LMFDB - Embedded newform 8040.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8040,2,Mod(1,8040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8040, 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("8040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8040.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:	$64.1997232251$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 37x^{8} + 132x^{7} + 358x^{6} - 1708x^{5} - 92x^{4} + 5969x^{3} - 3864x^{2} - 4752x + 3524 $ x^10 - 3x^9 - 37x^8 + 132x^7 + 358x^6 - 1708x^5 - 92x^4 + 5969x^3 - 3864x^2 - 4752*x + 3524
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.93365$ of defining polynomial
Character	$\chi$	$=$	8040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -4.46032 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -4.46032 q^{7} +1.00000 q^{9} -5.90680 q^{11} -2.55082 q^{13} -1.00000 q^{15} -2.72964 q^{17} +3.14020 q^{19} +4.46032 q^{21} -5.52537 q^{23} +1.00000 q^{25} -1.00000 q^{27} -0.163867 q^{29} -9.75109 q^{31} +5.90680 q^{33} -4.46032 q^{35} -4.56432 q^{37} +2.55082 q^{39} +1.87001 q^{41} -2.98660 q^{43} +1.00000 q^{45} -0.397720 q^{47} +12.8945 q^{49} +2.72964 q^{51} -6.93339 q^{53} -5.90680 q^{55} -3.14020 q^{57} +10.2268 q^{59} -9.41579 q^{61} -4.46032 q^{63} -2.55082 q^{65} -1.00000 q^{67} +5.52537 q^{69} +0.360961 q^{71} -9.38969 q^{73} -1.00000 q^{75} +26.3462 q^{77} -16.5449 q^{79} +1.00000 q^{81} -16.1988 q^{83} -2.72964 q^{85} +0.163867 q^{87} +7.61288 q^{89} +11.3775 q^{91} +9.75109 q^{93} +3.14020 q^{95} +16.4941 q^{97} -5.90680 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9	$ 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9} + 7 q^{11} - q^{13} - 10 q^{15} + 3 q^{17} + 4 q^{19} - q^{21} - 13 q^{23} + 10 q^{25} - 10 q^{27} + 18 q^{29} - 9 q^{31} - 7 q^{33} + q^{35} - 3 q^{37} + q^{39} + 19 q^{41} + 5 q^{43} + 10 q^{45} - 8 q^{47} + 43 q^{49} - 3 q^{51} + 17 q^{53} + 7 q^{55} - 4 q^{57} + 24 q^{59} + 21 q^{61} + q^{63} - q^{65} - 10 q^{67} + 13 q^{69} + 2 q^{71} + 25 q^{73} - 10 q^{75} + 15 q^{77} - q^{79} + 10 q^{81} - 6 q^{83} + 3 q^{85} - 18 q^{87} + 23 q^{89} + 29 q^{91} + 9 q^{93} + 4 q^{95} + 21 q^{97} + 7 q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9 + 7 * q^11 - q^13 - 10 * q^15 + 3 * q^17 + 4 * q^19 - q^21 - 13 * q^23 + 10 * q^25 - 10 * q^27 + 18 * q^29 - 9 * q^31 - 7 * q^33 + q^35 - 3 * q^37 + q^39 + 19 * q^41 + 5 * q^43 + 10 * q^45 - 8 * q^47 + 43 * q^49 - 3 * q^51 + 17 * q^53 + 7 * q^55 - 4 * q^57 + 24 * q^59 + 21 * q^61 + q^63 - q^65 - 10 * q^67 + 13 * q^69 + 2 * q^71 + 25 * q^73 - 10 * q^75 + 15 * q^77 - q^79 + 10 * q^81 - 6 * q^83 + 3 * q^85 - 18 * q^87 + 23 * q^89 + 29 * q^91 + 9 * q^93 + 4 * q^95 + 21 * q^97 + 7 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.46032	−1.68584	−0.842922	−	0.538036i	$-0.819166\pi$
$7$	−4.46032	−1.68584	−0.842922	+	0.538036i	$0.819166\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.90680	−1.78097	−0.890484	−	0.455015i	$-0.849634\pi$
$11$	−5.90680	−1.78097	−0.890484	+	0.455015i	$0.849634\pi$
$12$	0	0
$13$	−2.55082	−0.707470	−0.353735	−	0.935346i	$-0.615089\pi$
$13$	−2.55082	−0.707470	−0.353735	+	0.935346i	$0.615089\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−2.72964	−0.662036	−0.331018	−	0.943625i	$-0.607392\pi$
$17$	−2.72964	−0.662036	−0.331018	+	0.943625i	$0.607392\pi$
$18$	0	0
$19$	3.14020	0.720412	0.360206	−	0.932873i	$-0.382706\pi$
$19$	3.14020	0.720412	0.360206	+	0.932873i	$0.382706\pi$
$20$	0	0
$21$	4.46032	0.973322
$22$	0	0
$23$	−5.52537	−1.15212	−0.576060	−	0.817408i	$-0.695410\pi$
$23$	−5.52537	−1.15212	−0.576060	+	0.817408i	$0.695410\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.163867	−0.0304294	−0.0152147	−	0.999884i	$-0.504843\pi$
$29$	−0.163867	−0.0304294	−0.0152147	+	0.999884i	$0.504843\pi$
$30$	0	0
$31$	−9.75109	−1.75135	−0.875674	−	0.482904i	$-0.839582\pi$
$31$	−9.75109	−1.75135	−0.875674	+	0.482904i	$0.839582\pi$
$32$	0	0
$33$	5.90680	1.02824
$34$	0	0
$35$	−4.46032	−0.753932
$36$	0	0
$37$	−4.56432	−0.750370	−0.375185	−	0.926950i	$-0.622421\pi$
$37$	−4.56432	−0.750370	−0.375185	+	0.926950i	$0.622421\pi$
$38$	0	0
$39$	2.55082	0.408458
$40$	0	0
$41$	1.87001	0.292046	0.146023	−	0.989281i	$-0.453353\pi$
$41$	1.87001	0.292046	0.146023	+	0.989281i	$0.453353\pi$
$42$	0	0
$43$	−2.98660	−0.455453	−0.227726	−	0.973725i	$-0.573129\pi$
$43$	−2.98660	−0.455453	−0.227726	+	0.973725i	$0.573129\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−0.397720	−0.0580134	−0.0290067	−	0.999579i	$-0.509234\pi$
$47$	−0.397720	−0.0580134	−0.0290067	+	0.999579i	$0.509234\pi$
$48$	0	0
$49$	12.8945	1.84207
$50$	0	0
$51$	2.72964	0.382226
$52$	0	0
$53$	−6.93339	−0.952375	−0.476188	−	0.879344i	$-0.657982\pi$
