LMFDB - Embedded newform 7120.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7120,2,Mod(1,7120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7120 = 2^{4} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7120.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:	$56.8534862392$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 8x^{5} + 6x^{4} + 19x^{3} - 10x^{2} - 12x + 6 $ x^7 - x^6 - 8x^5 + 6x^4 + 19x^3 - 10x^2 - 12*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.07810$ of defining polynomial
Character	$\chi$	$=$	7120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.10226 q^{3} +1.00000 q^{5} +4.72699 q^{7} +1.41950 q^{9} +O(q^{10})$	$q-2.10226 q^{3} +1.00000 q^{5} +4.72699 q^{7} +1.41950 q^{9} +4.62787 q^{11} -4.47978 q^{13} -2.10226 q^{15} -5.66969 q^{17} -1.34308 q^{19} -9.93737 q^{21} -4.44187 q^{23} +1.00000 q^{25} +3.32263 q^{27} -4.27246 q^{29} -9.40035 q^{31} -9.72899 q^{33} +4.72699 q^{35} +6.89093 q^{37} +9.41765 q^{39} +5.69631 q^{41} +5.04294 q^{43} +1.41950 q^{45} +8.76271 q^{47} +15.3444 q^{49} +11.9192 q^{51} -8.23538 q^{53} +4.62787 q^{55} +2.82349 q^{57} +7.52173 q^{59} -6.94612 q^{61} +6.70996 q^{63} -4.47978 q^{65} +6.38703 q^{67} +9.33796 q^{69} -5.84983 q^{71} +11.0282 q^{73} -2.10226 q^{75} +21.8759 q^{77} -7.89145 q^{79} -11.2435 q^{81} +15.3880 q^{83} -5.66969 q^{85} +8.98182 q^{87} +1.00000 q^{89} -21.1759 q^{91} +19.7620 q^{93} -1.34308 q^{95} -0.674067 q^{97} +6.56926 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9}+O(q^{10}) $ 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9	$ 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9} + 10 q^{11} - 7 q^{13} + 8 q^{15} - 13 q^{17} + 7 q^{19} + 16 q^{21} + 13 q^{23} + 7 q^{25} + 23 q^{27} - 4 q^{29} - q^{31} - 6 q^{33} + 16 q^{35} - 5 q^{37} + 13 q^{39} + 5 q^{41} + 31 q^{43} + 11 q^{45} + 14 q^{47} + 19 q^{49} + q^{51} - 13 q^{53} + 10 q^{55} + 21 q^{57} + 14 q^{59} + 3 q^{61} + 54 q^{63} - 7 q^{65} - q^{67} + 31 q^{69} + 8 q^{71} + 9 q^{73} + 8 q^{75} + 42 q^{77} - 9 q^{79} + 35 q^{81} + 42 q^{83} - 13 q^{85} - 6 q^{87} + 7 q^{89} - 31 q^{91} + 24 q^{93} + 7 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) $ 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9 + 10 * q^11 - 7 * q^13 + 8 * q^15 - 13 * q^17 + 7 * q^19 + 16 * q^21 + 13 * q^23 + 7 * q^25 + 23 * q^27 - 4 * q^29 - q^31 - 6 * q^33 + 16 * q^35 - 5 * q^37 + 13 * q^39 + 5 * q^41 + 31 * q^43 + 11 * q^45 + 14 * q^47 + 19 * q^49 + q^51 - 13 * q^53 + 10 * q^55 + 21 * q^57 + 14 * q^59 + 3 * q^61 + 54 * q^63 - 7 * q^65 - q^67 + 31 * q^69 + 8 * q^71 + 9 * q^73 + 8 * q^75 + 42 * q^77 - 9 * q^79 + 35 * q^81 + 42 * q^83 - 13 * q^85 - 6 * q^87 + 7 * q^89 - 31 * q^91 + 24 * q^93 + 7 * q^95 - 7 * q^97 - 5 * 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
$3$	−2.10226	−1.21374	−0.606870	−	0.794801i	$-0.707575\pi$
$3$	−2.10226	−1.21374	−0.606870	+	0.794801i	$0.707575\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.72699	1.78663	0.893317	−	0.449426i	$-0.148372\pi$
$7$	4.72699	1.78663	0.893317	+	0.449426i	$0.148372\pi$
$8$	0	0
$9$	1.41950	0.473166
$10$	0	0
$11$	4.62787	1.39536	0.697678	−	0.716412i	$-0.254217\pi$
$11$	4.62787	1.39536	0.697678	+	0.716412i	$0.254217\pi$
$12$	0	0
$13$	−4.47978	−1.24247	−0.621233	−	0.783626i	$-0.713368\pi$
$13$	−4.47978	−1.24247	−0.621233	+	0.783626i	$0.713368\pi$
$14$	0	0
$15$	−2.10226	−0.542801
$16$	0	0
$17$	−5.66969	−1.37510	−0.687551	−	0.726136i	$-0.741314\pi$
$17$	−5.66969	−1.37510	−0.687551	+	0.726136i	$0.741314\pi$
$18$	0	0
$19$	−1.34308	−0.308123	−0.154061	−	0.988061i	$-0.549235\pi$
$19$	−1.34308	−0.308123	−0.154061	+	0.988061i	$0.549235\pi$
$20$	0	0
$21$	−9.93737	−2.16851
$22$	0	0
$23$	−4.44187	−0.926193	−0.463097	−	0.886308i	$-0.653262\pi$
$23$	−4.44187	−0.926193	−0.463097	+	0.886308i	$0.653262\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.32263	0.639440
$28$	0	0
$29$	−4.27246	−0.793376	−0.396688	−	0.917954i	$-0.629840\pi$
$29$	−4.27246	−0.793376	−0.396688	+	0.917954i	$0.629840\pi$
$30$	0	0
$31$	−9.40035	−1.68835	−0.844176	−	0.536066i	$-0.819910\pi$
$31$	−9.40035	−1.68835	−0.844176	+	0.536066i	$0.819910\pi$
$32$	0	0
$33$	−9.72899	−1.69360
$34$	0	0
$35$	4.72699	0.799007
$36$	0	0
$37$	6.89093	1.13286	0.566431	−	0.824109i	$-0.308324\pi$
$37$	6.89093	1.13286	0.566431	+	0.824109i	$0.308324\pi$
$38$	0	0
$39$	9.41765	1.50803
$40$	0	0
$41$	5.69631	0.889614	0.444807	−	0.895626i	$-0.353272\pi$
$41$	5.69631	0.889614	0.444807	+	0.895626i	$0.353272\pi$
$42$	0	0
$43$	5.04294	0.769041	0.384520	−	0.923117i	$-0.374367\pi$
$43$	5.04294	0.769041	0.384520	+	0.923117i	$0.374367\pi$
$44$	0	0
$45$	1.41950	0.211606
$46$	0	0
$47$	8.76271	1.27817	0.639087	−	0.769135i	$-0.279313\pi$
$47$	8.76271	1.27817	0.639087	+	0.769135i	$0.279313\pi$
$48$	0	0
$49$	15.3444	2.19206
