi$
$37$	5.55252	0.912829	0.456414	+	0.889767i	$0.349133\pi$
$38$	0	0
$39$	−0.539189	−0.0863393
$40$	0	0
$41$	−4.68035	−0.730947	−0.365474	−	0.930822i	$-0.619093\pi$
$41$	−4.68035	−0.730947	−0.365474	+	0.930822i	$0.619093\pi$
$42$	0	0
$43$	10.6381	1.62229	0.811146	−	0.584843i	$-0.198844\pi$
$43$	10.6381	1.62229	0.811146	+	0.584843i	$0.198844\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.29072	0.625867	0.312933	−	0.949775i	$-0.398688\pi$
$47$	4.29072	0.625867	0.312933	+	0.949775i	$0.398688\pi$
$48$	0	0
$49$	6.75872	0.965532
$50$	0	0
$51$	0.183417	0.0256836
$52$	0	0
$53$	9.07838	1.24701	0.623506	−	0.781819i	$-0.285708\pi$
$53$	9.07838	1.24701	0.623506	+	0.781819i	$0.285708\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.21235	0.160579
$58$	0	0
$59$	0.986669	0.128453	0.0642267	−	0.997935i	$-0.479542\pi$
$59$	0.986669	0.128453	0.0642267	+	0.997935i	$0.479542\pi$
$60$	0	0
$61$	−12.2329	−1.56626	−0.783129	−	0.621859i	$-0.786378\pi$
$61$	−12.2329	−1.56626	−0.783129	+	0.621859i	$0.786378\pi$
$62$	0	0
$63$	−10.0494	−1.26611
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.97107	0.607313	0.303656	−	0.952782i	$-0.401792\pi$
$67$	4.97107	0.607313	0.303656	+	0.952782i	$0.401792\pi$
$68$	0	0
$69$	−0.787653	−0.0948223
$70$	0	0
$71$	7.32684	0.869536	0.434768	−	0.900542i	$-0.356830\pi$
$71$	7.32684	0.869536	0.434768	+	0.900542i	$0.356830\pi$
$72$	0	0
$73$	11.1278	1.30241	0.651207	−	0.758900i	$-0.274263\pi$
$73$	11.1278	1.30241	0.651207	+	0.758900i	$0.274263\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	20.4391	2.32925
$78$	0	0
$79$	9.75872	1.09794	0.548971	−	0.835841i	$-0.315020\pi$
$79$	9.75872	1.09794	0.548971	+	0.835841i	$0.315020\pi$
$80$	0	0
$81$	6.46800	0.718667
$82$	0	0
$83$	−16.8865	−1.85354	−0.926770	−	0.375630i	$-0.877426\pi$
$83$	−16.8865	−1.85354	−0.926770	+	0.375630i	$0.877426\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.340173	−0.0364704
$88$	0	0
$89$	14.9939	1.58935	0.794673	−	0.607038i	$-0.207642\pi$
$89$	14.9939	1.58935	0.794673	+	0.607038i	$0.207642\pi$
$90$	0	0
$91$	3.70928	0.388838
$92$	0	0
$93$	3.55252	0.368379
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.4391	−1.26300	−0.631498	−	0.775377i	$-0.717560\pi$
$97$	−12.4391	−1.26300	−0.631498	+	0.775377i	$0.717560\pi$
$98$	0	0
$99$	−14.9288	−1.50040
$100$	0	0
$101$	7.75872	0.772022	0.386011	−	0.922494i	$-0.373853\pi$
$101$	7.75872	0.772022	0.386011	+	0.922494i	$0.373853\pi$
$102$	0	0
$103$	−7.06278	−0.695916	−0.347958	−	0.937510i	$-0.613125\pi$
$103$	−7.06278	−0.695916	−0.347958	+	0.937510i	$0.613125\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.24620	−0.893864	−0.446932	−	0.894568i	$-0.647483\pi$
$107$	−9.24620	−0.893864	−0.446932	+	0.894568i	$0.647483\pi$
$108$	0	0
$109$	−11.0205	−1.05557	−0.527787	−	0.849377i	$-0.676978\pi$
$109$	−11.0205	−1.05557	−0.527787	+	0.849377i	$0.676978\pi$
$110$	0	0
$111$	−2.99386	−0.284164
$112$	0	0
$113$	17.0205	1.60116	0.800578	−	0.599229i	$-0.204526\pi$
$113$	17.0205	1.60116	0.800578	+	0.599229i	$0.204526\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.70928	−0.250473
$118$	0	0
$119$	−1.26180	−0.115669
$120$	0	0
$121$	19.3630	1.76027
$122$	0	0
$123$	2.52359	0.227544
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.77432	−0.334917	−0.167458	−	0.985879i	$-0.553556\pi$
$127$	−3.77432	−0.334917	−0.167458	+	0.985879i	$0.553556\pi$
$128$	0	0
$129$	−5.73594	−0.505021
$130$	0	0
$131$	3.41855	0.298680	0.149340	−	0.988786i	$-0.452285\pi$
$131$	3.41855	0.298680	0.149340	+	0.988786i	$0.452285\pi$
$132$	0	0
$133$	−8.34017	−0.723185
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.4124	1.91482	0.957411	−	0.288730i	$-0.0932329\pi$
$137$	22.4124	1.91482	0.957411	+	0.288730i	$0.0932329\pi$
$138$	0	0
$139$	5.75872	0.488449	0.244224	−	0.969719i	$-0.421467\pi$
$139$	5.75872	0.488449	0.244224	+	0.969719i	$0.421467\pi$
$140$	0	0
$141$	−2.31351	−0.194833
$142$	0	0
$143$	5.51026	0.460791
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.64423	−0.300571
$148$	0	0
$149$	−3.23513	−0.265032	−0.132516	−	0.991181i	$-0.542306\pi$
$149$	−3.23513	−0.265032	−0.132516	+	0.991181i	$0.542306\pi$
$150$	0	0
$151$	−17.1701	−1.39728	−0.698641	−	0.715472i	$-0.746211\pi$
$151$	−17.1701	−1.39728	−0.698641	+	0.715472i	$0.746211\pi$
$152$	0	0
$153$	0.921622	0.0745087