$53$	−6.93339	−0.952375	−0.476188	+	0.879344i	$0.657982\pi$
$54$	0	0
$55$	−5.90680	−0.796473
$56$	0	0
$57$	−3.14020	−0.415930
$58$	0	0
$59$	10.2268	1.33142	0.665709	−	0.746212i	$-0.268129\pi$
$59$	10.2268	1.33142	0.665709	+	0.746212i	$0.268129\pi$
$60$	0	0
$61$	−9.41579	−1.20557	−0.602784	−	0.797904i	$-0.705942\pi$
$61$	−9.41579	−1.20557	−0.602784	+	0.797904i	$0.705942\pi$
$62$	0	0
$63$	−4.46032	−0.561948
$64$	0	0
$65$	−2.55082	−0.316390
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	5.52537	0.665177
$70$	0	0
$71$	0.360961	0.0428382	0.0214191	−	0.999771i	$-0.493182\pi$
$71$	0.360961	0.0428382	0.0214191	+	0.999771i	$0.493182\pi$
$72$	0	0
$73$	−9.38969	−1.09898	−0.549490	−	0.835500i	$-0.685178\pi$
$73$	−9.38969	−1.09898	−0.549490	+	0.835500i	$0.685178\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	26.3462	3.00243
$78$	0	0
$79$	−16.5449	−1.86145	−0.930725	−	0.365720i	$-0.880823\pi$
$79$	−16.5449	−1.86145	−0.930725	+	0.365720i	$0.880823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.1988	−1.77805	−0.889026	−	0.457857i	$-0.848617\pi$
$83$	−16.1988	−1.77805	−0.889026	+	0.457857i	$0.848617\pi$
$84$	0	0
$85$	−2.72964	−0.296071
$86$	0	0
$87$	0.163867	0.0175684
$88$	0	0
$89$	7.61288	0.806964	0.403482	−	0.914988i	$-0.367800\pi$
$89$	7.61288	0.806964	0.403482	+	0.914988i	$0.367800\pi$
$90$	0	0
$91$	11.3775	1.19268
$92$	0	0
$93$	9.75109	1.01114
$94$	0	0
$95$	3.14020	0.322178
$96$	0	0
$97$	16.4941	1.67472	0.837362	−	0.546648i	$-0.184096\pi$
$97$	16.4941	1.67472	0.837362	+	0.546648i	$0.184096\pi$
$98$	0	0
$99$	−5.90680	−0.593656
$100$	0	0
$101$	−10.4321	−1.03803	−0.519017	−	0.854764i	$-0.673702\pi$
$101$	−10.4321	−1.03803	−0.519017	+	0.854764i	$0.673702\pi$
$102$	0	0
$103$	5.17657	0.510062	0.255031	−	0.966933i	$-0.417914\pi$
$103$	5.17657	0.510062	0.255031	+	0.966933i	$0.417914\pi$
$104$	0	0
$105$	4.46032	0.435283
$106$	0	0
$107$	−2.14339	−0.207209	−0.103604	−	0.994619i	$-0.533038\pi$
$107$	−2.14339	−0.207209	−0.103604	+	0.994619i	$0.533038\pi$
$108$	0	0
$109$	−0.242099	−0.0231889	−0.0115945	−	0.999933i	$-0.503691\pi$
$109$	−0.242099	−0.0231889	−0.0115945	+	0.999933i	$0.503691\pi$
$110$	0	0
$111$	4.56432	0.433226
$112$	0	0
$113$	−2.80474	−0.263847	−0.131924	−	0.991260i	$-0.542115\pi$
$113$	−2.80474	−0.263847	−0.131924	+	0.991260i	$0.542115\pi$
$114$	0	0
$115$	−5.52537	−0.515244
$116$	0	0
$117$	−2.55082	−0.235823
$118$	0	0
$119$	12.1751	1.11609
$120$	0	0
$121$	23.8903	2.17184
$122$	0	0
$123$	−1.87001	−0.168613
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.37083	−0.210377	−0.105188	−	0.994452i	$-0.533545\pi$
$127$	−2.37083	−0.210377	−0.105188	+	0.994452i	$0.533545\pi$
$128$	0	0
$129$	2.98660	0.262956
$130$	0	0
$131$	2.98555	0.260849	0.130424	−	0.991458i	$-0.458366\pi$
$131$	2.98555	0.260849	0.130424	+	0.991458i	$0.458366\pi$
$132$	0	0
$133$	−14.0063	−1.21450
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	1.55037	0.132457	0.0662285	−	0.997804i	$-0.478903\pi$
$137$	1.55037	0.132457	0.0662285	+	0.997804i	$0.478903\pi$
$138$	0	0
$139$	2.21619	0.187975	0.0939876	−	0.995573i	$-0.470039\pi$
$139$	2.21619	0.187975	0.0939876	+	0.995573i	$0.470039\pi$
$140$	0	0
$141$	0.397720	0.0334940
$142$	0	0
$143$	15.0672	1.25998
$144$	0	0
$145$	−0.163867	−0.0136084
$146$	0	0
$147$	−12.8945	−1.06352
$148$	0	0
$149$	−0.349900	−0.0286650	−0.0143325	−	0.999897i	$-0.504562\pi$
$149$	−0.349900	−0.0286650	−0.0143325	+	0.999897i	$0.504562\pi$
$150$	0	0
$151$	11.8950	0.968003	0.484002	−	0.875067i	$-0.339183\pi$
$151$	11.8950	0.968003	0.484002	+	0.875067i	$0.339183\pi$
$152$	0	0
$153$	−2.72964	−0.220679
$154$	0	0
$155$	−9.75109	−0.783226
$156$	0	0
$157$	12.1063	0.966190	0.483095	−	0.875568i	$-0.339513\pi$
$157$	12.1063	0.966190	0.483095	+	0.875568i	$0.339513\pi$
$158$	0	0
$159$	6.93339	0.549854
$160$	0	0
$161$	24.6449	1.94229
$162$	0	0
$163$	7.16556	0.561250	0.280625	−	0.959818i	$-0.409458\pi$
$163$	7.16556	0.561250	0.280625	+	0.959818i	$0.409458\pi$
$164$	0	0
$165$	5.90680	0.459844
$166$	0	0
$167$	−22.8595	−1.76892	−0.884459	−	0.466618i	$-0.845472\pi$
$167$	−22.8595	−1.76892	−0.884459	+	0.466618i	$0.845472\pi$
$168$	0	0
$169$	−6.49332	−0.499486
$170$	0	0
$171$	3.14020	0.240137
$172$	0	0
$173$	−17.0540	−1.29659	−0.648295	−	0.761389i	$-0.724518\pi$
$173$	−17.0540	−1.29659	−0.648295	+	0.761389i	$0.724518\pi$
$174$	0	0
$175$	−4.46032	−0.337169
$176$	0	0
$177$	−10.2268	−0.768694
$178$	0	0
$179$	18.5315	1.38511	0.692555	−	0.721365i	$-0.256485\pi$
$179$	18.5315	1.38511	0.692555	+	0.721365i	$0.256485\pi$
$180$	0	0
$181$	7.57607	0.563125	0.281562	−	0.959543i	$-0.409147\pi$
$181$	7.57607	0.563125	0.281562	+	0.959543i	$0.409147\pi$
$182$	0	0
$183$	9.41579	0.696035
$184$	0	0
$185$	−4.56432	−0.335576
$186$	0	0
$187$	16.1235	1.17906
$188$	0	0