$50$	0	0
$51$	11.9192	1.66902
$52$	0	0
$53$	−8.23538	−1.13122	−0.565608	−	0.824674i	$-0.691358\pi$
$53$	−8.23538	−1.13122	−0.565608	+	0.824674i	$0.691358\pi$
$54$	0	0
$55$	4.62787	0.624022
$56$	0	0
$57$	2.82349	0.373981
$58$	0	0
$59$	7.52173	0.979246	0.489623	−	0.871934i	$-0.337134\pi$
$59$	7.52173	0.979246	0.489623	+	0.871934i	$0.337134\pi$
$60$	0	0
$61$	−6.94612	−0.889360	−0.444680	−	0.895689i	$-0.646683\pi$
$61$	−6.94612	−0.889360	−0.444680	+	0.895689i	$0.646683\pi$
$62$	0	0
$63$	6.70996	0.845375
$64$	0	0
$65$	−4.47978	−0.555648
$66$	0	0
$67$	6.38703	0.780300	0.390150	−	0.920751i	$-0.372423\pi$
$67$	6.38703	0.780300	0.390150	+	0.920751i	$0.372423\pi$
$68$	0	0
$69$	9.33796	1.12416
$70$	0	0
$71$	−5.84983	−0.694247	−0.347123	−	0.937820i	$-0.612841\pi$
$71$	−5.84983	−0.694247	−0.347123	+	0.937820i	$0.612841\pi$
$72$	0	0
$73$	11.0282	1.29076	0.645378	−	0.763863i	$-0.276700\pi$
$73$	11.0282	1.29076	0.645378	+	0.763863i	$0.276700\pi$
$74$	0	0
$75$	−2.10226	−0.242748
$76$	0	0
$77$	21.8759	2.49299
$78$	0	0
$79$	−7.89145	−0.887857	−0.443929	−	0.896062i	$-0.646416\pi$
$79$	−7.89145	−0.887857	−0.443929	+	0.896062i	$0.646416\pi$
$80$	0	0
$81$	−11.2435	−1.24928
$82$	0	0
$83$	15.3880	1.68905	0.844527	−	0.535513i	$-0.179882\pi$
$83$	15.3880	1.68905	0.844527	+	0.535513i	$0.179882\pi$
$84$	0	0
$85$	−5.66969	−0.614965
$86$	0	0
$87$	8.98182	0.962952
$88$	0	0
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−21.1759	−2.21983
$92$	0	0
$93$	19.7620	2.04922
$94$	0	0
$95$	−1.34308	−0.137797
$96$	0	0
$97$	−0.674067	−0.0684412	−0.0342206	−	0.999414i	$-0.510895\pi$
$97$	−0.674067	−0.0684412	−0.0342206	+	0.999414i	$0.510895\pi$
$98$	0	0
$99$	6.56926	0.660235
$100$	0	0
$101$	13.4407	1.33740	0.668699	−	0.743533i	$-0.266851\pi$
$101$	13.4407	1.33740	0.668699	+	0.743533i	$0.266851\pi$
$102$	0	0
$103$	11.4024	1.12352	0.561758	−	0.827302i	$-0.310125\pi$
$103$	11.4024	1.12352	0.561758	+	0.827302i	$0.310125\pi$
$104$	0	0
$105$	−9.93737	−0.969788
$106$	0	0
$107$	8.51924	0.823586	0.411793	−	0.911277i	$-0.364903\pi$
$107$	8.51924	0.823586	0.411793	+	0.911277i	$0.364903\pi$
$108$	0	0
$109$	12.7180	1.21816	0.609081	−	0.793108i	$-0.291538\pi$
$109$	12.7180	1.21816	0.609081	+	0.793108i	$0.291538\pi$
$110$	0	0
$111$	−14.4865	−1.37500
$112$	0	0
$113$	8.25522	0.776586	0.388293	−	0.921536i	$-0.373065\pi$
$113$	8.25522	0.776586	0.388293	+	0.921536i	$0.373065\pi$
$114$	0	0
$115$	−4.44187	−0.414206
$116$	0	0
$117$	−6.35903	−0.587893
$118$	0	0
$119$	−26.8006	−2.45681
$120$	0	0
$121$	10.4172	0.947018
$122$	0	0
$123$	−11.9751	−1.07976
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.77097	−0.157149	−0.0785743	−	0.996908i	$-0.525037\pi$
$127$	−1.77097	−0.157149	−0.0785743	+	0.996908i	$0.525037\pi$
$128$	0	0
$129$	−10.6016	−0.933416
$130$	0	0
$131$	7.28762	0.636722	0.318361	−	0.947969i	$-0.396868\pi$
$131$	7.28762	0.636722	0.318361	+	0.947969i	$0.396868\pi$
$132$	0	0
$133$	−6.34870	−0.550503
$134$	0	0
$135$	3.32263	0.285966
$136$	0	0
$137$	0.754924	0.0644975	0.0322488	−	0.999480i	$-0.489733\pi$
$137$	0.754924	0.0644975	0.0322488	+	0.999480i	$0.489733\pi$
$138$	0	0
$139$	6.29913	0.534285	0.267142	−	0.963657i	$-0.413921\pi$
$139$	6.29913	0.534285	0.267142	+	0.963657i	$0.413921\pi$
$140$	0	0
$141$	−18.4215	−1.55137
$142$	0	0
$143$	−20.7318	−1.73368
$144$	0	0
$145$	−4.27246	−0.354808
$146$	0	0
$147$	−32.2580	−2.66060
$148$	0	0
$149$	−17.7115	−1.45098	−0.725490	−	0.688233i	$-0.758387\pi$
$149$	−17.7115	−1.45098	−0.725490	+	0.688233i	$0.758387\pi$
$150$	0	0
$151$	−10.3991	−0.846263	−0.423132	−	0.906068i	$-0.639069\pi$
$151$	−10.3991	−0.846263	−0.423132	+	0.906068i	$0.639069\pi$
$152$	0	0
$153$	−8.04812	−0.650652
$154$	0	0
$155$	−9.40035	−0.755054
$156$	0	0
$157$	−12.0575	−0.962292	−0.481146	−	0.876641i	$-0.659779\pi$
$157$	−12.0575	−0.962292	−0.481146	+	0.876641i	$0.659779\pi$
$158$	0	0
$159$	17.3129	1.37300
$160$	0	0
$161$	−20.9967	−1.65477
$162$	0	0
$163$	−6.76614	−0.529965	−0.264982	−	0.964253i	$-0.585366\pi$
$163$	−6.76614	−0.529965	−0.264982	+	0.964253i	$0.585366\pi$
$164$	0	0
$165$	−9.72899	−0.757401
$166$	0	0
$167$	5.08722	0.393661	0.196830	−	0.980438i	$-0.436935\pi$
$167$	5.08722	0.393661	0.196830	+	0.980438i	$0.436935\pi$
$168$	0	0
$169$	7.06839	0.543722
$170$	0	0
$171$	−1.90649	−0.145793
$172$	0	0
$173$	1.63396	0.124228	0.0621139	−	0.998069i	$-0.480216\pi$
$173$	1.63396	0.124228	0.0621139	+	0.998069i	$0.480216\pi$
$174$	0	0
$175$	4.72699	0.357327
$176$	0	0
$177$	−15.8126	−1.18855
$178$	0	0
$179$	8.89727	0.665014	0.332507	−	0.943101i	$-0.392106\pi$
$179$	8.89727	0.665014	0.332507	+	0.943101i	$0.392106\pi$
$180$	0	0
$181$	−7.14852	−0.531345	−0.265673	−	0.964063i	$-0.585594\pi$
$181$	−7.14852	−0.531345	−0.265673	+	0.964063i	$0.585594\pi$
$182$	0	0
$183$	14.6026	1.07945
$184$	0	0
$185$	6.89093	0.506631
