LMFDB - Embedded newform 7800.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.86907 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.86907 q^{7} +1.00000 q^{9} +0.675131 q^{11} -1.00000 q^{13} -5.31265 q^{17} -0.806063 q^{19} -1.86907 q^{21} +8.73084 q^{23} -1.00000 q^{27} -3.96239 q^{29} -1.71274 q^{31} -0.675131 q^{33} +7.38058 q^{37} +1.00000 q^{39} -8.54420 q^{41} -10.4690 q^{43} -2.13093 q^{47} -3.50659 q^{49} +5.31265 q^{51} +6.19394 q^{53} +0.806063 q^{57} +14.4944 q^{59} -10.3503 q^{61} +1.86907 q^{63} -1.06300 q^{67} -8.73084 q^{69} -13.4314 q^{71} +8.34297 q^{73} +1.26187 q^{77} -6.93207 q^{79} +1.00000 q^{81} -10.5696 q^{83} +3.96239 q^{87} +11.9756 q^{89} -1.86907 q^{91} +1.71274 q^{93} -0.261865 q^{97} +0.675131 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + q^7 + 3 * q^9	$ 3 q - 3 q^{3} + q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} + 5 q^{17} - 2 q^{19} - q^{21} + 4 q^{23} - 3 q^{27} - q^{29} - 11 q^{31} + 3 q^{33} + 10 q^{37} + 3 q^{39} - 16 q^{41} - 11 q^{47} + 10 q^{49} - 5 q^{51} + 19 q^{53} + 2 q^{57} - 3 q^{59} - 21 q^{61} + q^{63} + q^{67} - 4 q^{69} + 2 q^{71} + 2 q^{73} + 13 q^{77} - 12 q^{79} + 3 q^{81} - 7 q^{83} + q^{87} - 16 q^{89} - q^{91} + 11 q^{93} - 10 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + q^7 + 3 * q^9 - 3 * q^11 - 3 * q^13 + 5 * q^17 - 2 * q^19 - q^21 + 4 * q^23 - 3 * q^27 - q^29 - 11 * q^31 + 3 * q^33 + 10 * q^37 + 3 * q^39 - 16 * q^41 - 11 * q^47 + 10 * q^49 - 5 * q^51 + 19 * q^53 + 2 * q^57 - 3 * q^59 - 21 * q^61 + q^63 + q^67 - 4 * q^69 + 2 * q^71 + 2 * q^73 + 13 * q^77 - 12 * q^79 + 3 * q^81 - 7 * q^83 + q^87 - 16 * q^89 - q^91 + 11 * q^93 - 10 * q^97 - 3 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.86907	0.706441	0.353221	−	0.935540i	$-0.385087\pi$
$7$	1.86907	0.706441	0.353221	+	0.935540i	$0.385087\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.675131	0.203560	0.101780	−	0.994807i	$-0.467546\pi$
$11$	0.675131	0.203560	0.101780	+	0.994807i	$0.467546\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.31265	−1.28851	−0.644253	−	0.764812i	$-0.722832\pi$
$17$	−5.31265	−1.28851	−0.644253	+	0.764812i	$0.722832\pi$
$18$	0	0
$19$	−0.806063	−0.184924	−0.0924618	−	0.995716i	$-0.529474\pi$
$19$	−0.806063	−0.184924	−0.0924618	+	0.995716i	$0.529474\pi$
$20$	0	0
$21$	−1.86907	−0.407864
$22$	0	0
$23$	8.73084	1.82051	0.910253	−	0.414052i	$-0.135887\pi$
$23$	8.73084	1.82051	0.910253	+	0.414052i	$0.135887\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.96239	−0.735797	−0.367899	−	0.929866i	$-0.619923\pi$
$29$	−3.96239	−0.735797	−0.367899	+	0.929866i	$0.619923\pi$
$30$	0	0
$31$	−1.71274	−0.307618	−0.153809	−	0.988101i	$-0.549154\pi$
$31$	−1.71274	−0.307618	−0.153809	+	0.988101i	$0.549154\pi$
$32$	0	0
$33$	−0.675131	−0.117525
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.38058	1.21336	0.606680	−	0.794946i	$-0.292501\pi$
$37$	7.38058	1.21336	0.606680	+	0.794946i	$0.292501\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−8.54420	−1.33438	−0.667190	−	0.744888i	$-0.732503\pi$
$41$	−8.54420	−1.33438	−0.667190	+	0.744888i	$0.732503\pi$
$42$	0	0
$43$	−10.4690	−1.59650	−0.798252	−	0.602324i	$-0.794242\pi$
$43$	−10.4690	−1.59650	−0.798252	+	0.602324i	$0.794242\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.13093	−0.310828	−0.155414	−	0.987849i	$-0.549671\pi$
$47$	−2.13093	−0.310828	−0.155414	+	0.987849i	$0.549671\pi$
$48$	0	0
$49$	−3.50659	−0.500941
$50$	0	0
$51$	5.31265	0.743920
$52$	0	0
$53$	6.19394	0.850803	0.425401	−	0.905005i	$-0.360133\pi$
$53$	6.19394	0.850803	0.425401	+	0.905005i	$0.360133\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.806063	0.106766
$58$	0	0
$59$	14.4944	1.88701	0.943503	−	0.331364i	$-0.107509\pi$
$59$	14.4944	1.88701	0.943503	+	0.331364i	$0.107509\pi$
$60$	0	0
$61$	−10.3503	−1.32522	−0.662608	−	0.748967i	$-0.730550\pi$
$61$	−10.3503	−1.32522	−0.662608	+	0.748967i	$0.730550\pi$
$62$	0	0
$63$	1.86907	0.235480
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.06300	−0.129867	−0.0649333	−	0.997890i	$-0.520683\pi$
$67$	−1.06300	−0.129867	−0.0649333	+	0.997890i	$0.520683\pi$
$68$	0	0
$69$	−8.73084	−1.05107
$70$	0	0
$71$	−13.4314	−1.59401	−0.797005	−	0.603973i	$-0.793583\pi$
$71$	−13.4314	−1.59401	−0.797005	+	0.603973i	$0.793583\pi$
$72$	0	0
$73$	8.34297	0.976470	0.488235	−	0.872712i	$-0.337641\pi$
$73$	8.34297	0.976470	0.488235	+	0.872712i	$0.337641\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.26187	0.143803
$78$	0	0
$79$	−6.93207	−0.779919	−0.389959	−	0.920832i	$-0.627511\pi$