$189$	4.46032	0.324441
$190$	0	0
$191$	−0.267644	−0.0193660	−0.00968301	−	0.999953i	$-0.503082\pi$
$191$	−0.267644	−0.0193660	−0.00968301	+	0.999953i	$0.503082\pi$
$192$	0	0
$193$	8.27309	0.595510	0.297755	−	0.954642i	$-0.403762\pi$
$193$	8.27309	0.595510	0.297755	+	0.954642i	$0.403762\pi$
$194$	0	0
$195$	2.55082	0.182668
$196$	0	0
$197$	14.8969	1.06136	0.530681	−	0.847572i	$-0.321936\pi$
$197$	14.8969	1.06136	0.530681	+	0.847572i	$0.321936\pi$
$198$	0	0
$199$	−14.5198	−1.02928	−0.514641	−	0.857406i	$-0.672075\pi$
$199$	−14.5198	−1.02928	−0.514641	+	0.857406i	$0.672075\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	0.730902	0.0512992
$204$	0	0
$205$	1.87001	0.130607
$206$	0	0
$207$	−5.52537	−0.384040
$208$	0	0
$209$	−18.5485	−1.28303
$210$	0	0
$211$	9.63978	0.663630	0.331815	−	0.943345i	$-0.392339\pi$
$211$	9.63978	0.663630	0.331815	+	0.943345i	$0.392339\pi$
$212$	0	0
$213$	−0.360961	−0.0247326
$214$	0	0
$215$	−2.98660	−0.203685
$216$	0	0
$217$	43.4930	2.95250
$218$	0	0
$219$	9.38969	0.634497
$220$	0	0
$221$	6.96283	0.468370
$222$	0	0
$223$	−0.0257084	−0.00172156	−0.000860780	−	1.00000i	$-0.500274\pi$
$223$	−0.0257084	−0.00172156	−0.000860780		1.00000i	$0.500274\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−7.64782	−0.507604	−0.253802	−	0.967256i	$-0.581681\pi$
$227$	−7.64782	−0.507604	−0.253802	+	0.967256i	$0.581681\pi$
$228$	0	0
$229$	3.55912	0.235193	0.117597	−	0.993061i	$-0.462481\pi$
$229$	3.55912	0.235193	0.117597	+	0.993061i	$0.462481\pi$
$230$	0	0
$231$	−26.3462	−1.73346
$232$	0	0
$233$	11.5075	0.753881	0.376940	−	0.926238i	$-0.376976\pi$
$233$	11.5075	0.753881	0.376940	+	0.926238i	$0.376976\pi$
$234$	0	0
$235$	−0.397720	−0.0259444
$236$	0	0
$237$	16.5449	1.07471
$238$	0	0
$239$	6.65835	0.430693	0.215346	−	0.976538i	$-0.430912\pi$
$239$	6.65835	0.430693	0.215346	+	0.976538i	$0.430912\pi$
$240$	0	0
$241$	6.84364	0.440838	0.220419	−	0.975405i	$-0.429258\pi$
$241$	6.84364	0.440838	0.220419	+	0.975405i	$0.429258\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	12.8945	0.823799
$246$	0	0
$247$	−8.01009	−0.509670
$248$	0	0
$249$	16.1988	1.02656
$250$	0	0
$251$	−10.7248	−0.676946	−0.338473	−	0.940976i	$-0.609910\pi$
$251$	−10.7248	−0.676946	−0.338473	+	0.940976i	$0.609910\pi$
$252$	0	0
$253$	32.6373	2.05189
$254$	0	0
$255$	2.72964	0.170937
$256$	0	0
$257$	−3.40789	−0.212578	−0.106289	−	0.994335i	$-0.533897\pi$
$257$	−3.40789	−0.212578	−0.106289	+	0.994335i	$0.533897\pi$
$258$	0	0
$259$	20.3584	1.26501
$260$	0	0
$261$	−0.163867	−0.0101431
$262$	0	0
$263$	22.5160	1.38840	0.694199	−	0.719783i	$-0.255759\pi$
$263$	22.5160	1.38840	0.694199	+	0.719783i	$0.255759\pi$
$264$	0	0
$265$	−6.93339	−0.425915
$266$	0	0
$267$	−7.61288	−0.465901
$268$	0	0
$269$	6.96094	0.424416	0.212208	−	0.977224i	$-0.431935\pi$
$269$	6.96094	0.424416	0.212208	+	0.977224i	$0.431935\pi$
$270$	0	0
$271$	1.98173	0.120381	0.0601907	−	0.998187i	$-0.480829\pi$
$271$	1.98173	0.120381	0.0601907	+	0.998187i	$0.480829\pi$
$272$	0	0
$273$	−11.3775	−0.688597
$274$	0	0
$275$	−5.90680	−0.356193
$276$	0	0
$277$	−10.1378	−0.609119	−0.304559	−	0.952493i	$-0.598509\pi$
$277$	−10.1378	−0.609119	−0.304559	+	0.952493i	$0.598509\pi$
$278$	0	0
$279$	−9.75109	−0.583782
$280$	0	0
$281$	−14.5574	−0.868424	−0.434212	−	0.900811i	$-0.642973\pi$
$281$	−14.5574	−0.868424	−0.434212	+	0.900811i	$0.642973\pi$
$282$	0	0
$283$	13.3939	0.796183	0.398091	−	0.917346i	$-0.369673\pi$
$283$	13.3939	0.796183	0.398091	+	0.917346i	$0.369673\pi$
$284$	0	0
$285$	−3.14020	−0.186009
$286$	0	0
$287$	−8.34085	−0.492345
$288$	0	0
$289$	−9.54905	−0.561709
$290$	0	0
$291$	−16.4941	−0.966903
$292$	0	0
$293$	23.6966	1.38437	0.692184	−	0.721721i	$-0.256648\pi$
$293$	23.6966	1.38437	0.692184	+	0.721721i	$0.256648\pi$
$294$	0	0
$295$	10.2268	0.595428
$296$	0	0
$297$	5.90680	0.342747
$298$	0	0
$299$	14.0942	0.815090
$300$	0	0
$301$	13.3212	0.767823
$302$	0	0
$303$	10.4321	0.599309
$304$	0	0
$305$	−9.41579	−0.539146
$306$	0	0
$307$	−31.0330	−1.77115	−0.885573	−	0.464501i	$-0.846234\pi$
$307$	−31.0330	−1.77115	−0.885573	+	0.464501i	$0.846234\pi$
$308$	0	0
$309$	−5.17657	−0.294485
$310$	0	0
$311$	13.9885	0.793217	0.396608	−	0.917988i	$-0.370187\pi$
$311$	13.9885	0.793217	0.396608	+	0.917988i	$0.370187\pi$
$312$	0	0
$313$	0.117445	0.00663839	0.00331920	−	0.999994i	$-0.498943\pi$
$313$	0.117445	0.00663839	0.00331920	+	0.999994i	$0.498943\pi$
$314$	0	0
$315$	−4.46032	−0.251311
$316$	0	0
$317$	10.1403	0.569536	0.284768	−	0.958596i	$-0.408083\pi$
$317$	10.1403	0.569536	0.284768	+	0.958596i	$0.408083\pi$
$318$	0	0
$319$	0.967932	0.0541938
$320$	0	0
$321$	2.14339	0.119632
$322$	0	0
$323$	−8.57163	−0.476938
$324$	0	0
$325$	−2.55082	−0.141494
$326$	0	0
$327$	0.242099	0.0133881
$328$	0	0