$186$	0	0
$187$	−26.2386	−1.91876
$188$	0	0
$189$	15.7060	1.14245
$190$	0	0
$191$	8.74055	0.632444	0.316222	−	0.948685i	$-0.397586\pi$
$191$	8.74055	0.632444	0.316222	+	0.948685i	$0.397586\pi$
$192$	0	0
$193$	3.38192	0.243436	0.121718	−	0.992565i	$-0.461160\pi$
$193$	3.38192	0.243436	0.121718	+	0.992565i	$0.461160\pi$
$194$	0	0
$195$	9.41765	0.674412
$196$	0	0
$197$	25.4934	1.81633	0.908164	−	0.418614i	$-0.137484\pi$
$197$	25.4934	1.81633	0.908164	+	0.418614i	$0.137484\pi$
$198$	0	0
$199$	7.06669	0.500945	0.250472	−	0.968124i	$-0.419414\pi$
$199$	7.06669	0.500945	0.250472	+	0.968124i	$0.419414\pi$
$200$	0	0
$201$	−13.4272	−0.947082
$202$	0	0
$203$	−20.1959	−1.41747
$204$	0	0
$205$	5.69631	0.397847
$206$	0	0
$207$	−6.30522	−0.438243
$208$	0	0
$209$	−6.21558	−0.429941
$210$	0	0
$211$	14.2330	0.979842	0.489921	−	0.871767i	$-0.337026\pi$
$211$	14.2330	0.979842	0.489921	+	0.871767i	$0.337026\pi$
$212$	0	0
$213$	12.2979	0.842635
$214$	0	0
$215$	5.04294	0.343925
$216$	0	0
$217$	−44.4354	−3.01647
$218$	0	0
$219$	−23.1842	−1.56664
$220$	0	0
$221$	25.3990	1.70852
$222$	0	0
$223$	3.11207	0.208400	0.104200	−	0.994556i	$-0.466772\pi$
$223$	3.11207	0.208400	0.104200	+	0.994556i	$0.466772\pi$
$224$	0	0
$225$	1.41950	0.0946332
$226$	0	0
$227$	3.57158	0.237054	0.118527	−	0.992951i	$-0.462183\pi$
$227$	3.57158	0.237054	0.118527	+	0.992951i	$0.462183\pi$
$228$	0	0
$229$	5.99340	0.396055	0.198027	−	0.980196i	$-0.436547\pi$
$229$	5.99340	0.396055	0.198027	+	0.980196i	$0.436547\pi$
$230$	0	0
$231$	−45.9889	−3.02584
$232$	0	0
$233$	−4.75231	−0.311334	−0.155667	−	0.987810i	$-0.549753\pi$
$233$	−4.75231	−0.311334	−0.155667	+	0.987810i	$0.549753\pi$
$234$	0	0
$235$	8.76271	0.571616
$236$	0	0
$237$	16.5899	1.07763
$238$	0	0
$239$	15.9181	1.02965	0.514827	−	0.857294i	$-0.327856\pi$
$239$	15.9181	1.02965	0.514827	+	0.857294i	$0.327856\pi$
$240$	0	0
$241$	12.3726	0.796987	0.398493	−	0.917171i	$-0.369533\pi$
$241$	12.3726	0.796987	0.398493	+	0.917171i	$0.369533\pi$
$242$	0	0
$243$	13.6689	0.876862
$244$	0	0
$245$	15.3444	0.980321
$246$	0	0
$247$	6.01668	0.382832
$248$	0	0
$249$	−32.3496	−2.05007
$250$	0	0
$251$	28.0777	1.77225	0.886125	−	0.463447i	$-0.153388\pi$
$251$	28.0777	1.77225	0.886125	+	0.463447i	$0.153388\pi$
$252$	0	0
$253$	−20.5564	−1.29237
$254$	0	0
$255$	11.9192	0.746408
$256$	0	0
$257$	−8.05379	−0.502382	−0.251191	−	0.967938i	$-0.580822\pi$
$257$	−8.05379	−0.502382	−0.251191	+	0.967938i	$0.580822\pi$
$258$	0	0
$259$	32.5734	2.02401
$260$	0	0
$261$	−6.06475	−0.375398
$262$	0	0
$263$	−28.3461	−1.74790	−0.873949	−	0.486018i	$-0.838449\pi$
$263$	−28.3461	−1.74790	−0.873949	+	0.486018i	$0.838449\pi$
$264$	0	0
$265$	−8.23538	−0.505895
$266$	0	0
$267$	−2.10226	−0.128656
$268$	0	0
$269$	−4.54674	−0.277220	−0.138610	−	0.990347i	$-0.544263\pi$
$269$	−4.54674	−0.277220	−0.138610	+	0.990347i	$0.544263\pi$
$270$	0	0
$271$	−12.5691	−0.763521	−0.381760	−	0.924261i	$-0.624682\pi$
$271$	−12.5691	−0.763521	−0.381760	+	0.924261i	$0.624682\pi$
$272$	0	0
$273$	44.5172	2.69430
$274$	0	0
$275$	4.62787	0.279071
$276$	0	0
$277$	16.6171	0.998426	0.499213	−	0.866479i	$-0.333622\pi$
$277$	16.6171	0.998426	0.499213	+	0.866479i	$0.333622\pi$
$278$	0	0
$279$	−13.3438	−0.798871
$280$	0	0
$281$	−32.3979	−1.93270	−0.966348	−	0.257237i	$-0.917188\pi$
$281$	−32.3979	−1.93270	−0.966348	+	0.257237i	$0.917188\pi$
$282$	0	0
$283$	−1.59565	−0.0948517	−0.0474258	−	0.998875i	$-0.515102\pi$
$283$	−1.59565	−0.0948517	−0.0474258	+	0.998875i	$0.515102\pi$
$284$	0	0
$285$	2.82349	0.167249
$286$	0	0
$287$	26.9264	1.58942
$288$	0	0
$289$	15.1454	0.890908
$290$	0	0
$291$	1.41707	0.0830698
$292$	0	0
$293$	12.2986	0.718491	0.359246	−	0.933243i	$-0.383034\pi$
$293$	12.2986	0.718491	0.359246	+	0.933243i	$0.383034\pi$
$294$	0	0
$295$	7.52173	0.437932
$296$	0	0
$297$	15.3767	0.892246
$298$	0	0
$299$	19.8986	1.15076
$300$	0	0
$301$	23.8379	1.37399
$302$	0	0
$303$	−28.2558	−1.62325
$304$	0	0
$305$	−6.94612	−0.397734
$306$	0	0
$307$	25.7367	1.46887	0.734436	−	0.678678i	$-0.237447\pi$
$307$	25.7367	1.46887	0.734436	+	0.678678i	$0.237447\pi$
$308$	0	0
$309$	−23.9709	−1.36366
$310$	0	0
$311$	−8.88817	−0.504002	−0.252001	−	0.967727i	$-0.581089\pi$
$311$	−8.88817	−0.504002	−0.252001	+	0.967727i	$0.581089\pi$
$312$	0	0
$313$	12.2041	0.689818	0.344909	−	0.938636i	$-0.387910\pi$
$313$	12.2041	0.689818	0.344909	+	0.938636i	$0.387910\pi$
$314$	0	0
$315$	6.70996	0.378063
$316$	0	0
$317$	14.6948	0.825341	0.412671	−	0.910880i	$-0.364596\pi$
$317$	14.6948	0.825341	0.412671	+	0.910880i	$0.364596\pi$
$318$	0	0
$319$	−19.7724	−1.10704
$320$	0	0
$321$	−17.9097	−0.999620
$322$	0	0
$323$	7.61483	0.423700
$324$	0	0
$325$	−4.47978	−0.248493
$326$	0	0
$327$	−26.7365	−1.47853
$328$	0	0
$329$	41.4213	2.28363
$330$	0	0
$331$	18.4337	1.01321	0.506603	−	0.862179i	$-0.330901\pi$