$79$	−6.93207	−0.779919	−0.389959	+	0.920832i	$0.627511\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.5696	−1.16016	−0.580082	−	0.814558i	$-0.696979\pi$
$83$	−10.5696	−1.16016	−0.580082	+	0.814558i	$0.696979\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.96239	0.424813
$88$	0	0
$89$	11.9756	1.26941	0.634704	−	0.772756i	$-0.281122\pi$
$89$	11.9756	1.26941	0.634704	+	0.772756i	$0.281122\pi$
$90$	0	0
$91$	−1.86907	−0.195932
$92$	0	0
$93$	1.71274	0.177603
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.261865	−0.0265884	−0.0132942	−	0.999912i	$-0.504232\pi$
$97$	−0.261865	−0.0265884	−0.0132942	+	0.999912i	$0.504232\pi$
$98$	0	0
$99$	0.675131	0.0678532
$100$	0	0
$101$	−16.6932	−1.66104	−0.830519	−	0.556990i	$-0.811956\pi$
$101$	−16.6932	−1.66104	−0.830519	+	0.556990i	$0.811956\pi$
$102$	0	0
$103$	−12.4690	−1.22860	−0.614302	−	0.789071i	$-0.710562\pi$
$103$	−12.4690	−1.22860	−0.614302	+	0.789071i	$0.710562\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.1866	0.984780	0.492390	−	0.870375i	$-0.336123\pi$
$107$	10.1866	0.984780	0.492390	+	0.870375i	$0.336123\pi$
$108$	0	0
$109$	15.9551	1.52822	0.764110	−	0.645085i	$-0.223178\pi$
$109$	15.9551	1.52822	0.764110	+	0.645085i	$0.223178\pi$
$110$	0	0
$111$	−7.38058	−0.700534
$112$	0	0
$113$	10.1260	0.952575	0.476287	−	0.879290i	$-0.341982\pi$
$113$	10.1260	0.952575	0.476287	+	0.879290i	$0.341982\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−9.92970	−0.910254
$120$	0	0
$121$	−10.5442	−0.958563
$122$	0	0
$123$	8.54420	0.770404
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.9551	−1.41579	−0.707893	−	0.706320i	$-0.750354\pi$
$127$	−15.9551	−1.41579	−0.707893	+	0.706320i	$0.750354\pi$
$128$	0	0
$129$	10.4690	0.921742
$130$	0	0
$131$	17.7440	1.55030	0.775151	−	0.631776i	$-0.217674\pi$
$131$	17.7440	1.55030	0.775151	+	0.631776i	$0.217674\pi$
$132$	0	0
$133$	−1.50659	−0.130638
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.96239	−0.765709	−0.382854	−	0.923809i	$-0.625059\pi$
$137$	−8.96239	−0.765709	−0.382854	+	0.923809i	$0.625059\pi$
$138$	0	0
$139$	0.0752228	0.00638031	0.00319016	−	0.999995i	$-0.498985\pi$
$139$	0.0752228	0.00638031	0.00319016	+	0.999995i	$0.498985\pi$
$140$	0	0
$141$	2.13093	0.179457
$142$	0	0
$143$	−0.675131	−0.0564573
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.50659	0.289218
$148$	0	0
$149$	−1.66291	−0.136231	−0.0681155	−	0.997677i	$-0.521699\pi$
$149$	−1.66291	−0.136231	−0.0681155	+	0.997677i	$0.521699\pi$
$150$	0	0
$151$	−19.6678	−1.60055	−0.800273	−	0.599636i	$-0.795312\pi$
$151$	−19.6678	−1.60055	−0.800273	+	0.599636i	$0.795312\pi$
$152$	0	0
$153$	−5.31265	−0.429502
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.35756	0.108345	0.0541724	−	0.998532i	$-0.482748\pi$
$157$	1.35756	0.108345	0.0541724	+	0.998532i	$0.482748\pi$
$158$	0	0
$159$	−6.19394	−0.491211
$160$	0	0
$161$	16.3185	1.28608
$162$	0	0
$163$	20.1319	1.57685	0.788426	−	0.615130i	$-0.210897\pi$
$163$	20.1319	1.57685	0.788426	+	0.615130i	$0.210897\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.3054	−1.02960	−0.514800	−	0.857310i	$-0.672134\pi$
$167$	−13.3054	−1.02960	−0.514800	+	0.857310i	$0.672134\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.806063	−0.0616412
$172$	0	0
$173$	2.91890	0.221920	0.110960	−	0.993825i	$-0.464608\pi$
$173$	2.91890	0.221920	0.110960	+	0.993825i	$0.464608\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.4944	−1.08946
$178$	0	0
$179$	−18.8568	−1.40943	−0.704714	−	0.709492i	$-0.748924\pi$
$179$	−18.8568	−1.40943	−0.704714	+	0.709492i	$0.748924\pi$
$180$	0	0
$181$	9.39375	0.698232	0.349116	−	0.937079i	$-0.386482\pi$
$181$	9.39375	0.698232	0.349116	+	0.937079i	$0.386482\pi$
$182$	0	0
$183$	10.3503	0.765113
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.58673	−0.262288
$188$	0	0
$189$	−1.86907	−0.135955
$190$	0	0
$191$	14.6253	1.05825	0.529125	−	0.848544i	$-0.322520\pi$
$191$	14.6253	1.05825	0.529125	+	0.848544i	$0.322520\pi$
$192$	0	0
$193$	12.7308	0.916386	0.458193	−	0.888853i	$-0.348497\pi$
$193$	12.7308	0.916386	0.458193	+	0.888853i	$0.348497\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.84367	−0.416345	−0.208172	−	0.978092i	$-0.566751\pi$
$197$	−5.84367	−0.416345	−0.208172	+	0.978092i	$0.566751\pi$
$198$	0	0
$199$	−21.7889	−1.54458	−0.772288	−	0.635273i	$-0.780888\pi$
$199$	−21.7889	−1.54458	−0.772288	+	0.635273i	$0.780888\pi$
$200$	0	0
$201$	1.06300	0.0749785
$202$	0	0
$203$	−7.40597	−0.519797
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.73084	0.606835