$329$	1.77396	0.0978015
$330$	0	0
$331$	−26.2948	−1.44529	−0.722647	−	0.691217i	$-0.757075\pi$
$331$	−26.2948	−1.44529	−0.722647	+	0.691217i	$0.757075\pi$
$332$	0	0
$333$	−4.56432	−0.250123
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−18.7724	−1.02260	−0.511300	−	0.859403i	$-0.670836\pi$
$337$	−18.7724	−1.02260	−0.511300	+	0.859403i	$0.670836\pi$
$338$	0	0
$339$	2.80474	0.152332
$340$	0	0
$341$	57.5977	3.11909
$342$	0	0
$343$	−26.2913	−1.41960
$344$	0	0
$345$	5.52537	0.297476
$346$	0	0
$347$	−28.7289	−1.54225	−0.771125	−	0.636683i	$-0.780306\pi$
$347$	−28.7289	−1.54225	−0.771125	+	0.636683i	$0.780306\pi$
$348$	0	0
$349$	11.2965	0.604687	0.302344	−	0.953199i	$-0.402231\pi$
$349$	11.2965	0.604687	0.302344	+	0.953199i	$0.402231\pi$
$350$	0	0
$351$	2.55082	0.136153
$352$	0	0
$353$	−24.4389	−1.30075	−0.650375	−	0.759613i	$-0.725388\pi$
$353$	−24.4389	−1.30075	−0.650375	+	0.759613i	$0.725388\pi$
$354$	0	0
$355$	0.360961	0.0191578
$356$	0	0
$357$	−12.1751	−0.644374
$358$	0	0
$359$	−0.243516	−0.0128523	−0.00642614	−	0.999979i	$-0.502046\pi$
$359$	−0.243516	−0.0128523	−0.00642614	+	0.999979i	$0.502046\pi$
$360$	0	0
$361$	−9.13914	−0.481007
$362$	0	0
$363$	−23.8903	−1.25391
$364$	0	0
$365$	−9.38969	−0.491479
$366$	0	0
$367$	23.9501	1.25018	0.625092	−	0.780551i	$-0.285061\pi$
$367$	23.9501	1.25018	0.625092	+	0.780551i	$0.285061\pi$
$368$	0	0
$369$	1.87001	0.0973488
$370$	0	0
$371$	30.9252	1.60556
$372$	0	0
$373$	−19.1455	−0.991318	−0.495659	−	0.868517i	$-0.665073\pi$
$373$	−19.1455	−0.991318	−0.495659	+	0.868517i	$0.665073\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0.417996	0.0215279
$378$	0	0
$379$	21.0273	1.08010	0.540049	−	0.841634i	$-0.318406\pi$
$379$	21.0273	1.08010	0.540049	+	0.841634i	$0.318406\pi$
$380$	0	0
$381$	2.37083	0.121461
$382$	0	0
$383$	−22.1358	−1.13109	−0.565544	−	0.824718i	$-0.691334\pi$
$383$	−22.1358	−1.13109	−0.565544	+	0.824718i	$0.691334\pi$
$384$	0	0
$385$	26.3462	1.34273
$386$	0	0
$387$	−2.98660	−0.151818
$388$	0	0
$389$	−18.9547	−0.961040	−0.480520	−	0.876984i	$-0.659552\pi$
$389$	−18.9547	−0.961040	−0.480520	+	0.876984i	$0.659552\pi$
$390$	0	0
$391$	15.0823	0.762744
$392$	0	0
$393$	−2.98555	−0.150601
$394$	0	0
$395$	−16.5449	−0.832466
$396$	0	0
$397$	−25.5270	−1.28116	−0.640582	−	0.767890i	$-0.721307\pi$
$397$	−25.5270	−1.28116	−0.640582	+	0.767890i	$0.721307\pi$
$398$	0	0
$399$	14.0063	0.701193
$400$	0	0
$401$	−10.1677	−0.507752	−0.253876	−	0.967237i	$-0.581706\pi$
$401$	−10.1677	−0.507752	−0.253876	+	0.967237i	$0.581706\pi$
$402$	0	0
$403$	24.8733	1.23903
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	26.9605	1.33638
$408$	0	0
$409$	−24.4814	−1.21053	−0.605264	−	0.796025i	$-0.706932\pi$
$409$	−24.4814	−1.21053	−0.605264	+	0.796025i	$0.706932\pi$
$410$	0	0
$411$	−1.55037	−0.0764741
$412$	0	0
$413$	−45.6149	−2.24456
$414$	0	0
$415$	−16.1988	−0.795169
$416$	0	0
$417$	−2.21619	−0.108528
$418$	0	0
$419$	8.58693	0.419499	0.209750	−	0.977755i	$-0.432735\pi$
$419$	8.58693	0.419499	0.209750	+	0.977755i	$0.432735\pi$
$420$	0	0
$421$	−17.2399	−0.840222	−0.420111	−	0.907473i	$-0.638009\pi$
$421$	−17.2399	−0.840222	−0.420111	+	0.907473i	$0.638009\pi$
$422$	0	0
$423$	−0.397720	−0.0193378
$424$	0	0
$425$	−2.72964	−0.132407
$426$	0	0
$427$	41.9975	2.03240
$428$	0	0
$429$	−15.0672	−0.727451
$430$	0	0
$431$	31.4457	1.51469	0.757343	−	0.653017i	$-0.226497\pi$
$431$	31.4457	1.51469	0.757343	+	0.653017i	$0.226497\pi$
$432$	0	0
$433$	−10.2078	−0.490554	−0.245277	−	0.969453i	$-0.578879\pi$
$433$	−10.2078	−0.490554	−0.245277	+	0.969453i	$0.578879\pi$
$434$	0	0
$435$	0.163867	0.00785684
$436$	0	0
$437$	−17.3508	−0.830000
$438$	0	0
$439$	−2.33894	−0.111632	−0.0558158	−	0.998441i	$-0.517776\pi$
$439$	−2.33894	−0.111632	−0.0558158	+	0.998441i	$0.517776\pi$
$440$	0	0
$441$	12.8945	0.614023
$442$	0	0
$443$	16.2058	0.769959	0.384980	−	0.922925i	$-0.374209\pi$
$443$	16.2058	0.769959	0.384980	+	0.922925i	$0.374209\pi$
$444$	0	0
$445$	7.61288	0.360885
$446$	0	0
$447$	0.349900	0.0165497
$448$	0	0
$449$	37.6373	1.77621	0.888107	−	0.459636i	$-0.152020\pi$
$449$	37.6373	1.77621	0.888107	+	0.459636i	$0.152020\pi$
$450$	0	0
$451$	−11.0458	−0.520125
$452$	0	0
$453$	−11.8950	−0.558877
$454$	0	0
$455$	11.3775	0.533385
$456$	0	0
$457$	0.343799	0.0160823	0.00804113	−	0.999968i	$-0.497440\pi$
$457$	0.343799	0.0160823	0.00804113	+	0.999968i	$0.497440\pi$
$458$	0	0
$459$	2.72964	0.127409
$460$	0	0
$461$	−3.72078	−0.173294	−0.0866469	−	0.996239i	$-0.527615\pi$
$461$	−3.72078	−0.173294	−0.0866469	+	0.996239i	$0.527615\pi$
$462$	0	0
$463$	−4.73372	−0.219994	−0.109997	−	0.993932i	$-0.535084\pi$
$463$	−4.73372	−0.219994	−0.109997	+	0.993932i	$0.535084\pi$
$464$	0	0
$465$	9.75109	0.452196
$466$	0	0