$331$	18.4337	1.01321	0.506603	+	0.862179i	$0.330901\pi$
$332$	0	0
$333$	9.78166	0.536032
$334$	0	0
$335$	6.38703	0.348961
$336$	0	0
$337$	−31.8582	−1.73543	−0.867714	−	0.497064i	$-0.834411\pi$
$337$	−31.8582	−1.73543	−0.867714	+	0.497064i	$0.834411\pi$
$338$	0	0
$339$	−17.3546	−0.942574
$340$	0	0
$341$	−43.5036	−2.35585
$342$	0	0
$343$	39.4441	2.12978
$344$	0	0
$345$	9.33796	0.502739
$346$	0	0
$347$	−5.94469	−0.319128	−0.159564	−	0.987188i	$-0.551009\pi$
$347$	−5.94469	−0.319128	−0.159564	+	0.987188i	$0.551009\pi$
$348$	0	0
$349$	12.4803	0.668053	0.334026	−	0.942564i	$-0.391593\pi$
$349$	12.4803	0.668053	0.334026	+	0.942564i	$0.391593\pi$
$350$	0	0
$351$	−14.8846	−0.794482
$352$	0	0
$353$	17.7880	0.946761	0.473380	−	0.880858i	$-0.343034\pi$
$353$	17.7880	0.946761	0.473380	+	0.880858i	$0.343034\pi$
$354$	0	0
$355$	−5.84983	−0.310476
$356$	0	0
$357$	56.3418	2.98193
$358$	0	0
$359$	−6.80968	−0.359401	−0.179700	−	0.983721i	$-0.557513\pi$
$359$	−6.80968	−0.359401	−0.179700	+	0.983721i	$0.557513\pi$
$360$	0	0
$361$	−17.1961	−0.905060
$362$	0	0
$363$	−21.8997	−1.14943
$364$	0	0
$365$	11.0282	0.577244
$366$	0	0
$367$	10.9282	0.570447	0.285224	−	0.958461i	$-0.407932\pi$
$367$	10.9282	0.570447	0.285224	+	0.958461i	$0.407932\pi$
$368$	0	0
$369$	8.08590	0.420935
$370$	0	0
$371$	−38.9285	−2.02107
$372$	0	0
$373$	−6.11907	−0.316834	−0.158417	−	0.987372i	$-0.550639\pi$
$373$	−6.11907	−0.316834	−0.158417	+	0.987372i	$0.550639\pi$
$374$	0	0
$375$	−2.10226	−0.108560
$376$	0	0
$377$	19.1397	0.985742
$378$	0	0
$379$	−28.8933	−1.48415	−0.742074	−	0.670318i	$-0.766158\pi$
$379$	−28.8933	−1.48415	−0.742074	+	0.670318i	$0.766158\pi$
$380$	0	0
$381$	3.72305	0.190738
$382$	0	0
$383$	15.9953	0.817322	0.408661	−	0.912686i	$-0.365996\pi$
$383$	15.9953	0.817322	0.408661	+	0.912686i	$0.365996\pi$
$384$	0	0
$385$	21.8759	1.11490
$386$	0	0
$387$	7.15844	0.363884
$388$	0	0
$389$	30.0561	1.52390	0.761952	−	0.647633i	$-0.224241\pi$
$389$	30.0561	1.52390	0.761952	+	0.647633i	$0.224241\pi$
$390$	0	0
$391$	25.1840	1.27361
$392$	0	0
$393$	−15.3205	−0.772816
$394$	0	0
$395$	−7.89145	−0.397062
$396$	0	0
$397$	25.5045	1.28004	0.640018	−	0.768360i	$-0.278927\pi$
$397$	25.5045	1.28004	0.640018	+	0.768360i	$0.278927\pi$
$398$	0	0
$399$	13.3466	0.668167
$400$	0	0
$401$	−19.4677	−0.972172	−0.486086	−	0.873911i	$-0.661576\pi$
$401$	−19.4677	−0.972172	−0.486086	+	0.873911i	$0.661576\pi$
$402$	0	0
$403$	42.1114	2.09772
$404$	0	0
$405$	−11.2435	−0.558695
$406$	0	0
$407$	31.8903	1.58075
$408$	0	0
$409$	23.2110	1.14771	0.573854	−	0.818957i	$-0.305448\pi$
$409$	23.2110	1.14771	0.573854	+	0.818957i	$0.305448\pi$
$410$	0	0
$411$	−1.58705	−0.0782833
$412$	0	0
$413$	35.5552	1.74956
$414$	0	0
$415$	15.3880	0.755368
$416$	0	0
$417$	−13.2424	−0.648483
$418$	0	0
$419$	22.3096	1.08990	0.544948	−	0.838470i	$-0.316549\pi$
$419$	22.3096	1.08990	0.544948	+	0.838470i	$0.316549\pi$
$420$	0	0
$421$	−29.2036	−1.42330	−0.711648	−	0.702536i	$-0.752051\pi$
$421$	−29.2036	−1.42330	−0.711648	+	0.702536i	$0.752051\pi$
$422$	0	0
$423$	12.4387	0.604788
$424$	0	0
$425$	−5.66969	−0.275021
$426$	0	0
$427$	−32.8343	−1.58896
$428$	0	0
$429$	43.5837	2.10424
$430$	0	0
$431$	−14.8466	−0.715134	−0.357567	−	0.933888i	$-0.616394\pi$
$431$	−14.8466	−0.715134	−0.357567	+	0.933888i	$0.616394\pi$
$432$	0	0
$433$	−13.0139	−0.625408	−0.312704	−	0.949851i	$-0.601235\pi$
$433$	−13.0139	−0.625408	−0.312704	+	0.949851i	$0.601235\pi$
$434$	0	0
$435$	8.98182	0.430645
$436$	0	0
$437$	5.96576	0.285381
$438$	0	0
$439$	12.1861	0.581609	0.290804	−	0.956783i	$-0.406077\pi$
$439$	12.1861	0.581609	0.290804	+	0.956783i	$0.406077\pi$
$440$	0	0
$441$	21.7814	1.03721
$442$	0	0
$443$	1.82580	0.0867465	0.0433732	−	0.999059i	$-0.486190\pi$
$443$	1.82580	0.0867465	0.0433732	+	0.999059i	$0.486190\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	0	0
$447$	37.2341	1.76111
$448$	0	0
$449$	−10.9646	−0.517449	−0.258725	−	0.965951i	$-0.583302\pi$
$449$	−10.9646	−0.517449	−0.258725	+	0.965951i	$0.583302\pi$
$450$	0	0
$451$	26.3618	1.24133
$452$	0	0
$453$	21.8615	1.02714
$454$	0	0
$455$	−21.1759	−0.992740
$456$	0	0
$457$	4.14517	0.193903	0.0969514	−	0.995289i	$-0.469091\pi$
$457$	4.14517	0.193903	0.0969514	+	0.995289i	$0.469091\pi$
$458$	0	0
$459$	−18.8383	−0.879295
$460$	0	0
$461$	−35.6431	−1.66006	−0.830031	−	0.557717i	$-0.811678\pi$
$461$	−35.6431	−1.66006	−0.830031	+	0.557717i	$0.811678\pi$
$462$	0	0
$463$	3.08964	0.143588	0.0717939	−	0.997419i	$-0.477128\pi$
$463$	3.08964	0.143588	0.0717939	+	0.997419i	$0.477128\pi$
$464$	0	0
$465$	19.7620	0.916440
$466$	0	0
$467$	5.13078	0.237424	0.118712	−	0.992929i	$-0.462123\pi$
$467$	5.13078	0.237424	0.118712	+	0.992929i	$0.462123\pi$
$468$	0	0
$469$	30.1914	1.39411
$470$	0	0
$471$	25.3480	1.16797
$472$	0	0