$208$	0	0
$209$	−0.544198	−0.0376430
$210$	0	0
$211$	−21.3561	−1.47022	−0.735109	−	0.677949i	$-0.762869\pi$
$211$	−21.3561	−1.47022	−0.735109	+	0.677949i	$0.762869\pi$
$212$	0	0
$213$	13.4314	0.920302
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.20123	−0.217314
$218$	0	0
$219$	−8.34297	−0.563765
$220$	0	0
$221$	5.31265	0.357368
$222$	0	0
$223$	−7.96968	−0.533689	−0.266845	−	0.963740i	$-0.585981\pi$
$223$	−7.96968	−0.533689	−0.266845	+	0.963740i	$0.585981\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.13823	−0.208291	−0.104146	−	0.994562i	$-0.533211\pi$
$227$	−3.13823	−0.208291	−0.104146	+	0.994562i	$0.533211\pi$
$228$	0	0
$229$	−9.84955	−0.650877	−0.325438	−	0.945563i	$-0.605512\pi$
$229$	−9.84955	−0.650877	−0.325438	+	0.945563i	$0.605512\pi$
$230$	0	0
$231$	−1.26187	−0.0830246
$232$	0	0
$233$	20.8265	1.36439	0.682196	−	0.731170i	$-0.261025\pi$
$233$	20.8265	1.36439	0.682196	+	0.731170i	$0.261025\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.93207	0.450286
$238$	0	0
$239$	−7.34534	−0.475130	−0.237565	−	0.971372i	$-0.576349\pi$
$239$	−7.34534	−0.475130	−0.237565	+	0.971372i	$0.576349\pi$
$240$	0	0
$241$	−11.2692	−0.725910	−0.362955	−	0.931807i	$-0.618232\pi$
$241$	−11.2692	−0.725910	−0.362955	+	0.931807i	$0.618232\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.806063	0.0512886
$248$	0	0
$249$	10.5696	0.669821
$250$	0	0
$251$	−29.0943	−1.83641	−0.918207	−	0.396100i	$-0.870363\pi$
$251$	−29.0943	−1.83641	−0.918207	+	0.396100i	$0.870363\pi$
$252$	0	0
$253$	5.89446	0.370582
$254$	0	0
$255$	0	0
$256$	0	0
$257$	29.5198	1.84139	0.920696	−	0.390280i	$-0.127622\pi$
$257$	29.5198	1.84139	0.920696	+	0.390280i	$0.127622\pi$
$258$	0	0
$259$	13.7948	0.857167
$260$	0	0
$261$	−3.96239	−0.245266
$262$	0	0
$263$	17.7685	1.09565	0.547825	−	0.836593i	$-0.315456\pi$
$263$	17.7685	1.09565	0.547825	+	0.836593i	$0.315456\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−11.9756	−0.732893
$268$	0	0
$269$	−1.38787	−0.0846201	−0.0423101	−	0.999105i	$-0.513472\pi$
$269$	−1.38787	−0.0846201	−0.0423101	+	0.999105i	$0.513472\pi$
$270$	0	0
$271$	14.4191	0.875901	0.437950	−	0.898999i	$-0.355705\pi$
$271$	14.4191	0.875901	0.437950	+	0.898999i	$0.355705\pi$
$272$	0	0
$273$	1.86907	0.113121
$274$	0	0
$275$	0	0
$276$	0	0
$277$	19.0640	1.14544	0.572721	−	0.819750i	$-0.305888\pi$
$277$	19.0640	1.14544	0.572721	+	0.819750i	$0.305888\pi$
$278$	0	0
$279$	−1.71274	−0.102539
$280$	0	0
$281$	−27.4617	−1.63823	−0.819113	−	0.573632i	$-0.805534\pi$
$281$	−27.4617	−1.63823	−0.819113	+	0.573632i	$0.805534\pi$
$282$	0	0
$283$	13.7889	0.819666	0.409833	−	0.912161i	$-0.365587\pi$
$283$	13.7889	0.819666	0.409833	+	0.912161i	$0.365587\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.9697	−0.942661
$288$	0	0
$289$	11.2243	0.660250
$290$	0	0
$291$	0.261865	0.0153508
$292$	0	0
$293$	21.4763	1.25466	0.627329	−	0.778755i	$-0.284148\pi$
$293$	21.4763	1.25466	0.627329	+	0.778755i	$0.284148\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.675131	−0.0391751
$298$	0	0
$299$	−8.73084	−0.504918
$300$	0	0
$301$	−19.5672	−1.12784
$302$	0	0
$303$	16.6932	0.959001
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−17.7685	−1.01410	−0.507050	−	0.861917i	$-0.669264\pi$
$307$	−17.7685	−1.01410	−0.507050	+	0.861917i	$0.669264\pi$
$308$	0	0
$309$	12.4690	0.709335
$310$	0	0
$311$	−1.16362	−0.0659828	−0.0329914	−	0.999456i	$-0.510503\pi$
$311$	−1.16362	−0.0659828	−0.0329914	+	0.999456i	$0.510503\pi$
$312$	0	0
$313$	−25.2677	−1.42822	−0.714109	−	0.700035i	$-0.753168\pi$
$313$	−25.2677	−1.42822	−0.714109	+	0.700035i	$0.753168\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.8192	1.33782	0.668911	−	0.743343i	$-0.266761\pi$
$317$	23.8192	1.33782	0.668911	+	0.743343i	$0.266761\pi$
$318$	0	0
$319$	−2.67513	−0.149779
$320$	0	0
$321$	−10.1866	−0.568563
$322$	0	0
$323$	4.28233	0.238275
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−15.9551	−0.882319
$328$	0	0
$329$	−3.98286	−0.219582
$330$	0	0
$331$	21.7440	1.19516	0.597580	−	0.801810i	$-0.296129\pi$
$331$	21.7440	1.19516	0.597580	+	0.801810i	$0.296129\pi$
$332$	0	0
$333$	7.38058	0.404453
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−17.4690	−0.951596	−0.475798	−	0.879555i	$-0.657841\pi$
$337$	−17.4690	−0.951596	−0.475798	+	0.879555i	$0.657841\pi$
$338$	0	0
$339$	−10.1260	−0.549969
$340$	0	0
$341$	−1.15633	−0.0626185
$342$	0	0
$343$	−19.6375	−1.06033
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.47627	−0.0792503	−0.0396252	−	0.999215i	$-0.512616\pi$