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.9939	0.877405	0.438703	−	0.898632i	$-0.355438\pi$
$157$	10.9939	0.877405	0.438703	+	0.898632i	$0.355438\pi$
$158$	0	0
$159$	−4.89496	−0.388196
$160$	0	0
$161$	5.41855	0.427042
$162$	0	0
$163$	−19.3112	−1.51257	−0.756287	−	0.654240i	$-0.772988\pi$
$163$	−19.3112	−1.51257	−0.756287	+	0.654240i	$0.772988\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.26406	−0.484728	−0.242364	−	0.970185i	$-0.577923\pi$
$167$	−6.26406	−0.484728	−0.242364	+	0.970185i	$0.577923\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.09171	0.465845
$172$	0	0
$173$	14.9216	1.13447	0.567235	−	0.823556i	$-0.308013\pi$
$173$	14.9216	1.13447	0.567235	+	0.823556i	$0.308013\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.532001	−0.0399876
$178$	0	0
$179$	−16.0144	−1.19697	−0.598485	−	0.801134i	$-0.704231\pi$
$179$	−16.0144	−1.19697	−0.598485	+	0.801134i	$0.704231\pi$
$180$	0	0
$181$	−16.3630	−1.21625	−0.608125	−	0.793842i	$-0.708078\pi$
$181$	−16.3630	−1.21625	−0.608125	+	0.793842i	$0.708078\pi$
$182$	0	0
$183$	6.59583	0.487577
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.87444	−0.137073
$188$	0	0
$189$	11.4186	0.830577
$190$	0	0
$191$	−3.31965	−0.240202	−0.120101	−	0.992762i	$-0.538322\pi$
$191$	−3.31965	−0.240202	−0.120101	+	0.992762i	$0.538322\pi$
$192$	0	0
$193$	−12.2557	−0.882181	−0.441091	−	0.897463i	$-0.645408\pi$
$193$	−12.2557	−0.882181	−0.441091	+	0.897463i	$0.645408\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.6742	−1.11674	−0.558370	−	0.829592i	$-0.688573\pi$
$197$	−15.6742	−1.11674	−0.558370	+	0.829592i	$0.688573\pi$
$198$	0	0
$199$	2.02666	0.143666	0.0718331	−	0.997417i	$-0.477115\pi$
$199$	2.02666	0.143666	0.0718331	+	0.997417i	$0.477115\pi$
$200$	0	0
$201$	−2.68035	−0.189057
$202$	0	0
$203$	2.34017	0.164248
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.95774	−0.275082
$208$	0	0
$209$	−12.3896	−0.857008
$210$	0	0
$211$	−20.8638	−1.43632	−0.718160	−	0.695878i	$-0.755016\pi$
$211$	−20.8638	−1.43632	−0.718160	+	0.695878i	$0.755016\pi$
$212$	0	0
$213$	−3.95055	−0.270687
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−24.4391	−1.65903
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−0.340173	−0.0228825
$222$	0	0
$223$	−19.4947	−1.30546	−0.652730	−	0.757591i	$-0.726376\pi$
$223$	−19.4947	−1.30546	−0.652730	+	0.757591i	$0.726376\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.2907	−0.815764	−0.407882	−	0.913035i	$-0.633733\pi$
$227$	−12.2907	−0.815764	−0.407882	+	0.913035i	$0.633733\pi$
$228$	0	0
$229$	11.5441	0.762856	0.381428	−	0.924399i	$-0.375432\pi$
$229$	11.5441	0.762856	0.381428	+	0.924399i	$0.375432\pi$
$230$	0	0
$231$	−11.0205	−0.725097
$232$	0	0
$233$	−8.77924	−0.575147	−0.287574	−	0.957759i	$-0.592849\pi$
$233$	−8.77924	−0.575147	−0.287574	+	0.957759i	$0.592849\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5.26180	−0.341790
$238$	0	0
$239$	22.1061	1.42992	0.714962	−	0.699163i	$-0.246444\pi$
$239$	22.1061	1.42992	0.714962	+	0.699163i	$0.246444\pi$
$240$	0	0
$241$	25.9421	1.67108	0.835540	−	0.549429i	$-0.185155\pi$
$241$	25.9421	1.67108	0.835540	+	0.549429i	$0.185155\pi$
$242$	0	0
$243$	−12.7226	−0.816156
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.24846	−0.143066
$248$	0	0
$249$	9.10504	0.577008
$250$	0	0
$251$	3.90110	0.246235	0.123118	−	0.992392i	$-0.460711\pi$
$251$	3.90110	0.246235	0.123118	+	0.992392i	$0.460711\pi$
$252$	0	0
$253$	8.04945	0.506064
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.73206	−0.232800	−0.116400	−	0.993202i	$-0.537135\pi$
$257$	−3.73206	−0.232800	−0.116400	+	0.993202i	$0.537135\pi$
$258$	0	0
$259$	20.5958	1.27976
$260$	0	0
$261$	−1.70928	−0.105801
$262$	0	0
$263$	20.0566	1.23675	0.618373	−	0.785885i	$-0.287792\pi$
$263$	20.0566	1.23675	0.618373	+	0.785885i	$0.287792\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.08452	−0.494765
$268$	0	0
$269$	−24.0722	−1.46771	−0.733855	−	0.679306i	$-0.762281\pi$
$269$	−24.0722	−1.46771	−0.733855	+	0.679306i	$0.762281\pi$
$270$	0	0
$271$	−4.95547	−0.301023	−0.150512	−	0.988608i	$-0.548092\pi$
$271$	−4.95547	−0.301023	−0.150512	+	0.988608i	$0.548092\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−28.8638	−1.73426	−0.867128	−	0.498086i	$-0.834036\pi$
$277$	−28.8638	−1.73426	−0.867128	+	0.498086i	$0.834036\pi$