$473$	23.3381	1.07309
$474$	0	0
$475$	−1.34308	−0.0616245
$476$	0	0
$477$	−11.6901	−0.535253
$478$	0	0
$479$	−6.76545	−0.309121	−0.154561	−	0.987983i	$-0.549396\pi$
$479$	−6.76545	−0.309121	−0.154561	+	0.987983i	$0.549396\pi$
$480$	0	0
$481$	−30.8698	−1.40754
$482$	0	0
$483$	44.1405	2.00846
$484$	0	0
$485$	−0.674067	−0.0306078
$486$	0	0
$487$	0.747648	0.0338792	0.0169396	−	0.999857i	$-0.494608\pi$
$487$	0.747648	0.0338792	0.0169396	+	0.999857i	$0.494608\pi$
$488$	0	0
$489$	14.2242	0.643240
$490$	0	0
$491$	−22.1110	−0.997857	−0.498928	−	0.866643i	$-0.666273\pi$
$491$	−22.1110	−0.997857	−0.498928	+	0.866643i	$0.666273\pi$
$492$	0	0
$493$	24.2235	1.09097
$494$	0	0
$495$	6.56926	0.295266
$496$	0	0
$497$	−27.6521	−1.24036
$498$	0	0
$499$	35.1425	1.57319	0.786597	−	0.617467i	$-0.211841\pi$
$499$	35.1425	1.57319	0.786597	+	0.617467i	$0.211841\pi$
$500$	0	0
$501$	−10.6947	−0.477802
$502$	0	0
$503$	14.8767	0.663320	0.331660	−	0.943399i	$-0.392391\pi$
$503$	14.8767	0.663320	0.331660	+	0.943399i	$0.392391\pi$
$504$	0	0
$505$	13.4407	0.598103
$506$	0	0
$507$	−14.8596	−0.659938
$508$	0	0
$509$	−13.5925	−0.602476	−0.301238	−	0.953549i	$-0.597400\pi$
$509$	−13.5925	−0.602476	−0.301238	+	0.953549i	$0.597400\pi$
$510$	0	0
$511$	52.1303	2.30611
$512$	0	0
$513$	−4.46254	−0.197026
$514$	0	0
$515$	11.4024	0.502451
$516$	0	0
$517$	40.5527	1.78351
$518$	0	0
$519$	−3.43502	−0.150780
$520$	0	0
$521$	21.2471	0.930851	0.465425	−	0.885087i	$-0.345901\pi$
$521$	21.2471	0.930851	0.465425	+	0.885087i	$0.345901\pi$
$522$	0	0
$523$	−37.9681	−1.66023	−0.830115	−	0.557592i	$-0.811725\pi$
$523$	−37.9681	−1.66023	−0.830115	+	0.557592i	$0.811725\pi$
$524$	0	0
$525$	−9.93737	−0.433702
$526$	0	0
$527$	53.2971	2.32166
$528$	0	0
$529$	−3.26981	−0.142166
$530$	0	0
$531$	10.6771	0.463346
$532$	0	0
$533$	−25.5182	−1.10532
$534$	0	0
$535$	8.51924	0.368319
$536$	0	0
$537$	−18.7044	−0.807154
$538$	0	0
$539$	71.0121	3.05871
$540$	0	0
$541$	24.0786	1.03522	0.517609	−	0.855617i	$-0.326822\pi$
$541$	24.0786	1.03522	0.517609	+	0.855617i	$0.326822\pi$
$542$	0	0
$543$	15.0280	0.644915
$544$	0	0
$545$	12.7180	0.544779
$546$	0	0
$547$	7.90482	0.337986	0.168993	−	0.985617i	$-0.445948\pi$
$547$	7.90482	0.337986	0.168993	+	0.985617i	$0.445948\pi$
$548$	0	0
$549$	−9.86001	−0.420815
$550$	0	0
$551$	5.73823	0.244457
$552$	0	0
$553$	−37.3028	−1.58628
$554$	0	0
$555$	−14.4865	−0.614919
$556$	0	0
$557$	−28.0206	−1.18727	−0.593635	−	0.804734i	$-0.702308\pi$
$557$	−28.0206	−1.18727	−0.593635	+	0.804734i	$0.702308\pi$
$558$	0	0
$559$	−22.5912	−0.955507
$560$	0	0
$561$	55.1604	2.32887
$562$	0	0
$563$	41.0773	1.73120	0.865601	−	0.500735i	$-0.166937\pi$
$563$	41.0773	1.73120	0.865601	+	0.500735i	$0.166937\pi$
$564$	0	0
$565$	8.25522	0.347300
$566$	0	0
$567$	−53.1480	−2.23201
$568$	0	0
$569$	−30.8521	−1.29339	−0.646693	−	0.762751i	$-0.723848\pi$
$569$	−30.8521	−1.29339	−0.646693	+	0.762751i	$0.723848\pi$
$570$	0	0
$571$	−27.2466	−1.14023	−0.570117	−	0.821564i	$-0.693102\pi$
$571$	−27.2466	−1.14023	−0.570117	+	0.821564i	$0.693102\pi$
$572$	0	0
$573$	−18.3749	−0.767623
$574$	0	0
$575$	−4.44187	−0.185239
$576$	0	0
$577$	26.1936	1.09045	0.545227	−	0.838288i	$-0.316443\pi$
$577$	26.1936	1.09045	0.545227	+	0.838288i	$0.316443\pi$
$578$	0	0
$579$	−7.10968	−0.295468
$580$	0	0
$581$	72.7390	3.01772
$582$	0	0
$583$	−38.1123	−1.57845
$584$	0	0
$585$	−6.35903	−0.262914
$586$	0	0
$587$	16.6599	0.687627	0.343814	−	0.939038i	$-0.388281\pi$
$587$	16.6599	0.687627	0.343814	+	0.939038i	$0.388281\pi$
$588$	0	0
$589$	12.6254	0.520219
$590$	0	0
$591$	−53.5937	−2.20455
$592$	0	0
$593$	−30.7704	−1.26359	−0.631794	−	0.775136i	$-0.717681\pi$
$593$	−30.7704	−1.26359	−0.631794	+	0.775136i	$0.717681\pi$
$594$	0	0
$595$	−26.8006	−1.09872
$596$	0	0
$597$	−14.8560	−0.608017
$598$	0	0
$599$	24.7821	1.01257	0.506285	−	0.862366i	$-0.331019\pi$
$599$	24.7821	1.01257	0.506285	+	0.862366i	$0.331019\pi$
$600$	0	0
$601$	2.25624	0.0920340	0.0460170	−	0.998941i	$-0.485347\pi$
$601$	2.25624	0.0920340	0.0460170	+	0.998941i	$0.485347\pi$
$602$	0	0
$603$	9.06638	0.369212
$604$	0	0
$605$	10.4172	0.423519
$606$	0	0
$607$	−20.1084	−0.816175	−0.408088	−	0.912943i	$-0.633804\pi$
$607$	−20.1084	−0.816175	−0.408088	+	0.912943i	$0.633804\pi$
$608$	0	0
$609$	42.4570	1.72044
$610$	0	0
$611$	−39.2550	−1.58809
$612$	0	0
$613$	−14.9346	−0.603204	−0.301602	−	0.953434i	$-0.597521\pi$
$613$	−14.9346	−0.603204	−0.301602	+	0.953434i	$0.597521\pi$
$614$	0	0
$615$	−11.9751	−0.482884
$616$	0	0
$617$	17.1073	0.688714	0.344357	−	0.938839i	$-0.388097\pi$
$617$	17.1073	0.688714	0.344357	+	0.938839i	$0.388097\pi$
$618$	0	0
$619$	−39.8548	−1.60190	−0.800950	−	0.598732i	$-0.795672\pi$
$619$	−39.8548	−1.60190	−0.800950	+	0.598732i	$0.795672\pi$
$620$	0	0
$621$	−14.7587	−0.592245
$622$	0	0
$623$	4.72699	0.189383