$347$	−1.47627	−0.0792503	−0.0396252	+	0.999215i	$0.512616\pi$
$348$	0	0
$349$	14.8568	0.795269	0.397634	−	0.917544i	$-0.369831\pi$
$349$	14.8568	0.795269	0.397634	+	0.917544i	$0.369831\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	0.604833	0.0321920	0.0160960	−	0.999870i	$-0.494876\pi$
$353$	0.604833	0.0321920	0.0160960	+	0.999870i	$0.494876\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	9.92970	0.525536
$358$	0	0
$359$	6.94288	0.366431	0.183215	−	0.983073i	$-0.441349\pi$
$359$	6.94288	0.366431	0.183215	+	0.983073i	$0.441349\pi$
$360$	0	0
$361$	−18.3503	−0.965803
$362$	0	0
$363$	10.5442	0.553427
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.0738	0.578048	0.289024	−	0.957322i	$-0.406669\pi$
$367$	11.0738	0.578048	0.289024	+	0.957322i	$0.406669\pi$
$368$	0	0
$369$	−8.54420	−0.444793
$370$	0	0
$371$	11.5769	0.601042
$372$	0	0
$373$	−26.9307	−1.39442	−0.697208	−	0.716869i	$-0.745575\pi$
$373$	−26.9307	−1.39442	−0.697208	+	0.716869i	$0.745575\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.96239	0.204073
$378$	0	0
$379$	5.12364	0.263184	0.131592	−	0.991304i	$-0.457991\pi$
$379$	5.12364	0.263184	0.131592	+	0.991304i	$0.457991\pi$
$380$	0	0
$381$	15.9551	0.817404
$382$	0	0
$383$	−17.4920	−0.893799	−0.446900	−	0.894584i	$-0.647472\pi$
$383$	−17.4920	−0.893799	−0.446900	+	0.894584i	$0.647472\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.4690	−0.532168
$388$	0	0
$389$	−23.2750	−1.18009	−0.590046	−	0.807370i	$-0.700890\pi$
$389$	−23.2750	−1.18009	−0.590046	+	0.807370i	$0.700890\pi$
$390$	0	0
$391$	−46.3839	−2.34573
$392$	0	0
$393$	−17.7440	−0.895067
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−35.7499	−1.79424	−0.897118	−	0.441791i	$-0.854343\pi$
$397$	−35.7499	−1.79424	−0.897118	+	0.441791i	$0.854343\pi$
$398$	0	0
$399$	1.50659	0.0754237
$400$	0	0
$401$	−24.0059	−1.19880	−0.599398	−	0.800451i	$-0.704593\pi$
$401$	−24.0059	−1.19880	−0.599398	+	0.800451i	$0.704593\pi$
$402$	0	0
$403$	1.71274	0.0853178
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.98286	0.246991
$408$	0	0
$409$	−17.7539	−0.877872	−0.438936	−	0.898518i	$-0.644645\pi$
$409$	−17.7539	−0.877872	−0.438936	+	0.898518i	$0.644645\pi$
$410$	0	0
$411$	8.96239	0.442082
$412$	0	0
$413$	27.0910	1.33306
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−0.0752228	−0.00368368
$418$	0	0
$419$	−12.6801	−0.619461	−0.309731	−	0.950824i	$-0.600239\pi$
$419$	−12.6801	−0.619461	−0.309731	+	0.950824i	$0.600239\pi$
$420$	0	0
$421$	−33.5633	−1.63577	−0.817886	−	0.575380i	$-0.804854\pi$
$421$	−33.5633	−1.63577	−0.817886	+	0.575380i	$0.804854\pi$
$422$	0	0
$423$	−2.13093	−0.103609
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−19.3453	−0.936186
$428$	0	0
$429$	0.675131	0.0325956
$430$	0	0
$431$	0.554047	0.0266875	0.0133438	−	0.999911i	$-0.495752\pi$
$431$	0.554047	0.0266875	0.0133438	+	0.999911i	$0.495752\pi$
$432$	0	0
$433$	−2.06537	−0.0992555	−0.0496278	−	0.998768i	$-0.515803\pi$
$433$	−2.06537	−0.0992555	−0.0496278	+	0.998768i	$0.515803\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.03761	−0.336655
$438$	0	0
$439$	−26.9076	−1.28423	−0.642116	−	0.766608i	$-0.721943\pi$
$439$	−26.9076	−1.28423	−0.642116	+	0.766608i	$0.721943\pi$
$440$	0	0
$441$	−3.50659	−0.166980
$442$	0	0
$443$	16.4544	0.781772	0.390886	−	0.920439i	$-0.372169\pi$
$443$	16.4544	0.781772	0.390886	+	0.920439i	$0.372169\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.66291	0.0786530
$448$	0	0
$449$	−3.28489	−0.155023	−0.0775117	−	0.996991i	$-0.524698\pi$
$449$	−3.28489	−0.155023	−0.0775117	+	0.996991i	$0.524698\pi$
$450$	0	0
$451$	−5.76845	−0.271626
$452$	0	0
$453$	19.6678	0.924076
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.6458	−1.33999	−0.669996	−	0.742364i	$-0.733704\pi$
$457$	−28.6458	−1.33999	−0.669996	+	0.742364i	$0.733704\pi$
$458$	0	0
$459$	5.31265	0.247973
$460$	0	0
$461$	9.78892	0.455915	0.227958	−	0.973671i	$-0.426795\pi$
$461$	9.78892	0.455915	0.227958	+	0.973671i	$0.426795\pi$
$462$	0	0
$463$	19.3757	0.900463	0.450232	−	0.892912i	$-0.351341\pi$
$463$	19.3757	0.900463	0.450232	+	0.892912i	$0.351341\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−39.0494	−1.80699	−0.903495	−	0.428599i	$-0.859007\pi$
$467$	−39.0494	−1.80699	−0.903495	+	0.428599i	$0.859007\pi$
$468$	0	0
$469$	−1.98683	−0.0917431
$470$	0	0
$471$	−1.35756	−0.0625529
$472$	0	0
$473$	−7.06793	−0.324984
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.19394	0.283601
$478$	0	0
$479$	−21.6375	−0.988643	−0.494322	−	0.869279i	$-0.664584\pi$
$479$	−21.6375	−0.988643	−0.494322	+	0.869279i	$0.664584\pi$