$278$	0	0
$279$	17.8504	1.06868
$280$	0	0
$281$	−23.4186	−1.39703	−0.698517	−	0.715594i	$-0.746156\pi$
$281$	−23.4186	−1.39703	−0.698517	+	0.715594i	$0.746156\pi$
$282$	0	0
$283$	−7.51867	−0.446939	−0.223469	−	0.974711i	$-0.571738\pi$
$283$	−7.51867	−0.446939	−0.223469	+	0.974711i	$0.571738\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−17.3607	−1.02477
$288$	0	0
$289$	−16.8843	−0.993193
$290$	0	0
$291$	6.70701	0.393172
$292$	0	0
$293$	15.4413	0.902093	0.451046	−	0.892501i	$-0.351051\pi$
$293$	15.4413	0.902093	0.451046	+	0.892501i	$0.351051\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.9627	0.984273
$298$	0	0
$299$	1.46081	0.0844809
$300$	0	0
$301$	39.4596	2.27441
$302$	0	0
$303$	−4.18342	−0.240331
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−17.4680	−0.996951	−0.498476	−	0.866904i	$-0.666107\pi$
$307$	−17.4680	−0.996951	−0.498476	+	0.866904i	$0.666107\pi$
$308$	0	0
$309$	3.80817	0.216639
$310$	0	0
$311$	−3.68649	−0.209042	−0.104521	−	0.994523i	$-0.533331\pi$
$311$	−3.68649	−0.209042	−0.104521	+	0.994523i	$0.533331\pi$
$312$	0	0
$313$	−9.60197	−0.542735	−0.271368	−	0.962476i	$-0.587476\pi$
$313$	−9.60197	−0.542735	−0.271368	+	0.962476i	$0.587476\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.38962	−0.471208	−0.235604	−	0.971849i	$-0.575707\pi$
$317$	−8.38962	−0.471208	−0.235604	+	0.971849i	$0.575707\pi$
$318$	0	0
$319$	3.47641	0.194642
$320$	0	0
$321$	4.98545	0.278260
$322$	0	0
$323$	0.764867	0.0425583
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.94214	0.328601
$328$	0	0
$329$	15.9155	0.877449
$330$	0	0
$331$	23.4836	1.29078	0.645388	−	0.763855i	$-0.276696\pi$
$331$	23.4836	1.29078	0.645388	+	0.763855i	$0.276696\pi$
$332$	0	0
$333$	−15.0433	−0.824368
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.5958	0.577191	0.288596	−	0.957451i	$-0.406812\pi$
$337$	10.5958	0.577191	0.288596	+	0.957451i	$0.406812\pi$
$338$	0	0
$339$	−9.17727	−0.498441
$340$	0	0
$341$	−36.3051	−1.96603
$342$	0	0
$343$	−0.894960	−0.0483233
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.9988	1.39569	0.697844	−	0.716250i	$-0.254143\pi$
$347$	25.9988	1.39569	0.697844	+	0.716250i	$0.254143\pi$
$348$	0	0
$349$	−26.6947	−1.42894	−0.714468	−	0.699668i	$-0.753331\pi$
$349$	−26.6947	−1.42894	−0.714468	+	0.699668i	$0.753331\pi$
$350$	0	0
$351$	3.07838	0.164312
$352$	0	0
$353$	20.7565	1.10475	0.552377	−	0.833594i	$-0.313721\pi$
$353$	20.7565	1.10475	0.552377	+	0.833594i	$0.313721\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.680346	0.0360077
$358$	0	0
$359$	−16.4585	−0.868649	−0.434324	−	0.900757i	$-0.643013\pi$
$359$	−16.4585	−0.868649	−0.434324	+	0.900757i	$0.643013\pi$
$360$	0	0
$361$	−13.9444	−0.733916
$362$	0	0
$363$	−10.4403	−0.547973
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.6114	0.553912	0.276956	−	0.960883i	$-0.410674\pi$
$367$	10.6114	0.553912	0.276956	+	0.960883i	$0.410674\pi$
$368$	0	0
$369$	12.6803	0.660112
$370$	0	0
$371$	33.6742	1.74828
$372$	0	0
$373$	−2.31351	−0.119789	−0.0598945	−	0.998205i	$-0.519076\pi$
$373$	−2.31351	−0.119789	−0.0598945	+	0.998205i	$0.519076\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.630898	0.0324929
$378$	0	0
$379$	4.56198	0.234333	0.117166	−	0.993112i	$-0.462619\pi$
$379$	4.56198	0.234333	0.117166	+	0.993112i	$0.462619\pi$
$380$	0	0
$381$	2.03507	0.104260
$382$	0	0
$383$	2.84551	0.145399	0.0726994	−	0.997354i	$-0.476839\pi$
$383$	2.84551	0.145399	0.0726994	+	0.997354i	$0.476839\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−28.8215	−1.46508
$388$	0	0
$389$	−23.3607	−1.18443	−0.592217	−	0.805778i	$-0.701747\pi$
$389$	−23.3607	−1.18443	−0.592217	+	0.805778i	$0.701747\pi$
$390$	0	0
$391$	−0.496928	−0.0251308
$392$	0	0
$393$	−1.84324	−0.0929794
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.66209	0.133607	0.0668033	−	0.997766i	$-0.478720\pi$
$397$	2.66209	0.133607	0.0668033	+	0.997766i	$0.478720\pi$
$398$	0	0
$399$	4.49693	0.225128
$400$	0	0
$401$	−16.9360	−0.845743	−0.422872	−	0.906190i	$-0.638978\pi$
$401$	−16.9360	−0.845743	−0.422872	+	0.906190i	$0.638978\pi$
$402$	0	0
$403$	−6.58864	−0.328203
$404$	0	0
$405$	0	0
$406$	0	0
$407$	30.5958	1.51658
$408$	0	0
$409$	−15.2039	−0.751786	−0.375893	−	0.926663i	$-0.622664\pi$
$409$	−15.2039	−0.751786	−0.375893	+	0.926663i	$0.622664\pi$