$467$	23.7459	1.09883	0.549414	−	0.835550i	$-0.314851\pi$
$467$	23.7459	1.09883	0.549414	+	0.835550i	$0.314851\pi$
$468$	0	0
$469$	4.46032	0.205959
$470$	0	0
$471$	−12.1063	−0.557830
$472$	0	0
$473$	17.6413	0.811147
$474$	0	0
$475$	3.14020	0.144082
$476$	0	0
$477$	−6.93339	−0.317458
$478$	0	0
$479$	−0.955414	−0.0436540	−0.0218270	−	0.999762i	$-0.506948\pi$
$479$	−0.955414	−0.0436540	−0.0218270	+	0.999762i	$0.506948\pi$
$480$	0	0
$481$	11.6428	0.530864
$482$	0	0
$483$	−24.6449	−1.12138
$484$	0	0
$485$	16.4941	0.748960
$486$	0	0
$487$	13.0055	0.589334	0.294667	−	0.955600i	$-0.404791\pi$
$487$	13.0055	0.589334	0.294667	+	0.955600i	$0.404791\pi$
$488$	0	0
$489$	−7.16556	−0.324038
$490$	0	0
$491$	−35.7135	−1.61173	−0.805863	−	0.592101i	$-0.798298\pi$
$491$	−35.7135	−1.61173	−0.805863	+	0.592101i	$0.798298\pi$
$492$	0	0
$493$	0.447299	0.0201453
$494$	0	0
$495$	−5.90680	−0.265491
$496$	0	0
$497$	−1.61000	−0.0722185
$498$	0	0
$499$	15.6563	0.700874	0.350437	−	0.936586i	$-0.386033\pi$
$499$	15.6563	0.700874	0.350437	+	0.936586i	$0.386033\pi$
$500$	0	0
$501$	22.8595	1.02129
$502$	0	0
$503$	−38.5416	−1.71849	−0.859243	−	0.511568i	$-0.829065\pi$
$503$	−38.5416	−1.71849	−0.859243	+	0.511568i	$0.829065\pi$
$504$	0	0
$505$	−10.4321	−0.464223
$506$	0	0
$507$	6.49332	0.288378
$508$	0	0
$509$	−1.73076	−0.0767144	−0.0383572	−	0.999264i	$-0.512212\pi$
$509$	−1.73076	−0.0767144	−0.0383572	+	0.999264i	$0.512212\pi$
$510$	0	0
$511$	41.8811	1.85271
$512$	0	0
$513$	−3.14020	−0.138643
$514$	0	0
$515$	5.17657	0.228107
$516$	0	0
$517$	2.34925	0.103320
$518$	0	0
$519$	17.0540	0.748587
$520$	0	0
$521$	41.1557	1.80306	0.901532	−	0.432713i	$-0.142443\pi$
$521$	41.1557	1.80306	0.901532	+	0.432713i	$0.142443\pi$
$522$	0	0
$523$	4.29821	0.187948	0.0939739	−	0.995575i	$-0.470043\pi$
$523$	4.29821	0.187948	0.0939739	+	0.995575i	$0.470043\pi$
$524$	0	0
$525$	4.46032	0.194664
$526$	0	0
$527$	26.6170	1.15945
$528$	0	0
$529$	7.52973	0.327380
$530$	0	0
$531$	10.2268	0.443806
$532$	0	0
$533$	−4.77006	−0.206614
$534$	0	0
$535$	−2.14339	−0.0926667
$536$	0	0
$537$	−18.5315	−0.799694
$538$	0	0
$539$	−76.1652	−3.28067
$540$	0	0
$541$	−36.7653	−1.58066	−0.790332	−	0.612679i	$-0.790092\pi$
$541$	−36.7653	−1.58066	−0.790332	+	0.612679i	$0.790092\pi$
$542$	0	0
$543$	−7.57607	−0.325120
$544$	0	0
$545$	−0.242099	−0.0103704
$546$	0	0
$547$	−3.28581	−0.140491	−0.0702456	−	0.997530i	$-0.522378\pi$
$547$	−3.28581	−0.140491	−0.0702456	+	0.997530i	$0.522378\pi$
$548$	0	0
$549$	−9.41579	−0.401856
$550$	0	0
$551$	−0.514577	−0.0219217
$552$	0	0
$553$	73.7957	3.13811
$554$	0	0
$555$	4.56432	0.193745
$556$	0	0
$557$	19.8542	0.841248	0.420624	−	0.907235i	$-0.361811\pi$
$557$	19.8542	0.841248	0.420624	+	0.907235i	$0.361811\pi$
$558$	0	0
$559$	7.61829	0.322219
$560$	0	0
$561$	−16.1235	−0.680733
$562$	0	0
$563$	46.2194	1.94792	0.973959	−	0.226726i	$-0.0728021\pi$
$563$	46.2194	1.94792	0.973959	+	0.226726i	$0.0728021\pi$
$564$	0	0
$565$	−2.80474	−0.117996
$566$	0	0
$567$	−4.46032	−0.187316
$568$	0	0
$569$	24.1873	1.01398	0.506992	−	0.861951i	$-0.330757\pi$
$569$	24.1873	1.01398	0.506992	+	0.861951i	$0.330757\pi$
$570$	0	0
$571$	39.7465	1.66334	0.831670	−	0.555271i	$-0.187385\pi$
$571$	39.7465	1.66334	0.831670	+	0.555271i	$0.187385\pi$
$572$	0	0
$573$	0.267644	0.0111810
$574$	0	0
$575$	−5.52537	−0.230424
$576$	0	0
$577$	−40.9880	−1.70635	−0.853176	−	0.521623i	$-0.825327\pi$
$577$	−40.9880	−1.70635	−0.853176	+	0.521623i	$0.825327\pi$
$578$	0	0
$579$	−8.27309	−0.343818
$580$	0	0
$581$	72.2520	2.99752
$582$	0	0
$583$	40.9542	1.69615
$584$	0	0
$585$	−2.55082	−0.105463
$586$	0	0
$587$	−11.2875	−0.465885	−0.232942	−	0.972491i	$-0.574835\pi$
$587$	−11.2875	−0.465885	−0.232942	+	0.972491i	$0.574835\pi$
$588$	0	0
$589$	−30.6204	−1.26169
$590$	0	0
$591$	−14.8969	−0.612778
$592$	0	0
$593$	−7.55998	−0.310451	−0.155225	−	0.987879i	$-0.549610\pi$
$593$	−7.55998	−0.310451	−0.155225	+	0.987879i	$0.549610\pi$
$594$	0	0
$595$	12.1751	0.499130
$596$	0	0
$597$	14.5198	0.594256
$598$	0	0
$599$	37.3937	1.52786	0.763932	−	0.645296i	$-0.223266\pi$
$599$	37.3937	1.52786	0.763932	+	0.645296i	$0.223266\pi$
$600$	0	0
$601$	−25.0875	−1.02334	−0.511669	−	0.859182i	$-0.670973\pi$
$601$	−25.0875	−1.02334	−0.511669	+	0.859182i	$0.670973\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	23.8903	0.971278
$606$	0	0
$607$	−21.7640	−0.883374	−0.441687	−	0.897169i	$-0.645620\pi$
$607$	−21.7640	−0.883374	−0.441687	+	0.897169i	$0.645620\pi$
$608$	0	0
$609$	−0.730902	−0.0296176
$610$	0	0
$611$	1.01451	0.0410427
$612$	0	0
$613$	47.2951	1.91023	0.955115	−	0.296235i	$-0.0957310\pi$
$613$	47.2951	1.91023	0.955115	+	0.296235i	$0.0957310\pi$
$614$	0	0
$615$	−1.87001	−0.0754061
$616$	0	0
$617$	3.18286	0.128137	0.0640686	−	0.997945i	$-0.479592\pi$