$480$	0	0
$481$	−7.38058	−0.336525
$482$	0	0
$483$	−16.3185	−0.742519
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−27.5623	−1.24897	−0.624483	−	0.781038i	$-0.714690\pi$
$487$	−27.5623	−1.24897	−0.624483	+	0.781038i	$0.714690\pi$
$488$	0	0
$489$	−20.1319	−0.910395
$490$	0	0
$491$	−4.17679	−0.188496	−0.0942480	−	0.995549i	$-0.530045\pi$
$491$	−4.17679	−0.188496	−0.0942480	+	0.995549i	$0.530045\pi$
$492$	0	0
$493$	21.0508	0.948080
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−25.1041	−1.12607
$498$	0	0
$499$	10.0459	0.449714	0.224857	−	0.974392i	$-0.427808\pi$
$499$	10.0459	0.449714	0.224857	+	0.974392i	$0.427808\pi$
$500$	0	0
$501$	13.3054	0.594439
$502$	0	0
$503$	−30.2071	−1.34687	−0.673434	−	0.739247i	$-0.735182\pi$
$503$	−30.2071	−1.34687	−0.673434	+	0.739247i	$0.735182\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	26.1319	1.15828	0.579138	−	0.815230i	$-0.303389\pi$
$509$	26.1319	1.15828	0.579138	+	0.815230i	$0.303389\pi$
$510$	0	0
$511$	15.5936	0.689819
$512$	0	0
$513$	0.806063	0.0355886
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.43866	−0.0632721
$518$	0	0
$519$	−2.91890	−0.128125
$520$	0	0
$521$	−11.2995	−0.495039	−0.247520	−	0.968883i	$-0.579615\pi$
$521$	−11.2995	−0.495039	−0.247520	+	0.968883i	$0.579615\pi$
$522$	0	0
$523$	−20.3634	−0.890431	−0.445215	−	0.895423i	$-0.646873\pi$
$523$	−20.3634	−0.890431	−0.445215	+	0.895423i	$0.646873\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.09920	0.396367
$528$	0	0
$529$	53.2276	2.31424
$530$	0	0
$531$	14.4944	0.629002
$532$	0	0
$533$	8.54420	0.370090
$534$	0	0
$535$	0	0
$536$	0	0
$537$	18.8568	0.813733
$538$	0	0
$539$	−2.36741	−0.101971
$540$	0	0
$541$	23.2057	0.997691	0.498845	−	0.866691i	$-0.333758\pi$
$541$	23.2057	0.997691	0.498845	+	0.866691i	$0.333758\pi$
$542$	0	0
$543$	−9.39375	−0.403125
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.40693	0.231183	0.115592	−	0.993297i	$-0.463124\pi$
$547$	5.40693	0.231183	0.115592	+	0.993297i	$0.463124\pi$
$548$	0	0
$549$	−10.3503	−0.441738
$550$	0	0
$551$	3.19394	0.136066
$552$	0	0
$553$	−12.9565	−0.550967
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.9960	−0.847259	−0.423630	−	0.905836i	$-0.639244\pi$
$557$	−19.9960	−0.847259	−0.423630	+	0.905836i	$0.639244\pi$
$558$	0	0
$559$	10.4690	0.442790
$560$	0	0
$561$	3.58673	0.151432
$562$	0	0
$563$	−32.3693	−1.36420	−0.682102	−	0.731257i	$-0.738934\pi$
$563$	−32.3693	−1.36420	−0.682102	+	0.731257i	$0.738934\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.86907	0.0784935
$568$	0	0
$569$	−2.47039	−0.103564	−0.0517821	−	0.998658i	$-0.516490\pi$
$569$	−2.47039	−0.103564	−0.0517821	+	0.998658i	$0.516490\pi$
$570$	0	0
$571$	−30.8178	−1.28969	−0.644843	−	0.764315i	$-0.723077\pi$
$571$	−30.8178	−1.28969	−0.644843	+	0.764315i	$0.723077\pi$
$572$	0	0
$573$	−14.6253	−0.610981
$574$	0	0
$575$	0	0
$576$	0	0
$577$	24.6155	1.02475	0.512377	−	0.858760i	$-0.328765\pi$
$577$	24.6155	1.02475	0.512377	+	0.858760i	$0.328765\pi$
$578$	0	0
$579$	−12.7308	−0.529076
$580$	0	0
$581$	−19.7553	−0.819587
$582$	0	0
$583$	4.18172	0.173189
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.28726	−0.176954	−0.0884770	−	0.996078i	$-0.528200\pi$
$587$	−4.28726	−0.176954	−0.0884770	+	0.996078i	$0.528200\pi$
$588$	0	0
$589$	1.38058	0.0568858
$590$	0	0
$591$	5.84367	0.240377
$592$	0	0
$593$	13.0132	0.534387	0.267193	−	0.963643i	$-0.413904\pi$
$593$	13.0132	0.534387	0.267193	+	0.963643i	$0.413904\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21.7889	0.891761
$598$	0	0
$599$	−10.2677	−0.419529	−0.209764	−	0.977752i	$-0.567270\pi$
$599$	−10.2677	−0.419529	−0.209764	+	0.977752i	$0.567270\pi$
$600$	0	0
$601$	46.9814	1.91641	0.958206	−	0.286077i	$-0.0923515\pi$
$601$	46.9814	1.91641	0.958206	+	0.286077i	$0.0923515\pi$
$602$	0	0
$603$	−1.06300	−0.0432889
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−33.3503	−1.35365	−0.676823	−	0.736146i	$-0.736644\pi$
$607$	−33.3503	−1.35365	−0.676823	+	0.736146i	$0.736644\pi$
$608$	0	0
$609$	7.40597	0.300105
$610$	0	0
$611$	2.13093	0.0862083
$612$	0	0
$613$	−18.1016	−0.731116	−0.365558	−	0.930789i	$-0.619122\pi$
$613$	−18.1016	−0.731116	−0.365558	+	0.930789i	$0.619122\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.27504	0.292882	0.146441	−	0.989219i	$-0.453218\pi$
$617$	7.27504	0.292882	0.146441	+	0.989219i	$0.453218\pi$
$618$	0	0
$619$	−22.7064	−0.912647	−0.456324	−	0.889814i	$-0.650834\pi$
$619$	−22.7064	−0.912647	−0.456324	+	0.889814i	$0.650834\pi$
$620$	0	0
$621$	−8.73084	−0.350357