$410$	0	0
$411$	−12.0845	−0.596085
$412$	0	0
$413$	3.65983	0.180088
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3.10504	−0.152054
$418$	0	0
$419$	−10.0579	−0.491359	−0.245679	−	0.969351i	$-0.579011\pi$
$419$	−10.0579	−0.491359	−0.245679	+	0.969351i	$0.579011\pi$
$420$	0	0
$421$	10.6537	0.519229	0.259614	−	0.965712i	$-0.416405\pi$
$421$	10.6537	0.519229	0.259614	+	0.965712i	$0.416405\pi$
$422$	0	0
$423$	−11.6248	−0.565215
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−45.3751	−2.19585
$428$	0	0
$429$	−2.97107	−0.143445
$430$	0	0
$431$	22.7721	1.09689	0.548446	−	0.836186i	$-0.315220\pi$
$431$	22.7721	1.09689	0.548446	+	0.836186i	$0.315220\pi$
$432$	0	0
$433$	−15.6332	−0.751282	−0.375641	−	0.926765i	$-0.622577\pi$
$433$	−15.6332	−0.751282	−0.375641	+	0.926765i	$0.622577\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.28458	−0.157123
$438$	0	0
$439$	0.711543	0.0339601	0.0169800	−	0.999856i	$-0.494595\pi$
$439$	0.711543	0.0339601	0.0169800	+	0.999856i	$0.494595\pi$
$440$	0	0
$441$	−18.3112	−0.871964
$442$	0	0
$443$	−23.7887	−1.13024	−0.565118	−	0.825010i	$-0.691169\pi$
$443$	−23.7887	−1.13024	−0.565118	+	0.825010i	$0.691169\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.74435	0.0825048
$448$	0	0
$449$	11.2039	0.528746	0.264373	−	0.964420i	$-0.414835\pi$
$449$	11.2039	0.528746	0.264373	+	0.964420i	$0.414835\pi$
$450$	0	0
$451$	−25.7899	−1.21440
$452$	0	0
$453$	9.25792	0.434975
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.60197	0.262049	0.131025	−	0.991379i	$-0.458173\pi$
$457$	5.60197	0.262049	0.131025	+	0.991379i	$0.458173\pi$
$458$	0	0
$459$	−1.04718	−0.0488782
$460$	0	0
$461$	24.1256	1.12364	0.561820	−	0.827260i	$-0.310101\pi$
$461$	24.1256	1.12364	0.561820	+	0.827260i	$0.310101\pi$
$462$	0	0
$463$	−20.7877	−0.966084	−0.483042	−	0.875597i	$-0.660468\pi$
$463$	−20.7877	−0.966084	−0.483042	+	0.875597i	$0.660468\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−38.5802	−1.78528	−0.892640	−	0.450770i	$-0.851149\pi$
$467$	−38.5802	−1.78528	−0.892640	+	0.450770i	$0.851149\pi$
$468$	0	0
$469$	18.4391	0.851437
$470$	0	0
$471$	−5.92777	−0.273137
$472$	0	0
$473$	58.6186	2.69529
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−24.5958	−1.12617
$478$	0	0
$479$	8.27513	0.378100	0.189050	−	0.981967i	$-0.439459\pi$
$479$	8.27513	0.378100	0.189050	+	0.981967i	$0.439459\pi$
$480$	0	0
$481$	5.55252	0.253173
$482$	0	0
$483$	−2.92162	−0.132938
$484$	0	0
$485$	0	0
$486$	0	0
$487$	34.3630	1.55713	0.778567	−	0.627561i	$-0.215947\pi$
$487$	34.3630	1.55713	0.778567	+	0.627561i	$0.215947\pi$
$488$	0	0
$489$	10.4124	0.470865
$490$	0	0
$491$	−6.70701	−0.302683	−0.151342	−	0.988482i	$-0.548359\pi$
$491$	−6.70701	−0.302683	−0.151342	+	0.988482i	$0.548359\pi$
$492$	0	0
$493$	−0.214614	−0.00966574
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27.1773	1.21907
$498$	0	0
$499$	−23.6092	−1.05689	−0.528445	−	0.848967i	$-0.677225\pi$
$499$	−23.6092	−1.05689	−0.528445	+	0.848967i	$0.677225\pi$
$500$	0	0
$501$	3.37751	0.150896
$502$	0	0
$503$	−6.24005	−0.278230	−0.139115	−	0.990276i	$-0.544426\pi$
$503$	−6.24005	−0.278230	−0.139115	+	0.990276i	$0.544426\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.539189	−0.0239462
$508$	0	0
$509$	28.1256	1.24664	0.623322	−	0.781965i	$-0.285783\pi$
$509$	28.1256	1.24664	0.623322	+	0.781965i	$0.285783\pi$
$510$	0	0
$511$	41.2762	1.82595
$512$	0	0
$513$	−6.92162	−0.305597
$514$	0	0
$515$	0	0
$516$	0	0
$517$	23.6430	1.03982
$518$	0	0
$519$	−8.04557	−0.353161
$520$	0	0
$521$	−17.4101	−0.762752	−0.381376	−	0.924420i	$-0.624550\pi$
$521$	−17.4101	−0.762752	−0.381376	+	0.924420i	$0.624550\pi$
$522$	0	0
$523$	17.9311	0.784071	0.392036	−	0.919950i	$-0.371771\pi$
$523$	17.9311	0.784071	0.392036	+	0.919950i	$0.371771\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.24128	0.0976315
$528$	0	0
$529$	−20.8660	−0.907219
$530$	0	0
$531$	−2.67316	−0.116005
$532$	0	0
$533$	−4.68035	−0.202728
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.63477	0.372618
$538$	0	0
$539$	37.2423	1.60414
$540$	0	0
$541$	−4.72383	−0.203093	−0.101547	−	0.994831i	$-0.532379\pi$
$541$	−4.72383	−0.203093	−0.101547	+	0.994831i	$0.532379\pi$
$542$	0	0
$543$	8.82273	0.378619
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21.8420	0.933897	0.466949	−	0.884284i	$-0.345353\pi$