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	13.0668	0.521836
$628$	0	0
$629$	−39.0695	−1.55780
$630$	0	0
$631$	1.26371	0.0503077	0.0251538	−	0.999684i	$-0.491992\pi$
$631$	1.26371	0.0503077	0.0251538	+	0.999684i	$0.491992\pi$
$632$	0	0
$633$	−29.9215	−1.18927
$634$	0	0
$635$	−1.77097	−0.0702790
$636$	0	0
$637$	−68.7397	−2.72356
$638$	0	0
$639$	−8.30382	−0.328494
$640$	0	0
$641$	10.4319	0.412037	0.206018	−	0.978548i	$-0.433949\pi$
$641$	10.4319	0.412037	0.206018	+	0.978548i	$0.433949\pi$
$642$	0	0
$643$	−35.6759	−1.40692	−0.703460	−	0.710735i	$-0.748363\pi$
$643$	−35.6759	−1.40692	−0.703460	+	0.710735i	$0.748363\pi$
$644$	0	0
$645$	−10.6016	−0.417436
$646$	0	0
$647$	27.5263	1.08217	0.541085	−	0.840968i	$-0.318014\pi$
$647$	27.5263	1.08217	0.541085	+	0.840968i	$0.318014\pi$
$648$	0	0
$649$	34.8096	1.36640
$650$	0	0
$651$	93.4147	3.66121
$652$	0	0
$653$	−1.79316	−0.0701718	−0.0350859	−	0.999384i	$-0.511170\pi$
$653$	−1.79316	−0.0701718	−0.0350859	+	0.999384i	$0.511170\pi$
$654$	0	0
$655$	7.28762	0.284751
$656$	0	0
$657$	15.6545	0.610742
$658$	0	0
$659$	−5.13756	−0.200131	−0.100065	−	0.994981i	$-0.531905\pi$
$659$	−5.13756	−0.200131	−0.100065	+	0.994981i	$0.531905\pi$
$660$	0	0
$661$	−18.0419	−0.701748	−0.350874	−	0.936423i	$-0.614115\pi$
$661$	−18.0419	−0.701748	−0.350874	+	0.936423i	$0.614115\pi$
$662$	0	0
$663$	−53.3952	−2.07370
$664$	0	0
$665$	−6.34870	−0.246192
$666$	0	0
$667$	18.9777	0.734819
$668$	0	0
$669$	−6.54238	−0.252943
$670$	0	0
$671$	−32.1458	−1.24097
$672$	0	0
$673$	−1.02851	−0.0396461	−0.0198231	−	0.999804i	$-0.506310\pi$
$673$	−1.02851	−0.0396461	−0.0198231	+	0.999804i	$0.506310\pi$
$674$	0	0
$675$	3.32263	0.127888
$676$	0	0
$677$	36.0528	1.38562	0.692812	−	0.721119i	$-0.256372\pi$
$677$	36.0528	1.38562	0.692812	+	0.721119i	$0.256372\pi$
$678$	0	0
$679$	−3.18631	−0.122279
$680$	0	0
$681$	−7.50839	−0.287722
$682$	0	0
$683$	−14.0836	−0.538895	−0.269447	−	0.963015i	$-0.586841\pi$
$683$	−14.0836	−0.538895	−0.269447	+	0.963015i	$0.586841\pi$
$684$	0	0
$685$	0.754924	0.0288442
$686$	0	0
$687$	−12.5997	−0.480708
$688$	0	0
$689$	36.8926	1.40550
$690$	0	0
$691$	−15.6187	−0.594163	−0.297081	−	0.954852i	$-0.596013\pi$
$691$	−15.6187	−0.594163	−0.297081	+	0.954852i	$0.596013\pi$
$692$	0	0
$693$	31.0528	1.17960
$694$	0	0
$695$	6.29913	0.238939
$696$	0	0
$697$	−32.2963	−1.22331
$698$	0	0
$699$	9.99060	0.377879
$700$	0	0
$701$	23.8768	0.901814	0.450907	−	0.892571i	$-0.351101\pi$
$701$	23.8768	0.901814	0.450907	+	0.892571i	$0.351101\pi$
$702$	0	0
$703$	−9.25504	−0.349060
$704$	0	0
$705$	−18.4215	−0.693794
$706$	0	0
$707$	63.5340	2.38944
$708$	0	0
$709$	−13.7608	−0.516798	−0.258399	−	0.966038i	$-0.583195\pi$
$709$	−13.7608	−0.516798	−0.258399	+	0.966038i	$0.583195\pi$
$710$	0	0
$711$	−11.2019	−0.420104
$712$	0	0
$713$	41.7551	1.56374
$714$	0	0
$715$	−20.7318	−0.775326
$716$	0	0
$717$	−33.4639	−1.24973
$718$	0	0
$719$	18.7708	0.700032	0.350016	−	0.936744i	$-0.386176\pi$
$719$	18.7708	0.700032	0.350016	+	0.936744i	$0.386176\pi$
$720$	0	0
$721$	53.8992	2.00731
$722$	0	0
$723$	−26.0103	−0.967335
$724$	0	0
$725$	−4.27246	−0.158675
$726$	0	0
$727$	−26.9412	−0.999192	−0.499596	−	0.866258i	$-0.666518\pi$
$727$	−26.9412	−0.999192	−0.499596	+	0.866258i	$0.666518\pi$
$728$	0	0
$729$	4.99491	0.184997
$730$	0	0
$731$	−28.5919	−1.05751
$732$	0	0
$733$	17.5711	0.649004	0.324502	−	0.945885i	$-0.394803\pi$
$733$	17.5711	0.649004	0.324502	+	0.945885i	$0.394803\pi$
$734$	0	0
$735$	−32.2580	−1.18985
$736$	0	0
$737$	29.5584	1.08880
$738$	0	0
$739$	−11.5994	−0.426689	−0.213345	−	0.976977i	$-0.568436\pi$
$739$	−11.5994	−0.426689	−0.213345	+	0.976977i	$0.568436\pi$
$740$	0	0
$741$	−12.6486	−0.464659
$742$	0	0
$743$	−8.10055	−0.297180	−0.148590	−	0.988899i	$-0.547474\pi$
$743$	−8.10055	−0.297180	−0.148590	+	0.988899i	$0.547474\pi$
$744$	0	0
$745$	−17.7115	−0.648898
$746$	0	0
$747$	21.8433	0.799203
$748$	0	0
$749$	40.2704	1.47145
$750$	0	0
$751$	−2.23998	−0.0817381	−0.0408690	−	0.999165i	$-0.513013\pi$
$751$	−2.23998	−0.0817381	−0.0408690	+	0.999165i	$0.513013\pi$
$752$	0	0
$753$	−59.0267	−2.15105
$754$	0	0
$755$	−10.3991	−0.378461
$756$	0	0
$757$	−13.5325	−0.491846	−0.245923	−	0.969289i	$-0.579091\pi$
$757$	−13.5325	−0.491846	−0.245923	+	0.969289i	$0.579091\pi$
$758$	0	0
$759$	43.2149	1.56860
$760$	0	0
$761$	37.5201	1.36010	0.680051	−	0.733165i	$-0.261958\pi$
$761$	37.5201	1.36010	0.680051	+	0.733165i	$0.261958\pi$
$762$	0	0
$763$	60.1178	2.17641
$764$	0	0
$765$	−8.04812	−0.290980
$766$	0	0
$767$	−33.6957	−1.21668
$768$	0	0
$769$	7.95922	0.287017	0.143508	−	0.989649i	$-0.454162\pi$
$769$	7.95922	0.287017	0.143508	+	0.989649i	$0.454162\pi$
$770$	0	0
$771$	16.9312	0.609761
$772$	0	0
$773$	40.5378	1.45804	0.729021	−	0.684491i	$-0.239976\pi$
$773$	40.5378	1.45804	0.729021	+	0.684491i	$0.239976\pi$