$617$	3.18286	0.128137	0.0640686	+	0.997945i	$0.479592\pi$
$618$	0	0
$619$	−29.8064	−1.19802	−0.599011	−	0.800741i	$-0.704439\pi$
$619$	−29.8064	−1.19802	−0.599011	+	0.800741i	$0.704439\pi$
$620$	0	0
$621$	5.52537	0.221726
$622$	0	0
$623$	−33.9559	−1.36042
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	18.5485	0.740757
$628$	0	0
$629$	12.4590	0.496772
$630$	0	0
$631$	−30.7569	−1.22441	−0.612206	−	0.790698i	$-0.709718\pi$
$631$	−30.7569	−1.22441	−0.612206	+	0.790698i	$0.709718\pi$
$632$	0	0
$633$	−9.63978	−0.383147
$634$	0	0
$635$	−2.37083	−0.0940834
$636$	0	0
$637$	−32.8915	−1.30321
$638$	0	0
$639$	0.360961	0.0142794
$640$	0	0
$641$	48.6011	1.91963	0.959813	−	0.280640i	$-0.0905466\pi$
$641$	48.6011	1.91963	0.959813	+	0.280640i	$0.0905466\pi$
$642$	0	0
$643$	30.5455	1.20460	0.602299	−	0.798271i	$-0.294251\pi$
$643$	30.5455	1.20460	0.602299	+	0.798271i	$0.294251\pi$
$644$	0	0
$645$	2.98660	0.117597
$646$	0	0
$647$	7.12822	0.280239	0.140120	−	0.990135i	$-0.455251\pi$
$647$	7.12822	0.280239	0.140120	+	0.990135i	$0.455251\pi$
$648$	0	0
$649$	−60.4078	−2.37121
$650$	0	0
$651$	−43.4930	−1.70463
$652$	0	0
$653$	−10.6528	−0.416876	−0.208438	−	0.978036i	$-0.566838\pi$
$653$	−10.6528	−0.416876	−0.208438	+	0.978036i	$0.566838\pi$
$654$	0	0
$655$	2.98555	0.116655
$656$	0	0
$657$	−9.38969	−0.366327
$658$	0	0
$659$	45.6743	1.77922	0.889609	−	0.456724i	$-0.150977\pi$
$659$	45.6743	1.77922	0.889609	+	0.456724i	$0.150977\pi$
$660$	0	0
$661$	−3.18380	−0.123836	−0.0619178	−	0.998081i	$-0.519722\pi$
$661$	−3.18380	−0.123836	−0.0619178	+	0.998081i	$0.519722\pi$
$662$	0	0
$663$	−6.96283	−0.270414
$664$	0	0
$665$	−14.0063	−0.543142
$666$	0	0
$667$	0.905428	0.0350583
$668$	0	0
$669$	0.0257084	0.000993944 0
$670$	0	0
$671$	55.6172	2.14708
$672$	0	0
$673$	−12.7078	−0.489850	−0.244925	−	0.969542i	$-0.578763\pi$
$673$	−12.7078	−0.489850	−0.244925	+	0.969542i	$0.578763\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	42.7045	1.64127	0.820633	−	0.571456i	$-0.193621\pi$
$677$	42.7045	1.64127	0.820633	+	0.571456i	$0.193621\pi$
$678$	0	0
$679$	−73.5691	−2.82332
$680$	0	0
$681$	7.64782	0.293065
$682$	0	0
$683$	6.34885	0.242932	0.121466	−	0.992596i	$-0.461240\pi$
$683$	6.34885	0.242932	0.121466	+	0.992596i	$0.461240\pi$
$684$	0	0
$685$	1.55037	0.0592366
$686$	0	0
$687$	−3.55912	−0.135789
$688$	0	0
$689$	17.6858	0.673777
$690$	0	0
$691$	14.8580	0.565226	0.282613	−	0.959234i	$-0.408799\pi$
$691$	14.8580	0.565226	0.282613	+	0.959234i	$0.408799\pi$
$692$	0	0
$693$	26.3462	1.00081
$694$	0	0
$695$	2.21619	0.0840650
$696$	0	0
$697$	−5.10446	−0.193345
$698$	0	0
$699$	−11.5075	−0.435253
$700$	0	0
$701$	18.3178	0.691852	0.345926	−	0.938262i	$-0.387565\pi$
$701$	18.3178	0.691852	0.345926	+	0.938262i	$0.387565\pi$
$702$	0	0
$703$	−14.3329	−0.540575
$704$	0	0
$705$	0.397720	0.0149790
$706$	0	0
$707$	46.5306	1.74996
$708$	0	0
$709$	−26.4636	−0.993863	−0.496932	−	0.867790i	$-0.665540\pi$
$709$	−26.4636	−0.993863	−0.496932	+	0.867790i	$0.665540\pi$
$710$	0	0
$711$	−16.5449	−0.620483
$712$	0	0
$713$	53.8784	2.01776
$714$	0	0
$715$	15.0672	0.563481
$716$	0	0
$717$	−6.65835	−0.248661
$718$	0	0
$719$	−20.8393	−0.777176	−0.388588	−	0.921412i	$-0.627037\pi$
$719$	−20.8393	−0.777176	−0.388588	+	0.921412i	$0.627037\pi$
$720$	0	0
$721$	−23.0892	−0.859886
$722$	0	0
$723$	−6.84364	−0.254518
$724$	0	0
$725$	−0.163867	−0.00608588
$726$	0	0
$727$	−5.30378	−0.196706	−0.0983531	−	0.995152i	$-0.531357\pi$
$727$	−5.30378	−0.196706	−0.0983531	+	0.995152i	$0.531357\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.15236	0.301526
$732$	0	0
$733$	17.1684	0.634130	0.317065	−	0.948404i	$-0.397303\pi$
$733$	17.1684	0.634130	0.317065	+	0.948404i	$0.397303\pi$
$734$	0	0
$735$	−12.8945	−0.475620
$736$	0	0
$737$	5.90680	0.217580
$738$	0	0
$739$	48.9129	1.79929	0.899645	−	0.436623i	$-0.143826\pi$
$739$	48.9129	1.79929	0.899645	+	0.436623i	$0.143826\pi$
$740$	0	0
$741$	8.01009	0.294258
$742$	0	0
$743$	−24.2901	−0.891119	−0.445559	−	0.895252i	$-0.646995\pi$
$743$	−24.2901	−0.891119	−0.445559	+	0.895252i	$0.646995\pi$
$744$	0	0
$745$	−0.349900	−0.0128194
$746$	0	0
$747$	−16.1988	−0.592684
$748$	0	0
$749$	9.56020	0.349322
$750$	0	0
$751$	−13.4584	−0.491103	−0.245551	−	0.969384i	$-0.578969\pi$
$751$	−13.4584	−0.491103	−0.245551	+	0.969384i	$0.578969\pi$
$752$	0	0
$753$	10.7248	0.390835
$754$	0	0
$755$	11.8950	0.432904
$756$	0	0
$757$	−20.3774	−0.740629	−0.370315	−	0.928906i	$-0.620750\pi$
$757$	−20.3774	−0.740629	−0.370315	+	0.928906i	$0.620750\pi$
$758$	0	0
$759$	−32.6373	−1.18466
$760$	0	0
$761$	29.2227	1.05932	0.529660	−	0.848210i	$-0.322319\pi$
$761$	29.2227	1.05932	0.529660	+	0.848210i	$0.322319\pi$
$762$	0	0
$763$	1.07984	0.0390929
$764$	0	0
$765$	−2.72964	−0.0986904