$547$	21.8420	0.933897	0.466949	+	0.884284i	$0.345353\pi$
$548$	0	0
$549$	33.1422	1.41447
$550$	0	0
$551$	−1.41855	−0.0604323
$552$	0	0
$553$	36.1978	1.53929
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.7093	1.34357	0.671783	−	0.740748i	$-0.265529\pi$
$557$	31.7093	1.34357	0.671783	+	0.740748i	$0.265529\pi$
$558$	0	0
$559$	10.6381	0.449943
$560$	0	0
$561$	1.01068	0.0426709
$562$	0	0
$563$	−25.9265	−1.09267	−0.546337	−	0.837566i	$-0.683978\pi$
$563$	−25.9265	−1.09267	−0.546337	+	0.837566i	$0.683978\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	23.9916	1.00755
$568$	0	0
$569$	−0.447480	−0.0187593	−0.00937967	−	0.999956i	$-0.502986\pi$
$569$	−0.447480	−0.0187593	−0.00937967	+	0.999956i	$0.502986\pi$
$570$	0	0
$571$	3.23513	0.135386	0.0676931	−	0.997706i	$-0.478436\pi$
$571$	3.23513	0.135386	0.0676931	+	0.997706i	$0.478436\pi$
$572$	0	0
$573$	1.78992	0.0747750
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.44134	0.309787	0.154893	−	0.987931i	$-0.450497\pi$
$577$	7.44134	0.309787	0.154893	+	0.987931i	$0.450497\pi$
$578$	0	0
$579$	6.60811	0.274624
$580$	0	0
$581$	−62.6369	−2.59861
$582$	0	0
$583$	50.0242	2.07179
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.3112	−0.631963	−0.315981	−	0.948765i	$-0.602334\pi$
$587$	−15.3112	−0.631963	−0.315981	+	0.948765i	$0.602334\pi$
$588$	0	0
$589$	14.8143	0.610413
$590$	0	0
$591$	8.45136	0.347642
$592$	0	0
$593$	0.868298	0.0356567	0.0178284	−	0.999841i	$-0.494325\pi$
$593$	0.868298	0.0356567	0.0178284	+	0.999841i	$0.494325\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.09275	−0.0447234
$598$	0	0
$599$	−23.1506	−0.945908	−0.472954	−	0.881087i	$-0.656812\pi$
$599$	−23.1506	−0.945908	−0.472954	+	0.881087i	$0.656812\pi$
$600$	0	0
$601$	−20.7838	−0.847788	−0.423894	−	0.905712i	$-0.639337\pi$
$601$	−20.7838	−0.847788	−0.423894	+	0.905712i	$0.639337\pi$
$602$	0	0
$603$	−13.4680	−0.548459
$604$	0	0
$605$	0	0
$606$	0	0
$607$	33.5597	1.36215	0.681073	−	0.732215i	$-0.261514\pi$
$607$	33.5597	1.36215	0.681073	+	0.732215i	$0.261514\pi$
$608$	0	0
$609$	−1.26180	−0.0511305
$610$	0	0
$611$	4.29072	0.173584
$612$	0	0
$613$	−18.8104	−0.759746	−0.379873	−	0.925039i	$-0.624032\pi$
$613$	−18.8104	−0.759746	−0.379873	+	0.925039i	$0.624032\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.79153	0.192900	0.0964498	−	0.995338i	$-0.469251\pi$
$617$	4.79153	0.192900	0.0964498	+	0.995338i	$0.469251\pi$
$618$	0	0
$619$	−14.9600	−0.601293	−0.300647	−	0.953736i	$-0.597203\pi$
$619$	−14.9600	−0.601293	−0.300647	+	0.953736i	$0.597203\pi$
$620$	0	0
$621$	4.49693	0.180456
$622$	0	0
$623$	55.6163	2.22822
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.68035	0.266787
$628$	0	0
$629$	−1.88882	−0.0753121
$630$	0	0
$631$	20.6465	0.821924	0.410962	−	0.911652i	$-0.365193\pi$
$631$	20.6465	0.821924	0.410962	+	0.911652i	$0.365193\pi$
$632$	0	0
$633$	11.2495	0.447128
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.75872	0.267790
$638$	0	0
$639$	−19.8504	−0.785271
$640$	0	0
$641$	38.2511	1.51083	0.755414	−	0.655248i	$-0.227436\pi$
$641$	38.2511	1.51083	0.755414	+	0.655248i	$0.227436\pi$
$642$	0	0
$643$	13.9649	0.550723	0.275361	−	0.961341i	$-0.411202\pi$
$643$	13.9649	0.550723	0.275361	+	0.961341i	$0.411202\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.5330	1.63283	0.816416	−	0.577464i	$-0.195958\pi$
$647$	41.5330	1.63283	0.816416	+	0.577464i	$0.195958\pi$
$648$	0	0
$649$	5.43680	0.213413
$650$	0	0
$651$	13.1773	0.516458
$652$	0	0
$653$	19.1629	0.749902	0.374951	−	0.927045i	$-0.377660\pi$
$653$	19.1629	0.749902	0.374951	+	0.927045i	$0.377660\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−30.1483	−1.17620
$658$	0	0
$659$	−0.634773	−0.0247273	−0.0123636	−	0.999924i	$-0.503936\pi$
$659$	−0.634773	−0.0247273	−0.0123636	+	0.999924i	$0.503936\pi$
$660$	0	0
$661$	40.1133	1.56023	0.780113	−	0.625639i	$-0.215162\pi$
$661$	40.1133	1.56023	0.780113	+	0.625639i	$0.215162\pi$
$662$	0	0
$663$	0.183417	0.00712334
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.921622	0.0356854
$668$	0	0
$669$	10.5113	0.406391
$670$	0	0
$671$	−67.4063	−2.60219
$672$	0	0
$673$	−48.6369	−1.87481	−0.937407	−	0.348237i	$-0.886781\pi$
$673$	−48.6369	−1.87481	−0.937407	+	0.348237i	$0.886781\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24.0144	−0.922947	−0.461474	−	0.887154i	$-0.652679\pi$