$774$	0	0
$775$	−9.40035	−0.337670
$776$	0	0
$777$	−68.4777	−2.45662
$778$	0	0
$779$	−7.65057	−0.274110
$780$	0	0
$781$	−27.0722	−0.968721
$782$	0	0
$783$	−14.1958	−0.507316
$784$	0	0
$785$	−12.0575	−0.430350
$786$	0	0
$787$	39.0735	1.39282	0.696409	−	0.717645i	$-0.254780\pi$
$787$	39.0735	1.39282	0.696409	+	0.717645i	$0.254780\pi$
$788$	0	0
$789$	59.5910	2.12149
$790$	0	0
$791$	39.0224	1.38748
$792$	0	0
$793$	31.1171	1.10500
$794$	0	0
$795$	17.3129	0.614025
$796$	0	0
$797$	−1.92101	−0.0680456	−0.0340228	−	0.999421i	$-0.510832\pi$
$797$	−1.92101	−0.0680456	−0.0340228	+	0.999421i	$0.510832\pi$
$798$	0	0
$799$	−49.6819	−1.75762
$800$	0	0
$801$	1.41950	0.0501555
$802$	0	0
$803$	51.0372	1.80106
$804$	0	0
$805$	−20.9967	−0.740035
$806$	0	0
$807$	9.55843	0.336473
$808$	0	0
$809$	−35.3476	−1.24276	−0.621378	−	0.783511i	$-0.713427\pi$
$809$	−35.3476	−1.24276	−0.621378	+	0.783511i	$0.713427\pi$
$810$	0	0
$811$	42.5750	1.49501	0.747506	−	0.664255i	$-0.231251\pi$
$811$	42.5750	1.49501	0.747506	+	0.664255i	$0.231251\pi$
$812$	0	0
$813$	26.4236	0.926716
$814$	0	0
$815$	−6.76614	−0.237007
$816$	0	0
$817$	−6.77304	−0.236959
$818$	0	0
$819$	−30.0591	−1.05035
$820$	0	0
$821$	−15.8671	−0.553766	−0.276883	−	0.960904i	$-0.589301\pi$
$821$	−15.8671	−0.553766	−0.276883	+	0.960904i	$0.589301\pi$
$822$	0	0
$823$	13.2314	0.461217	0.230609	−	0.973047i	$-0.425928\pi$
$823$	13.2314	0.461217	0.230609	+	0.973047i	$0.425928\pi$
$824$	0	0
$825$	−9.72899	−0.338720
$826$	0	0
$827$	26.0034	0.904228	0.452114	−	0.891960i	$-0.350670\pi$
$827$	26.0034	0.904228	0.452114	+	0.891960i	$0.350670\pi$
$828$	0	0
$829$	−4.11404	−0.142886	−0.0714432	−	0.997445i	$-0.522760\pi$
$829$	−4.11404	−0.142886	−0.0714432	+	0.997445i	$0.522760\pi$
$830$	0	0
$831$	−34.9335	−1.21183
$832$	0	0
$833$	−86.9983	−3.01431
$834$	0	0
$835$	5.08722	0.176050
$836$	0	0
$837$	−31.2338	−1.07960
$838$	0	0
$839$	−34.8946	−1.20469	−0.602347	−	0.798234i	$-0.705768\pi$
$839$	−34.8946	−1.20469	−0.602347	+	0.798234i	$0.705768\pi$
$840$	0	0
$841$	−10.7461	−0.370555
$842$	0	0
$843$	68.1088	2.34579
$844$	0	0
$845$	7.06839	0.243160
$846$	0	0
$847$	49.2420	1.69198
$848$	0	0
$849$	3.35448	0.115125
$850$	0	0
$851$	−30.6086	−1.04925
$852$	0	0
$853$	2.21046	0.0756847	0.0378423	−	0.999284i	$-0.487952\pi$
$853$	2.21046	0.0756847	0.0378423	+	0.999284i	$0.487952\pi$
$854$	0	0
$855$	−1.90649	−0.0652007
$856$	0	0
$857$	0.0315501	0.00107773	0.000538865	−	1.00000i	$-0.499828\pi$
$857$	0.0315501	0.00107773	0.000538865		1.00000i	$0.499828\pi$
$858$	0	0
$859$	−47.3550	−1.61573	−0.807867	−	0.589365i	$-0.799378\pi$
$859$	−47.3550	−1.61573	−0.807867	+	0.589365i	$0.799378\pi$
$860$	0	0
$861$	−56.6063	−1.92914
$862$	0	0
$863$	−2.08780	−0.0710696	−0.0355348	−	0.999368i	$-0.511313\pi$
$863$	−2.08780	−0.0710696	−0.0355348	+	0.999368i	$0.511313\pi$
$864$	0	0
$865$	1.63396	0.0555564
$866$	0	0
$867$	−31.8396	−1.08133
$868$	0	0
$869$	−36.5206	−1.23888
$870$	0	0
$871$	−28.6125	−0.969496
$872$	0	0
$873$	−0.956838	−0.0323840
$874$	0	0
$875$	4.72699	0.159801
$876$	0	0
$877$	−49.8979	−1.68493	−0.842467	−	0.538748i	$-0.818898\pi$
$877$	−49.8979	−1.68493	−0.842467	+	0.538748i	$0.818898\pi$
$878$	0	0
$879$	−25.8548	−0.872062
$880$	0	0
$881$	35.1975	1.18583	0.592917	−	0.805264i	$-0.297976\pi$
$881$	35.1975	1.18583	0.592917	+	0.805264i	$0.297976\pi$
$882$	0	0
$883$	47.7239	1.60604	0.803019	−	0.595953i	$-0.203226\pi$
$883$	47.7239	1.60604	0.803019	+	0.595953i	$0.203226\pi$
$884$	0	0
$885$	−15.8126	−0.531536
$886$	0	0
$887$	20.4982	0.688263	0.344132	−	0.938921i	$-0.388173\pi$
$887$	20.4982	0.688263	0.344132	+	0.938921i	$0.388173\pi$
$888$	0	0
$889$	−8.37138	−0.280767
$890$	0	0
$891$	−52.0336	−1.74319
$892$	0	0
$893$	−11.7690	−0.393834
$894$	0	0
$895$	8.89727	0.297403
$896$	0	0
$897$	−41.8320	−1.39673
$898$	0	0
$899$	40.1626	1.33950
$900$	0	0
$901$	46.6921	1.55554
$902$	0	0
$903$	−50.1135	−1.66767
$904$	0	0
$905$	−7.14852	−0.237625
$906$	0	0
$907$	−16.1289	−0.535551	−0.267775	−	0.963481i	$-0.586288\pi$
$907$	−16.1289	−0.535551	−0.267775	+	0.963481i	$0.586288\pi$
$908$	0	0
$909$	19.0790	0.632811
$910$	0	0
$911$	5.86877	0.194441	0.0972204	−	0.995263i	$-0.469005\pi$
$911$	5.86877	0.194441	0.0972204	+	0.995263i	$0.469005\pi$
$912$	0	0
$913$	71.2138	2.35683
$914$	0	0
$915$	14.6026	0.482746
$916$	0	0
$917$	34.4485	1.13759
$918$	0	0
$919$	52.0408	1.71667	0.858333	−	0.513093i	$-0.171500\pi$
$919$	52.0408	1.71667	0.858333	+	0.513093i	$0.171500\pi$
$920$	0	0
$921$	−54.1052	−1.78283
$922$	0	0
$923$	26.2059	0.862578
$924$	0	0
$925$	6.89093	0.226572
$926$	0	0
$927$	16.1857	0.531609
$928$	0	0
$929$	−9.81663	−0.322073	−0.161037	−	0.986948i	$-0.551484\pi$
$929$	−9.81663	−0.322073	−0.161037	+	0.986948i	$0.551484\pi$
$930$	0	0
$931$	−20.6087	−0.675424
$932$	0	0
$933$	18.6853	0.611728