$766$	0	0
$767$	−26.0868	−0.941939
$768$	0	0
$769$	−49.9983	−1.80299	−0.901493	−	0.432794i	$-0.857528\pi$
$769$	−49.9983	−1.80299	−0.901493	+	0.432794i	$0.857528\pi$
$770$	0	0
$771$	3.40789	0.122732
$772$	0	0
$773$	−35.0882	−1.26204	−0.631018	−	0.775768i	$-0.717363\pi$
$773$	−35.0882	−1.26204	−0.631018	+	0.775768i	$0.717363\pi$
$774$	0	0
$775$	−9.75109	−0.350269
$776$	0	0
$777$	−20.3584	−0.730352
$778$	0	0
$779$	5.87221	0.210394
$780$	0	0
$781$	−2.13213	−0.0762934
$782$	0	0
$783$	0.163867	0.00585614
$784$	0	0
$785$	12.1063	0.432093
$786$	0	0
$787$	−33.8278	−1.20583	−0.602916	−	0.797805i	$-0.705995\pi$
$787$	−33.8278	−1.20583	−0.602916	+	0.797805i	$0.705995\pi$
$788$	0	0
$789$	−22.5160	−0.801592
$790$	0	0
$791$	12.5100	0.444806
$792$	0	0
$793$	24.0180	0.852904
$794$	0	0
$795$	6.93339	0.245902
$796$	0	0
$797$	−12.2777	−0.434898	−0.217449	−	0.976072i	$-0.569774\pi$
$797$	−12.2777	−0.434898	−0.217449	+	0.976072i	$0.569774\pi$
$798$	0	0
$799$	1.08563	0.0384069
$800$	0	0
$801$	7.61288	0.268988
$802$	0	0
$803$	55.4630	1.95725
$804$	0	0
$805$	24.6449	0.868620
$806$	0	0
$807$	−6.96094	−0.245037
$808$	0	0
$809$	−25.2650	−0.888271	−0.444135	−	0.895960i	$-0.646489\pi$
$809$	−25.2650	−0.888271	−0.444135	+	0.895960i	$0.646489\pi$
$810$	0	0
$811$	−11.4469	−0.401955	−0.200977	−	0.979596i	$-0.564412\pi$
$811$	−11.4469	−0.401955	−0.200977	+	0.979596i	$0.564412\pi$
$812$	0	0
$813$	−1.98173	−0.0695022
$814$	0	0
$815$	7.16556	0.250998
$816$	0	0
$817$	−9.37854	−0.328114
$818$	0	0
$819$	11.3775	0.397561
$820$	0	0
$821$	28.9570	1.01061	0.505304	−	0.862942i	$-0.331381\pi$
$821$	28.9570	1.01061	0.505304	+	0.862942i	$0.331381\pi$
$822$	0	0
$823$	9.30859	0.324477	0.162238	−	0.986752i	$-0.448129\pi$
$823$	9.30859	0.324477	0.162238	+	0.986752i	$0.448129\pi$
$824$	0	0
$825$	5.90680	0.205648
$826$	0	0
$827$	−4.46571	−0.155288	−0.0776440	−	0.996981i	$-0.524740\pi$
$827$	−4.46571	−0.155288	−0.0776440	+	0.996981i	$0.524740\pi$
$828$	0	0
$829$	43.7654	1.52003	0.760017	−	0.649903i	$-0.225190\pi$
$829$	43.7654	1.52003	0.760017	+	0.649903i	$0.225190\pi$
$830$	0	0
$831$	10.1378	0.351675
$832$	0	0
$833$	−35.1973	−1.21952
$834$	0	0
$835$	−22.8595	−0.791084
$836$	0	0
$837$	9.75109	0.337047
$838$	0	0
$839$	14.4674	0.499470	0.249735	−	0.968314i	$-0.419656\pi$
$839$	14.4674	0.499470	0.249735	+	0.968314i	$0.419656\pi$
$840$	0	0
$841$	−28.9731	−0.999074
$842$	0	0
$843$	14.5574	0.501385
$844$	0	0
$845$	−6.49332	−0.223377
$846$	0	0
$847$	−106.558	−3.66139
$848$	0	0
$849$	−13.3939	−0.459676
$850$	0	0
$851$	25.2196	0.864516
$852$	0	0
$853$	19.5641	0.669862	0.334931	−	0.942243i	$-0.391287\pi$
$853$	19.5641	0.669862	0.334931	+	0.942243i	$0.391287\pi$
$854$	0	0
$855$	3.14020	0.107393
$856$	0	0
$857$	28.5695	0.975917	0.487958	−	0.872867i	$-0.337742\pi$
$857$	28.5695	0.975917	0.487958	+	0.872867i	$0.337742\pi$
$858$	0	0
$859$	55.8316	1.90495	0.952475	−	0.304618i	$-0.0985288\pi$
$859$	55.8316	1.90495	0.952475	+	0.304618i	$0.0985288\pi$
$860$	0	0
$861$	8.34085	0.284255
$862$	0	0
$863$	19.3523	0.658759	0.329380	−	0.944198i	$-0.393160\pi$
$863$	19.3523	0.658759	0.329380	+	0.944198i	$0.393160\pi$
$864$	0	0
$865$	−17.0540	−0.579853
$866$	0	0
$867$	9.54905	0.324303
$868$	0	0
$869$	97.7276	3.31518
$870$	0	0
$871$	2.55082	0.0864312
$872$	0	0
$873$	16.4941	0.558242
$874$	0	0
$875$	−4.46032	−0.150786
$876$	0	0
$877$	−52.8070	−1.78317	−0.891583	−	0.452858i	$-0.850404\pi$
$877$	−52.8070	−1.78317	−0.891583	+	0.452858i	$0.850404\pi$
$878$	0	0
$879$	−23.6966	−0.799266
$880$	0	0
$881$	−24.3656	−0.820896	−0.410448	−	0.911884i	$-0.634628\pi$
$881$	−24.3656	−0.820896	−0.410448	+	0.911884i	$0.634628\pi$
$882$	0	0
$883$	−40.1116	−1.34986	−0.674932	−	0.737880i	$-0.735827\pi$
$883$	−40.1116	−1.34986	−0.674932	+	0.737880i	$0.735827\pi$
$884$	0	0
$885$	−10.2268	−0.343771
$886$	0	0
$887$	−31.7344	−1.06554	−0.532768	−	0.846261i	$-0.678848\pi$
$887$	−31.7344	−1.06554	−0.532768	+	0.846261i	$0.678848\pi$
$888$	0	0
$889$	10.5747	0.354663
$890$	0	0
$891$	−5.90680	−0.197885
$892$	0	0
$893$	−1.24892	−0.0417935
$894$	0	0
$895$	18.5315	0.619440
$896$	0	0
$897$	−14.0942	−0.470593
$898$	0	0
$899$	1.59789	0.0532925
$900$	0	0
$901$	18.9257	0.630506
$902$	0	0
$903$	−13.3212	−0.443303
$904$	0	0
$905$	7.57607	0.251837
$906$	0	0
$907$	14.4510	0.479837	0.239918	−	0.970793i	$-0.422879\pi$
$907$	14.4510	0.479837	0.239918	+	0.970793i	$0.422879\pi$
$908$	0	0
$909$	−10.4321	−0.346011
$910$	0	0
$911$	−6.83166	−0.226343	−0.113172	−	0.993575i	$-0.536101\pi$
$911$	−6.83166	−0.226343	−0.113172	+	0.993575i	$0.536101\pi$
$912$	0	0
$913$	95.6832	3.16665
$914$	0	0
$915$	9.41579	0.311276
$916$	0	0
$917$	−13.3165	−0.439751
$918$	0	0
$919$	−35.1243	−1.15864	−0.579322	−	0.815099i	$-0.696683\pi$
$919$	−35.1243	−1.15864	−0.579322	+	0.815099i	$0.696683\pi$