$677$	−24.0144	−0.922947	−0.461474	+	0.887154i	$0.652679\pi$
$678$	0	0
$679$	−46.1399	−1.77069
$680$	0	0
$681$	6.62702	0.253948
$682$	0	0
$683$	28.1750	1.07809	0.539043	−	0.842278i	$-0.318786\pi$
$683$	28.1750	1.07809	0.539043	+	0.842278i	$0.318786\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6.22446	−0.237478
$688$	0	0
$689$	9.07838	0.345859
$690$	0	0
$691$	−40.2772	−1.53222	−0.766109	−	0.642711i	$-0.777810\pi$
$691$	−40.2772	−1.53222	−0.766109	+	0.642711i	$0.777810\pi$
$692$	0	0
$693$	−55.3751	−2.10352
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.59213	0.0603061
$698$	0	0
$699$	4.73367	0.179044
$700$	0	0
$701$	0.752581	0.0284246	0.0142123	−	0.999899i	$-0.495476\pi$
$701$	0.752581	0.0284246	0.0142123	+	0.999899i	$0.495476\pi$
$702$	0	0
$703$	−12.4846	−0.470867
$704$	0	0
$705$	0	0
$706$	0	0
$707$	28.7792	1.08235
$708$	0	0
$709$	7.92777	0.297733	0.148867	−	0.988857i	$-0.452437\pi$
$709$	7.92777	0.297733	0.148867	+	0.988857i	$0.452437\pi$
$710$	0	0
$711$	−26.4391	−0.991543
$712$	0	0
$713$	−9.62475	−0.360450
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.9194	−0.445136
$718$	0	0
$719$	−8.46573	−0.315719	−0.157859	−	0.987462i	$-0.550459\pi$
$719$	−8.46573	−0.315719	−0.157859	+	0.987462i	$0.550459\pi$
$720$	0	0
$721$	−26.1978	−0.975657
$722$	0	0
$723$	−13.9877	−0.520209
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−30.0254	−1.11358	−0.556791	−	0.830653i	$-0.687968\pi$
$727$	−30.0254	−1.11358	−0.556791	+	0.830653i	$0.687968\pi$
$728$	0	0
$729$	−12.5441	−0.464597
$730$	0	0
$731$	−3.61879	−0.133846
$732$	0	0
$733$	10.9939	0.406067	0.203034	−	0.979172i	$-0.434920\pi$
$733$	10.9939	0.406067	0.203034	+	0.979172i	$0.434920\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	27.3919	1.00899
$738$	0	0
$739$	−16.1951	−0.595748	−0.297874	−	0.954605i	$-0.596278\pi$
$739$	−16.1951	−0.595748	−0.297874	+	0.954605i	$0.596278\pi$
$740$	0	0
$741$	1.21235	0.0445367
$742$	0	0
$743$	11.1590	0.409385	0.204692	−	0.978826i	$-0.434381\pi$
$743$	11.1590	0.409385	0.204692	+	0.978826i	$0.434381\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	45.7503	1.67392
$748$	0	0
$749$	−34.2967	−1.25317
$750$	0	0
$751$	24.8326	0.906153	0.453077	−	0.891472i	$-0.350326\pi$
$751$	24.8326	0.906153	0.453077	+	0.891472i	$0.350326\pi$
$752$	0	0
$753$	−2.10343	−0.0766533
$754$	0	0
$755$	0	0
$756$	0	0
$757$	32.1666	1.16911	0.584557	−	0.811352i	$-0.301268\pi$
$757$	32.1666	1.16911	0.584557	+	0.811352i	$0.301268\pi$
$758$	0	0
$759$	−4.34017	−0.157538
$760$	0	0
$761$	25.0517	0.908124	0.454062	−	0.890970i	$-0.349974\pi$
$761$	25.0517	0.908124	0.454062	+	0.890970i	$0.349974\pi$
$762$	0	0
$763$	−40.8781	−1.47989
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.986669	0.0356266
$768$	0	0
$769$	−2.77924	−0.100222	−0.0501110	−	0.998744i	$-0.515958\pi$
$769$	−2.77924	−0.100222	−0.0501110	+	0.998744i	$0.515958\pi$
$770$	0	0
$771$	2.01229	0.0724707
$772$	0	0
$773$	−19.3256	−0.695094	−0.347547	−	0.937663i	$-0.612985\pi$
$773$	−19.3256	−0.695094	−0.347547	+	0.937663i	$0.612985\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−11.1050	−0.398391
$778$	0	0
$779$	10.5236	0.377047
$780$	0	0
$781$	40.3728	1.44465
$782$	0	0
$783$	1.94214	0.0694065
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−40.1361	−1.43070	−0.715348	−	0.698768i	$-0.753732\pi$
$787$	−40.1361	−1.43070	−0.715348	+	0.698768i	$0.753732\pi$
$788$	0	0
$789$	−10.8143	−0.385000
$790$	0	0
$791$	63.1338	2.24478
$792$	0	0
$793$	−12.2329	−0.434402
$794$	0	0
$795$	0	0
$796$	0	0
$797$	50.2967	1.78160	0.890800	−	0.454395i	$-0.150145\pi$
$797$	50.2967	1.78160	0.890800	+	0.454395i	$0.150145\pi$
$798$	0	0
$799$	−1.45959	−0.0516365
$800$	0	0
$801$	−40.6225	−1.43533
$802$	0	0
$803$	61.3172	2.16384
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.9795	0.456899
$808$	0	0
$809$	24.2472	0.852488	0.426244	−	0.904608i	$-0.359837\pi$
$809$	24.2472	0.852488	0.426244	+	0.904608i	$0.359837\pi$
$810$	0	0
$811$	−41.1266	−1.44415	−0.722075	−	0.691815i	$-0.756812\pi$
$811$	−41.1266	−1.44415	−0.722075	+	0.691815i	$0.756812\pi$
$812$	0	0
$813$	2.67194	0.0937089
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−23.9194	−0.836832
$818$	0	0
$819$	−10.0494	−0.351156
$820$	0	0