$622$	0	0
$623$	22.3831	0.896761
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.544198	0.0217332
$628$	0	0
$629$	−39.2104	−1.56342
$630$	0	0
$631$	−12.9927	−0.517231	−0.258616	−	0.965980i	$-0.583266\pi$
$631$	−12.9927	−0.517231	−0.258616	+	0.965980i	$0.583266\pi$
$632$	0	0
$633$	21.3561	0.848830
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.50659	0.138936
$638$	0	0
$639$	−13.4314	−0.531337
$640$	0	0
$641$	−11.3127	−0.446823	−0.223411	−	0.974724i	$-0.571719\pi$
$641$	−11.3127	−0.446823	−0.223411	+	0.974724i	$0.571719\pi$
$642$	0	0
$643$	−24.0303	−0.947663	−0.473832	−	0.880615i	$-0.657129\pi$
$643$	−24.0303	−0.947663	−0.473832	+	0.880615i	$0.657129\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.6009	0.652647	0.326324	−	0.945258i	$-0.394190\pi$
$647$	16.6009	0.652647	0.326324	+	0.945258i	$0.394190\pi$
$648$	0	0
$649$	9.78560	0.384118
$650$	0	0
$651$	3.20123	0.125466
$652$	0	0
$653$	33.6107	1.31529	0.657644	−	0.753329i	$-0.271553\pi$
$653$	33.6107	1.31529	0.657644	+	0.753329i	$0.271553\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.34297	0.325490
$658$	0	0
$659$	−4.14174	−0.161339	−0.0806696	−	0.996741i	$-0.525706\pi$
$659$	−4.14174	−0.161339	−0.0806696	+	0.996741i	$0.525706\pi$
$660$	0	0
$661$	−18.2579	−0.710150	−0.355075	−	0.934838i	$-0.615545\pi$
$661$	−18.2579	−0.710150	−0.355075	+	0.934838i	$0.615545\pi$
$662$	0	0
$663$	−5.31265	−0.206326
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−34.5950	−1.33952
$668$	0	0
$669$	7.96968	0.308126
$670$	0	0
$671$	−6.98778	−0.269760
$672$	0	0
$673$	−47.4953	−1.83081	−0.915405	−	0.402534i	$-0.868130\pi$
$673$	−47.4953	−1.83081	−0.915405	+	0.402534i	$0.868130\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19.4617	−0.747973	−0.373987	−	0.927434i	$-0.622009\pi$
$677$	−19.4617	−0.747973	−0.373987	+	0.927434i	$0.622009\pi$
$678$	0	0
$679$	−0.489444	−0.0187831
$680$	0	0
$681$	3.13823	0.120257
$682$	0	0
$683$	7.61308	0.291306	0.145653	−	0.989336i	$-0.453472\pi$
$683$	7.61308	0.291306	0.145653	+	0.989336i	$0.453472\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.84955	0.375784
$688$	0	0
$689$	−6.19394	−0.235970
$690$	0	0
$691$	−12.7807	−0.486200	−0.243100	−	0.970001i	$-0.578164\pi$
$691$	−12.7807	−0.486200	−0.243100	+	0.970001i	$0.578164\pi$
$692$	0	0
$693$	1.26187	0.0479343
$694$	0	0
$695$	0	0
$696$	0	0
$697$	45.3923	1.71936
$698$	0	0
$699$	−20.8265	−0.787732
$700$	0	0
$701$	−4.56134	−0.172279	−0.0861397	−	0.996283i	$-0.527453\pi$
$701$	−4.56134	−0.172279	−0.0861397	+	0.996283i	$0.527453\pi$
$702$	0	0
$703$	−5.94921	−0.224379
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−31.2008	−1.17343
$708$	0	0
$709$	39.8094	1.49507	0.747536	−	0.664221i	$-0.231237\pi$
$709$	39.8094	1.49507	0.747536	+	0.664221i	$0.231237\pi$
$710$	0	0
$711$	−6.93207	−0.259973
$712$	0	0
$713$	−14.9537	−0.560020
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.34534	0.274317
$718$	0	0
$719$	−7.47039	−0.278599	−0.139299	−	0.990250i	$-0.544485\pi$
$719$	−7.47039	−0.278599	−0.139299	+	0.990250i	$0.544485\pi$
$720$	0	0
$721$	−23.3054	−0.867937
$722$	0	0
$723$	11.2692	0.419105
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−37.4967	−1.39068	−0.695339	−	0.718682i	$-0.744746\pi$
$727$	−37.4967	−1.39068	−0.695339	+	0.718682i	$0.744746\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	55.6180	2.05711
$732$	0	0
$733$	−7.83780	−0.289496	−0.144748	−	0.989469i	$-0.546237\pi$
$733$	−7.83780	−0.289496	−0.144748	+	0.989469i	$0.546237\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.717667	−0.0264356
$738$	0	0
$739$	18.3282	0.674214	0.337107	−	0.941466i	$-0.390552\pi$
$739$	18.3282	0.674214	0.337107	+	0.941466i	$0.390552\pi$
$740$	0	0
$741$	−0.806063	−0.0296115
$742$	0	0
$743$	41.7899	1.53312	0.766561	−	0.642172i	$-0.221966\pi$
$743$	41.7899	1.53312	0.766561	+	0.642172i	$0.221966\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−10.5696	−0.386721
$748$	0	0
$749$	19.0395	0.695689
$750$	0	0
$751$	37.6688	1.37455	0.687277	−	0.726396i	$-0.258806\pi$
$751$	37.6688	1.37455	0.687277	+	0.726396i	$0.258806\pi$
$752$	0	0
$753$	29.0943	1.06025
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.7480	0.390642	0.195321	−	0.980739i	$-0.437425\pi$
$757$	10.7480	0.390642	0.195321	+	0.980739i	$0.437425\pi$
$758$	0	0
$759$	−5.89446	−0.213955
$760$	0	0
$761$	−42.9829	−1.55813	−0.779064	−	0.626945i	$-0.784305\pi$
$761$	−42.9829	−1.55813	−0.779064	+	0.626945i	$0.784305\pi$
$762$	0	0
$763$	29.8211	1.07960
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.4944	−0.523361