$920$	0	0
$921$	31.0330	1.02257
$922$	0	0
$923$	−0.920747	−0.0303068
$924$	0	0
$925$	−4.56432	−0.150074
$926$	0	0
$927$	5.17657	0.170021
$928$	0	0
$929$	−49.9011	−1.63720	−0.818601	−	0.574363i	$-0.805250\pi$
$929$	−49.9011	−1.63720	−0.818601	+	0.574363i	$0.805250\pi$
$930$	0	0
$931$	40.4913	1.32705
$932$	0	0
$933$	−13.9885	−0.457964
$934$	0	0
$935$	16.1235	0.527293
$936$	0	0
$937$	58.0287	1.89572	0.947858	−	0.318694i	$-0.103244\pi$
$937$	58.0287	1.89572	0.947858	+	0.318694i	$0.103244\pi$
$938$	0	0
$939$	−0.117445	−0.00383268
$940$	0	0
$941$	−12.5715	−0.409819	−0.204910	−	0.978781i	$-0.565690\pi$
$941$	−12.5715	−0.409819	−0.204910	+	0.978781i	$0.565690\pi$
$942$	0	0
$943$	−10.3325	−0.336472
$944$	0	0
$945$	4.46032	0.145094
$946$	0	0
$947$	−37.8205	−1.22900	−0.614501	−	0.788916i	$-0.710643\pi$
$947$	−37.8205	−1.22900	−0.614501	+	0.788916i	$0.710643\pi$
$948$	0	0
$949$	23.9514	0.777496
$950$	0	0
$951$	−10.1403	−0.328822
$952$	0	0
$953$	−15.9101	−0.515378	−0.257689	−	0.966228i	$-0.582961\pi$
$953$	−15.9101	−0.515378	−0.257689	+	0.966228i	$0.582961\pi$
$954$	0	0
$955$	−0.267644	−0.00866074
$956$	0	0
$957$	−0.967932	−0.0312888
$958$	0	0
$959$	−6.91515	−0.223302
$960$	0	0
$961$	64.0837	2.06722
$962$	0	0
$963$	−2.14339	−0.0690697
$964$	0	0
$965$	8.27309	0.266320
$966$	0	0
$967$	−61.5001	−1.97771	−0.988855	−	0.148882i	$-0.952433\pi$
$967$	−61.5001	−1.97771	−0.988855	+	0.148882i	$0.952433\pi$
$968$	0	0
$969$	8.57163	0.275360
$970$	0	0
$971$	−38.4585	−1.23419	−0.617096	−	0.786888i	$-0.711691\pi$
$971$	−38.4585	−1.23419	−0.617096	+	0.786888i	$0.711691\pi$
$972$	0	0
$973$	−9.88494	−0.316897
$974$	0	0
$975$	2.55082	0.0816916
$976$	0	0
$977$	−1.35734	−0.0434253	−0.0217126	−	0.999764i	$-0.506912\pi$
$977$	−1.35734	−0.0434253	−0.0217126	+	0.999764i	$0.506912\pi$
$978$	0	0
$979$	−44.9678	−1.43718
$980$	0	0
$981$	−0.242099	−0.00772964
$982$	0	0
$983$	20.4386	0.651889	0.325944	−	0.945389i	$-0.394318\pi$
$983$	20.4386	0.651889	0.325944	+	0.945389i	$0.394318\pi$
$984$	0	0
$985$	14.8969	0.474655
$986$	0	0
$987$	−1.77396	−0.0564657
$988$	0	0
$989$	16.5021	0.524736
$990$	0	0
$991$	−33.6512	−1.06896	−0.534482	−	0.845180i	$-0.679493\pi$
$991$	−33.6512	−1.06896	−0.534482	+	0.845180i	$0.679493\pi$
$992$	0	0
$993$	26.2948	0.834441
$994$	0	0
$995$	−14.5198	−0.460309
$996$	0	0
$997$	−31.4088	−0.994727	−0.497364	−	0.867542i	$-0.665699\pi$
$997$	−31.4088	−0.994727	−0.497364	+	0.867542i	$0.665699\pi$
$998$	0	0
$999$	4.56432	0.144409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.bc.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.bc.1.1	✓	10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -4.46032 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -4.46032 q^{7} +1.00000 q^{9} -5.90680 q^{11} -2.55082 q^{13} -1.00000 q^{15} -2.72964 q^{17} +3.14020 q^{19} +4.46032 q^{21} -5.52537 q^{23} +1.00000 q^{25} -1.00000 q^{27} -0.163867 q^{29} -9.75109 q^{31} +5.90680 q^{33} -4.46032 q^{35} -4.56432 q^{37} +2.55082 q^{39} +1.87001 q^{41} -2.98660 q^{43} +1.00000 q^{45} -0.397720 q^{47} +12.8945 q^{49} +2.72964 q^{51} -6.93339 q^{53} -5.90680 q^{55} -3.14020 q^{57} +10.2268 q^{59} -9.41579 q^{61} -4.46032 q^{63} -2.55082 q^{65} -1.00000 q^{67} +5.52537 q^{69} +0.360961 q^{71} -9.38969 q^{73} -1.00000 q^{75} +26.3462 q^{77} -16.5449 q^{79} +1.00000 q^{81} -16.1988 q^{83} -2.72964 q^{85} +0.163867 q^{87} +7.61288 q^{89} +11.3775 q^{91} +9.75109 q^{93} +3.14020 q^{95} +16.4941 q^{97} -5.90680 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9	\( 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9} + 7 q^{11} - q^{13} - 10 q^{15} + 3 q^{17} + 4 q^{19} - q^{21} - 13 q^{23} + 10 q^{25} - 10 q^{27} + 18 q^{29} - 9 q^{31} - 7 q^{33} + q^{35} - 3 q^{37} + q^{39} + 19 q^{41} + 5 q^{43} + 10 q^{45} - 8 q^{47} + 43 q^{49} - 3 q^{51} + 17 q^{53} + 7 q^{55} - 4 q^{57} + 24 q^{59} + 21 q^{61} + q^{63} - q^{65} - 10 q^{67} + 13 q^{69} + 2 q^{71} + 25 q^{73} - 10 q^{75} + 15 q^{77} - q^{79} + 10 q^{81} - 6 q^{83} + 3 q^{85} - 18 q^{87} + 23 q^{89} + 29 q^{91} + 9 q^{93} + 4 q^{95} + 21 q^{97} + 7 q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9 + 7 * q^11 - q^13 - 10 * q^15 + 3 * q^17 + 4 * q^19 - q^21 - 13 * q^23 + 10 * q^25 - 10 * q^27 + 18 * q^29 - 9 * q^31 - 7 * q^33 + q^35 - 3 * q^37 + q^39 + 19 * q^41 + 5 * q^43 + 10 * q^45 - 8 * q^47 + 43 * q^49 - 3 * q^51 + 17 * q^53 + 7 * q^55 - 4 * q^57 + 24 * q^59 + 21 * q^61 + q^63 - q^65 - 10 * q^67 + 13 * q^69 + 2 * q^71 + 25 * q^73 - 10 * q^75 + 15 * q^77 - q^79 + 10 * q^81 - 6 * q^83 + 3 * q^85 - 18 * q^87 + 23 * q^89 + 29 * q^91 + 9 * q^93 + 4 * q^95 + 21 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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