$821$	34.3857	1.20007	0.600035	−	0.799973i	$-0.295153\pi$
$821$	34.3857	1.20007	0.600035	+	0.799973i	$0.295153\pi$
$822$	0	0
$823$	37.0361	1.29100	0.645499	−	0.763761i	$-0.276649\pi$
$823$	37.0361	1.29100	0.645499	+	0.763761i	$0.276649\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21.3691	−0.743076	−0.371538	−	0.928418i	$-0.621170\pi$
$827$	−21.3691	−0.743076	−0.371538	+	0.928418i	$0.621170\pi$
$828$	0	0
$829$	−33.2800	−1.15586	−0.577932	−	0.816085i	$-0.696140\pi$
$829$	−33.2800	−1.15586	−0.577932	+	0.816085i	$0.696140\pi$
$830$	0	0
$831$	15.5630	0.539875
$832$	0	0
$833$	−2.29914	−0.0796603
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−20.2823	−0.701059
$838$	0	0
$839$	−22.0072	−0.759773	−0.379886	−	0.925033i	$-0.624037\pi$
$839$	−22.0072	−0.759773	−0.379886	+	0.925033i	$0.624037\pi$
$840$	0	0
$841$	−28.6020	−0.986275
$842$	0	0
$843$	12.6270	0.434898
$844$	0	0
$845$	0	0
$846$	0	0
$847$	71.8225	2.46785
$848$	0	0
$849$	4.05398	0.139132
$850$	0	0
$851$	8.11118	0.278048
$852$	0	0
$853$	30.1627	1.03275	0.516376	−	0.856362i	$-0.327281\pi$
$853$	30.1627	1.03275	0.516376	+	0.856362i	$0.327281\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.15676	0.210311	0.105155	−	0.994456i	$-0.466466\pi$
$857$	6.15676	0.210311	0.105155	+	0.994456i	$0.466466\pi$
$858$	0	0
$859$	56.5257	1.92863	0.964316	−	0.264755i	$-0.0852911\pi$
$859$	56.5257	1.92863	0.964316	+	0.264755i	$0.0852911\pi$
$860$	0	0
$861$	9.36069	0.319012
$862$	0	0
$863$	19.5936	0.666972	0.333486	−	0.942755i	$-0.391775\pi$
$863$	19.5936	0.666972	0.333486	+	0.942755i	$0.391775\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.10382	0.309182
$868$	0	0
$869$	53.7731	1.82413
$870$	0	0
$871$	4.97107	0.168438
$872$	0	0
$873$	33.7009	1.14060
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.17113	0.140849	0.0704245	−	0.997517i	$-0.477565\pi$
$877$	4.17113	0.140849	0.0704245	+	0.997517i	$0.477565\pi$
$878$	0	0
$879$	−8.32580	−0.280822
$880$	0	0
$881$	14.1750	0.477568	0.238784	−	0.971073i	$-0.423251\pi$
$881$	14.1750	0.477568	0.238784	+	0.971073i	$0.423251\pi$
$882$	0	0
$883$	10.4859	0.352877	0.176439	−	0.984312i	$-0.443542\pi$
$883$	10.4859	0.352877	0.176439	+	0.984312i	$0.443542\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.41524	0.114672	0.0573362	−	0.998355i	$-0.481739\pi$
$887$	3.41524	0.114672	0.0573362	+	0.998355i	$0.481739\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	0	0
$891$	35.6404	1.19400
$892$	0	0
$893$	−9.64754	−0.322843
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.787653	−0.0262990
$898$	0	0
$899$	−4.15676	−0.138636
$900$	0	0
$901$	−3.08822	−0.102883
$902$	0	0
$903$	−21.2762	−0.708027
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−20.0566	−0.665970	−0.332985	−	0.942932i	$-0.608056\pi$
$907$	−20.0566	−0.665970	−0.332985	+	0.942932i	$0.608056\pi$
$908$	0	0
$909$	−21.0205	−0.697207
$910$	0	0
$911$	−17.5441	−0.581262	−0.290631	−	0.956835i	$-0.593865\pi$
$911$	−17.5441	−0.581262	−0.290631	+	0.956835i	$0.593865\pi$
$912$	0	0
$913$	−93.0493	−3.07948
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.6803	0.418742
$918$	0	0
$919$	−8.36683	−0.275996	−0.137998	−	0.990432i	$-0.544067\pi$
$919$	−8.36683	−0.275996	−0.137998	+	0.990432i	$0.544067\pi$
$920$	0	0
$921$	9.41855	0.310352
$922$	0	0
$923$	7.32684	0.241166
$924$	0	0
$925$	0	0
$926$	0	0
$927$	19.1350	0.628476
$928$	0	0
$929$	−21.2618	−0.697577	−0.348788	−	0.937201i	$-0.613407\pi$
$929$	−21.2618	−0.697577	−0.348788	+	0.937201i	$0.613407\pi$
$930$	0	0
$931$	−15.1967	−0.498053
$932$	0	0
$933$	1.98771	0.0650748
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.3090	0.859477	0.429738	−	0.902953i	$-0.358606\pi$
$937$	26.3090	0.859477	0.429738	+	0.902953i	$0.358606\pi$
$938$	0	0
$939$	5.17727	0.168954
$940$	0	0
$941$	−1.02052	−0.0332680	−0.0166340	−	0.999862i	$-0.505295\pi$
$941$	−1.02052	−0.0332680	−0.0166340	+	0.999862i	$0.505295\pi$
$942$	0	0
$943$	−6.83710	−0.222647
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.37525	−0.142176	−0.0710882	−	0.997470i	$-0.522647\pi$
$947$	−4.37525	−0.142176	−0.0710882	+	0.997470i	$0.522647\pi$
$948$	0	0
$949$	11.1278	0.361225
$950$	0	0
$951$	4.52359	0.146687
$952$	0	0
$953$	−3.51745	−0.113941	−0.0569706	−	0.998376i	$-0.518144\pi$
$953$	−3.51745	−0.113941	−0.0569706	+	0.998376i	$0.518144\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.87444	−0.0605921