$768$	0	0
$769$	4.71037	0.169860	0.0849302	−	0.996387i	$-0.472933\pi$
$769$	4.71037	0.169860	0.0849302	+	0.996387i	$0.472933\pi$
$770$	0	0
$771$	−29.5198	−1.06313
$772$	0	0
$773$	30.9624	1.11364	0.556820	−	0.830633i	$-0.312021\pi$
$773$	30.9624	1.11364	0.556820	+	0.830633i	$0.312021\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−13.7948	−0.494886
$778$	0	0
$779$	6.88717	0.246758
$780$	0	0
$781$	−9.06793	−0.324476
$782$	0	0
$783$	3.96239	0.141604
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.7377	1.30956	0.654778	−	0.755821i	$-0.272762\pi$
$787$	36.7377	1.30956	0.654778	+	0.755821i	$0.272762\pi$
$788$	0	0
$789$	−17.7685	−0.632574
$790$	0	0
$791$	18.9262	0.672938
$792$	0	0
$793$	10.3503	0.367549
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.3488	−0.543684	−0.271842	−	0.962342i	$-0.587633\pi$
$797$	−15.3488	−0.543684	−0.271842	+	0.962342i	$0.587633\pi$
$798$	0	0
$799$	11.3209	0.400505
$800$	0	0
$801$	11.9756	0.423136
$802$	0	0
$803$	5.63259	0.198770
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.38787	0.0488554
$808$	0	0
$809$	−8.88717	−0.312456	−0.156228	−	0.987721i	$-0.549933\pi$
$809$	−8.88717	−0.312456	−0.156228	+	0.987721i	$0.549933\pi$
$810$	0	0
$811$	−18.4582	−0.648154	−0.324077	−	0.946031i	$-0.605054\pi$
$811$	−18.4582	−0.648154	−0.324077	+	0.946031i	$0.605054\pi$
$812$	0	0
$813$	−14.4191	−0.505702
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.43866	0.295231
$818$	0	0
$819$	−1.86907	−0.0653105
$820$	0	0
$821$	39.3923	1.37480	0.687401	−	0.726278i	$-0.258752\pi$
$821$	39.3923	1.37480	0.687401	+	0.726278i	$0.258752\pi$
$822$	0	0
$823$	24.8726	0.867004	0.433502	−	0.901153i	$-0.357278\pi$
$823$	24.8726	0.867004	0.433502	+	0.901153i	$0.357278\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.4495	0.711097	0.355549	−	0.934658i	$-0.384294\pi$
$827$	20.4495	0.711097	0.355549	+	0.934658i	$0.384294\pi$
$828$	0	0
$829$	32.2144	1.11885	0.559426	−	0.828880i	$-0.311022\pi$
$829$	32.2144	1.11885	0.559426	+	0.828880i	$0.311022\pi$
$830$	0	0
$831$	−19.0640	−0.661321
$832$	0	0
$833$	18.6293	0.645466
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.71274	0.0592010
$838$	0	0
$839$	41.0289	1.41648	0.708238	−	0.705974i	$-0.249491\pi$
$839$	41.0289	1.41648	0.708238	+	0.705974i	$0.249491\pi$
$840$	0	0
$841$	−13.2995	−0.458603
$842$	0	0
$843$	27.4617	0.945831
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−19.7078	−0.677169
$848$	0	0
$849$	−13.7889	−0.473235
$850$	0	0
$851$	64.4387	2.20893
$852$	0	0
$853$	−23.5975	−0.807964	−0.403982	−	0.914767i	$-0.632374\pi$
$853$	−23.5975	−0.807964	−0.403982	+	0.914767i	$0.632374\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.4010	0.936002	0.468001	−	0.883728i	$-0.344974\pi$
$857$	27.4010	0.936002	0.468001	+	0.883728i	$0.344974\pi$
$858$	0	0
$859$	−2.99271	−0.102110	−0.0510549	−	0.998696i	$-0.516258\pi$
$859$	−2.99271	−0.102110	−0.0510549	+	0.998696i	$0.516258\pi$
$860$	0	0
$861$	15.9697	0.544245
$862$	0	0
$863$	−24.7767	−0.843409	−0.421704	−	0.906733i	$-0.638568\pi$
$863$	−24.7767	−0.843409	−0.421704	+	0.906733i	$0.638568\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−11.2243	−0.381196
$868$	0	0
$869$	−4.68006	−0.158760
$870$	0	0
$871$	1.06300	0.0360185
$872$	0	0
$873$	−0.261865	−0.00886279
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.71179	0.260409	0.130204	−	0.991487i	$-0.458437\pi$
$877$	7.71179	0.260409	0.130204	+	0.991487i	$0.458437\pi$
$878$	0	0
$879$	−21.4763	−0.724377
$880$	0	0
$881$	24.9405	0.840267	0.420133	−	0.907462i	$-0.361983\pi$
$881$	24.9405	0.840267	0.420133	+	0.907462i	$0.361983\pi$
$882$	0	0
$883$	−12.6907	−0.427075	−0.213538	−	0.976935i	$-0.568499\pi$
$883$	−12.6907	−0.427075	−0.213538	+	0.976935i	$0.568499\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.99412	0.335570	0.167785	−	0.985824i	$-0.446339\pi$
$887$	9.99412	0.335570	0.167785	+	0.985824i	$0.446339\pi$
$888$	0	0
$889$	−29.8211	−1.00017
$890$	0	0
$891$	0.675131	0.0226177
$892$	0	0
$893$	1.71767	0.0574795
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.73084	0.291514
$898$	0	0
$899$	6.78655	0.226344
$900$	0	0
$901$	−32.9062	−1.09627
$902$	0	0
$903$	19.5672	0.651156
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−15.5975	−0.517908	−0.258954	−	0.965890i	$-0.583378\pi$
$907$	−15.5975	−0.517908	−0.258954	+	0.965890i	$0.583378\pi$
$908$	0	0
$909$	−16.6932	−0.553679
$910$	0	0
$911$	6.43278	0.213127	0.106564	−	0.994306i	$-0.466015\pi$
$911$	6.43278	0.213127	0.106564	+	0.994306i	$0.466015\pi$
$912$	0	0
$913$	−7.13586	−0.236162