$934$	0	0
$935$	−26.2386	−0.858095
$936$	0	0
$937$	37.5704	1.22737	0.613686	−	0.789550i	$-0.289686\pi$
$937$	37.5704	1.22737	0.613686	+	0.789550i	$0.289686\pi$
$938$	0	0
$939$	−25.6563	−0.837261
$940$	0	0
$941$	−0.0461835	−0.00150554	−0.000752769	−	1.00000i	$-0.500240\pi$
$941$	−0.0461835	−0.00150554	−0.000752769		1.00000i	$0.500240\pi$
$942$	0	0
$943$	−25.3023	−0.823955
$944$	0	0
$945$	15.7060	0.510917
$946$	0	0
$947$	26.0877	0.847735	0.423867	−	0.905724i	$-0.360672\pi$
$947$	26.0877	0.847735	0.423867	+	0.905724i	$0.360672\pi$
$948$	0	0
$949$	−49.4040	−1.60372
$950$	0	0
$951$	−30.8923	−1.00175
$952$	0	0
$953$	35.0541	1.13551	0.567757	−	0.823196i	$-0.307811\pi$
$953$	35.0541	1.13551	0.567757	+	0.823196i	$0.307811\pi$
$954$	0	0
$955$	8.74055	0.282838
$956$	0	0
$957$	41.5667	1.34366
$958$	0	0
$959$	3.56852	0.115234
$960$	0	0
$961$	57.3665	1.85053
$962$	0	0
$963$	12.0931	0.389693
$964$	0	0
$965$	3.38192	0.108868
$966$	0	0
$967$	14.8997	0.479141	0.239570	−	0.970879i	$-0.422993\pi$
$967$	14.8997	0.479141	0.239570	+	0.970879i	$0.422993\pi$
$968$	0	0
$969$	−16.0083	−0.514262
$970$	0	0
$971$	9.37217	0.300767	0.150384	−	0.988628i	$-0.451949\pi$
$971$	9.37217	0.300767	0.150384	+	0.988628i	$0.451949\pi$
$972$	0	0
$973$	29.7759	0.954572
$974$	0	0
$975$	9.41765	0.301606
$976$	0	0
$977$	−10.5242	−0.336698	−0.168349	−	0.985727i	$-0.553844\pi$
$977$	−10.5242	−0.336698	−0.168349	+	0.985727i	$0.553844\pi$
$978$	0	0
$979$	4.62787	0.147907
$980$	0	0
$981$	18.0532	0.576393
$982$	0	0
$983$	1.32593	0.0422905	0.0211453	−	0.999776i	$-0.493269\pi$
$983$	1.32593	0.0422905	0.0211453	+	0.999776i	$0.493269\pi$
$984$	0	0
$985$	25.4934	0.812287
$986$	0	0
$987$	−87.0783	−2.77173
$988$	0	0
$989$	−22.4001	−0.712280
$990$	0	0
$991$	3.04920	0.0968609	0.0484305	−	0.998827i	$-0.484578\pi$
$991$	3.04920	0.0968609	0.0484305	+	0.998827i	$0.484578\pi$
$992$	0	0
$993$	−38.7524	−1.22977
$994$	0	0
$995$	7.06669	0.224029
$996$	0	0
$997$	−25.7222	−0.814630	−0.407315	−	0.913288i	$-0.633535\pi$
$997$	−25.7222	−0.814630	−0.407315	+	0.913288i	$0.633535\pi$
$998$	0	0
$999$	22.8960	0.724397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7120.2.a.bj.1.1		7
4.3	odd	2		445.2.a.f.1.3	✓	7
12.11	even	2		4005.2.a.o.1.5		7
20.19	odd	2		2225.2.a.k.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.f.1.3	✓	7	4.3	odd	2
2225.2.a.k.1.5		7	20.19	odd	2
4005.2.a.o.1.5		7	12.11	even	2
7120.2.a.bj.1.1		7	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:	\( N \)	\(=\)	\( 7120 = 2^{4} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7120.a (trivial)

\(f(q)\)	\(=\)	\(q-2.10226 q^{3} +1.00000 q^{5} +4.72699 q^{7} +1.41950 q^{9} +O(q^{10})\)	\(q-2.10226 q^{3} +1.00000 q^{5} +4.72699 q^{7} +1.41950 q^{9} +4.62787 q^{11} -4.47978 q^{13} -2.10226 q^{15} -5.66969 q^{17} -1.34308 q^{19} -9.93737 q^{21} -4.44187 q^{23} +1.00000 q^{25} +3.32263 q^{27} -4.27246 q^{29} -9.40035 q^{31} -9.72899 q^{33} +4.72699 q^{35} +6.89093 q^{37} +9.41765 q^{39} +5.69631 q^{41} +5.04294 q^{43} +1.41950 q^{45} +8.76271 q^{47} +15.3444 q^{49} +11.9192 q^{51} -8.23538 q^{53} +4.62787 q^{55} +2.82349 q^{57} +7.52173 q^{59} -6.94612 q^{61} +6.70996 q^{63} -4.47978 q^{65} +6.38703 q^{67} +9.33796 q^{69} -5.84983 q^{71} +11.0282 q^{73} -2.10226 q^{75} +21.8759 q^{77} -7.89145 q^{79} -11.2435 q^{81} +15.3880 q^{83} -5.66969 q^{85} +8.98182 q^{87} +1.00000 q^{89} -21.1759 q^{91} +19.7620 q^{93} -1.34308 q^{95} -0.674067 q^{97} +6.56926 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9}+O(q^{10}) \) 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9	\( 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9} + 10 q^{11} - 7 q^{13} + 8 q^{15} - 13 q^{17} + 7 q^{19} + 16 q^{21} + 13 q^{23} + 7 q^{25} + 23 q^{27} - 4 q^{29} - q^{31} - 6 q^{33} + 16 q^{35} - 5 q^{37} + 13 q^{39} + 5 q^{41} + 31 q^{43} + 11 q^{45} + 14 q^{47} + 19 q^{49} + q^{51} - 13 q^{53} + 10 q^{55} + 21 q^{57} + 14 q^{59} + 3 q^{61} + 54 q^{63} - 7 q^{65} - q^{67} + 31 q^{69} + 8 q^{71} + 9 q^{73} + 8 q^{75} + 42 q^{77} - 9 q^{79} + 35 q^{81} + 42 q^{83} - 13 q^{85} - 6 q^{87} + 7 q^{89} - 31 q^{91} + 24 q^{93} + 7 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) \) 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9 + 10 * q^11 - 7 * q^13 + 8 * q^15 - 13 * q^17 + 7 * q^19 + 16 * q^21 + 13 * q^23 + 7 * q^25 + 23 * q^27 - 4 * q^29 - q^31 - 6 * q^33 + 16 * q^35 - 5 * q^37 + 13 * q^39 + 5 * q^41 + 31 * q^43 + 11 * q^45 + 14 * q^47 + 19 * q^49 + q^51 - 13 * q^53 + 10 * q^55 + 21 * q^57 + 14 * q^59 + 3 * q^61 + 54 * q^63 - 7 * q^65 - q^67 + 31 * q^69 + 8 * q^71 + 9 * q^73 + 8 * q^75 + 42 * q^77 - 9 * q^79 + 35 * q^81 + 42 * q^83 - 13 * q^85 - 6 * q^87 + 7 * q^89 - 31 * q^91 + 24 * q^93 + 7 * q^95 - 7 * q^97 - 5 * q^99

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