$958$	0	0
$959$	83.1338	2.68453
$960$	0	0
$961$	12.4101	0.400327
$962$	0	0
$963$	25.0505	0.807241
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−39.6925	−1.27642	−0.638212	−	0.769861i	$-0.720326\pi$
$967$	−39.6925	−1.27642	−0.638212	+	0.769861i	$0.720326\pi$
$968$	0	0
$969$	−0.412408	−0.0132484
$970$	0	0
$971$	−43.3340	−1.39066	−0.695328	−	0.718693i	$-0.744741\pi$
$971$	−43.3340	−1.39066	−0.695328	+	0.718693i	$0.744741\pi$
$972$	0	0
$973$	21.3607	0.684792
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−44.9009	−1.43651	−0.718254	−	0.695781i	$-0.755058\pi$
$977$	−44.9009	−1.43651	−0.718254	+	0.695781i	$0.755058\pi$
$978$	0	0
$979$	82.6200	2.64055
$980$	0	0
$981$	29.8576	0.953280
$982$	0	0
$983$	−8.30510	−0.264892	−0.132446	−	0.991190i	$-0.542283\pi$
$983$	−8.30510	−0.264892	−0.132446	+	0.991190i	$0.542283\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−8.58145	−0.273151
$988$	0	0
$989$	15.5402	0.494151
$990$	0	0
$991$	−34.0722	−1.08234	−0.541170	−	0.840913i	$-0.682018\pi$
$991$	−34.0722	−1.08234	−0.541170	+	0.840913i	$0.682018\pi$
$992$	0	0
$993$	−12.6621	−0.401819
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−35.6475	−1.12897	−0.564484	−	0.825444i	$-0.690925\pi$
$997$	−35.6475	−1.12897	−0.564484	+	0.825444i	$0.690925\pi$
$998$	0	0
$999$	17.0928	0.540791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.a.k.1.2		3
4.3	odd	2		5200.2.a.cd.1.2		3
5.2	odd	4		260.2.c.a.209.4	yes	6
5.3	odd	4		260.2.c.a.209.3	✓	6
5.4	even	2		1300.2.a.j.1.2		3
15.2	even	4		2340.2.h.e.469.4		6
15.8	even	4		2340.2.h.e.469.3		6
20.3	even	4		1040.2.d.d.209.4		6
20.7	even	4		1040.2.d.d.209.3		6
20.19	odd	2		5200.2.a.cg.1.2		3
65.8	even	4		3380.2.d.a.1689.3		6
65.12	odd	4		3380.2.c.c.2029.4		6
65.18	even	4		3380.2.d.b.1689.3		6
65.38	odd	4		3380.2.c.c.2029.3		6
65.47	even	4		3380.2.d.b.1689.4		6
65.57	even	4		3380.2.d.a.1689.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.c.a.209.3	✓	6	5.3	odd	4
260.2.c.a.209.4	yes	6	5.2	odd	4
1040.2.d.d.209.3		6	20.7	even	4
1040.2.d.d.209.4		6	20.3	even	4
1300.2.a.j.1.2		3	5.4	even	2
1300.2.a.k.1.2		3	1.1	even	1	trivial
2340.2.h.e.469.3		6	15.8	even	4
2340.2.h.e.469.4		6	15.2	even	4
3380.2.c.c.2029.3		6	65.38	odd	4
3380.2.c.c.2029.4		6	65.12	odd	4
3380.2.d.a.1689.3		6	65.8	even	4
3380.2.d.a.1689.4		6	65.57	even	4
3380.2.d.b.1689.3		6	65.18	even	4
3380.2.d.b.1689.4		6	65.47	even	4
5200.2.a.cd.1.2		3	4.3	odd	2
5200.2.a.cg.1.2		3	20.19	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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.a (trivial)

\(f(q)\)	\(=\)	\(q-0.539189 q^{3} +3.70928 q^{7} -2.70928 q^{9} +O(q^{10})\)	\(q-0.539189 q^{3} +3.70928 q^{7} -2.70928 q^{9} +5.51026 q^{11} +1.00000 q^{13} -0.340173 q^{17} -2.24846 q^{19} -2.00000 q^{21} +1.46081 q^{23} +3.07838 q^{27} +0.630898 q^{29} -6.58864 q^{31} -2.97107 q^{33} +5.55252 q^{37} -0.539189 q^{39} -4.68035 q^{41} +10.6381 q^{43} +4.29072 q^{47} +6.75872 q^{49} +0.183417 q^{51} +9.07838 q^{53} +1.21235 q^{57} +0.986669 q^{59} -12.2329 q^{61} -10.0494 q^{63} +4.97107 q^{67} -0.787653 q^{69} +7.32684 q^{71} +11.1278 q^{73} +20.4391 q^{77} +9.75872 q^{79} +6.46800 q^{81} -16.8865 q^{83} -0.340173 q^{87} +14.9939 q^{89} +3.70928 q^{91} +3.55252 q^{93} -12.4391 q^{97} -14.9288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{7} - q^{9}+O(q^{10}) \) 3 * q + 4 * q^7 - q^9	\( 3 q + 4 q^{7} - q^{9} + 3 q^{13} + 10 q^{17} + 2 q^{19} - 6 q^{21} + 6 q^{23} + 6 q^{27} - 2 q^{29} + 6 q^{33} + 16 q^{37} + 8 q^{41} - 6 q^{43} + 20 q^{47} - 5 q^{49} - 4 q^{51} + 24 q^{53} + 14 q^{57} + 2 q^{59} - 14 q^{61} - 12 q^{63} + 8 q^{69} + 10 q^{71} + 12 q^{73} + 14 q^{77} + 4 q^{79} - 13 q^{81} - 4 q^{83} + 10 q^{87} + 10 q^{89} + 4 q^{91} + 10 q^{93} + 10 q^{97} - 14 q^{99}+O(q^{100}) \) 3 * q + 4 * q^7 - q^9 + 3 * q^13 + 10 * q^17 + 2 * q^19 - 6 * q^21 + 6 * q^23 + 6 * q^27 - 2 * q^29 + 6 * q^33 + 16 * q^37 + 8 * q^41 - 6 * q^43 + 20 * q^47 - 5 * q^49 - 4 * q^51 + 24 * q^53 + 14 * q^57 + 2 * q^59 - 14 * q^61 - 12 * q^63 + 8 * q^69 + 10 * q^71 + 12 * q^73 + 14 * q^77 + 4 * q^79 - 13 * q^81 - 4 * q^83 + 10 * q^87 + 10 * q^89 + 4 * q^91 + 10 * q^93 + 10 * q^97 - 14 * q^99

Introduction

L-functions

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