$914$	0	0
$915$	0	0
$916$	0	0
$917$	33.1648	1.09520
$918$	0	0
$919$	30.2276	0.997116	0.498558	−	0.866856i	$-0.333863\pi$
$919$	30.2276	0.997116	0.498558	+	0.866856i	$0.333863\pi$
$920$	0	0
$921$	17.7685	0.585490
$922$	0	0
$923$	13.4314	0.442099
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−12.4690	−0.409535
$928$	0	0
$929$	50.7875	1.66628	0.833142	−	0.553059i	$-0.186540\pi$
$929$	50.7875	1.66628	0.833142	+	0.553059i	$0.186540\pi$
$930$	0	0
$931$	2.82653	0.0926358
$932$	0	0
$933$	1.16362	0.0380952
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.94921	0.0963466	0.0481733	−	0.998839i	$-0.484660\pi$
$937$	2.94921	0.0963466	0.0481733	+	0.998839i	$0.484660\pi$
$938$	0	0
$939$	25.2677	0.824582
$940$	0	0
$941$	0.432779	0.0141082	0.00705409	−	0.999975i	$-0.497755\pi$
$941$	0.432779	0.0141082	0.00705409	+	0.999975i	$0.497755\pi$
$942$	0	0
$943$	−74.5980	−2.42925
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35.5524	1.15530	0.577650	−	0.816285i	$-0.303970\pi$
$947$	35.5524	1.15530	0.577650	+	0.816285i	$0.303970\pi$
$948$	0	0
$949$	−8.34297	−0.270824
$950$	0	0
$951$	−23.8192	−0.772392
$952$	0	0
$953$	−23.9781	−0.776727	−0.388364	−	0.921506i	$-0.626960\pi$
$953$	−23.9781	−0.776727	−0.388364	+	0.921506i	$0.626960\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.67513	0.0864747
$958$	0	0
$959$	−16.7513	−0.540928
$960$	0	0
$961$	−28.0665	−0.905371
$962$	0	0
$963$	10.1866	0.328260
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−26.8472	−0.863347	−0.431674	−	0.902030i	$-0.642077\pi$
$967$	−26.8472	−0.863347	−0.431674	+	0.902030i	$0.642077\pi$
$968$	0	0
$969$	−4.28233	−0.137568
$970$	0	0
$971$	−20.8265	−0.668355	−0.334178	−	0.942510i	$-0.608459\pi$
$971$	−20.8265	−0.668355	−0.334178	+	0.942510i	$0.608459\pi$
$972$	0	0
$973$	0.140596	0.00450732
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46.9643	1.50252	0.751261	−	0.660006i	$-0.229446\pi$
$977$	46.9643	1.50252	0.751261	+	0.660006i	$0.229446\pi$
$978$	0	0
$979$	8.08507	0.258400
$980$	0	0
$981$	15.9551	0.509407
$982$	0	0
$983$	−32.8472	−1.04766	−0.523831	−	0.851822i	$-0.675498\pi$
$983$	−32.8472	−1.04766	−0.523831	+	0.851822i	$0.675498\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.98286	0.126776
$988$	0	0
$989$	−91.4030	−2.90644
$990$	0	0
$991$	0.820652	0.0260689	0.0130344	−	0.999915i	$-0.495851\pi$
$991$	0.820652	0.0260689	0.0130344	+	0.999915i	$0.495851\pi$
$992$	0	0
$993$	−21.7440	−0.690025
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−12.5125	−0.396274	−0.198137	−	0.980174i	$-0.563489\pi$
$997$	−12.5125	−0.396274	−0.198137	+	0.980174i	$0.563489\pi$
$998$	0	0
$999$	−7.38058	−0.233511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bl.1.2	✓	3
5.4	even	2		7800.2.a.bo.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bl.1.2	✓	3	1.1	even	1	trivial
7800.2.a.bo.1.2	yes	3	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.86907 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.86907 q^{7} +1.00000 q^{9} +0.675131 q^{11} -1.00000 q^{13} -5.31265 q^{17} -0.806063 q^{19} -1.86907 q^{21} +8.73084 q^{23} -1.00000 q^{27} -3.96239 q^{29} -1.71274 q^{31} -0.675131 q^{33} +7.38058 q^{37} +1.00000 q^{39} -8.54420 q^{41} -10.4690 q^{43} -2.13093 q^{47} -3.50659 q^{49} +5.31265 q^{51} +6.19394 q^{53} +0.806063 q^{57} +14.4944 q^{59} -10.3503 q^{61} +1.86907 q^{63} -1.06300 q^{67} -8.73084 q^{69} -13.4314 q^{71} +8.34297 q^{73} +1.26187 q^{77} -6.93207 q^{79} +1.00000 q^{81} -10.5696 q^{83} +3.96239 q^{87} +11.9756 q^{89} -1.86907 q^{91} +1.71274 q^{93} -0.261865 q^{97} +0.675131 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + q^7 + 3 * q^9	\( 3 q - 3 q^{3} + q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} + 5 q^{17} - 2 q^{19} - q^{21} + 4 q^{23} - 3 q^{27} - q^{29} - 11 q^{31} + 3 q^{33} + 10 q^{37} + 3 q^{39} - 16 q^{41} - 11 q^{47} + 10 q^{49} - 5 q^{51} + 19 q^{53} + 2 q^{57} - 3 q^{59} - 21 q^{61} + q^{63} + q^{67} - 4 q^{69} + 2 q^{71} + 2 q^{73} + 13 q^{77} - 12 q^{79} + 3 q^{81} - 7 q^{83} + q^{87} - 16 q^{89} - q^{91} + 11 q^{93} - 10 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + q^7 + 3 * q^9 - 3 * q^11 - 3 * q^13 + 5 * q^17 - 2 * q^19 - q^21 + 4 * q^23 - 3 * q^27 - q^29 - 11 * q^31 + 3 * q^33 + 10 * q^37 + 3 * q^39 - 16 * q^41 - 11 * q^47 + 10 * q^49 - 5 * q^51 + 19 * q^53 + 2 * q^57 - 3 * q^59 - 21 * q^61 + q^63 + q^67 - 4 * q^69 + 2 * q^71 + 2 * q^73 + 13 * q^77 - 12 * q^79 + 3 * q^81 - 7 * q^83 + q^87 - 16 * q^89 - q^91 + 11 * q^93 - 10 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more