LMFDB - Embedded newform 7120.2.a.bc.1.3

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:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.09529$ 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)

$f(q)$	$=$	$q+1.29496 q^{3} -1.00000 q^{5} +3.19059 q^{7} -1.32307 q^{9} +O(q^{10})$	$q+1.29496 q^{3} -1.00000 q^{5} +3.19059 q^{7} -1.32307 q^{9} +2.70504 q^{11} +2.03640 q^{13} -1.29496 q^{15} +2.67180 q^{17} -1.45989 q^{19} +4.13169 q^{21} +5.62632 q^{23} +1.00000 q^{25} -5.59822 q^{27} +3.44007 q^{29} +8.62237 q^{31} +3.50292 q^{33} -3.19059 q^{35} +9.46573 q^{37} +2.63706 q^{39} -10.1119 q^{41} -2.51878 q^{43} +1.32307 q^{45} -1.05810 q^{47} +3.17985 q^{49} +3.45989 q^{51} -1.72802 q^{53} -2.70504 q^{55} -1.89050 q^{57} +3.39934 q^{59} -14.9071 q^{61} -4.22137 q^{63} -2.03640 q^{65} +2.16888 q^{67} +7.28588 q^{69} -6.88221 q^{71} -8.79954 q^{73} +1.29496 q^{75} +8.63066 q^{77} -5.97939 q^{79} -3.28027 q^{81} +7.29685 q^{83} -2.67180 q^{85} +4.45476 q^{87} -1.00000 q^{89} +6.49731 q^{91} +11.1656 q^{93} +1.45989 q^{95} +15.6505 q^{97} -3.57896 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 4 * q^9	$ 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9} + 14 q^{11} - 5 q^{13} - 2 q^{15} - 3 q^{17} + q^{19} - 4 q^{21} + 3 q^{23} + 4 q^{25} - q^{27} - 10 q^{29} + 11 q^{31} + 2 q^{35} + 3 q^{37} - 11 q^{39} - 3 q^{41} - 9 q^{43} + 4 q^{45} + 24 q^{47} + 7 q^{51} + 3 q^{53} - 14 q^{55} + 19 q^{57} + 22 q^{59} - 3 q^{61} - 6 q^{63} + 5 q^{65} + 9 q^{67} + 7 q^{69} - 16 q^{71} + 3 q^{73} + 2 q^{75} - 4 q^{77} + 27 q^{79} - 8 q^{81} - 6 q^{83} + 3 q^{85} - 4 q^{87} - 4 q^{89} + 29 q^{91} - 2 q^{93} - q^{95} + 41 q^{97} - 21 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 4 * q^9 + 14 * q^11 - 5 * q^13 - 2 * q^15 - 3 * q^17 + q^19 - 4 * q^21 + 3 * q^23 + 4 * q^25 - q^27 - 10 * q^29 + 11 * q^31 + 2 * q^35 + 3 * q^37 - 11 * q^39 - 3 * q^41 - 9 * q^43 + 4 * q^45 + 24 * q^47 + 7 * q^51 + 3 * q^53 - 14 * q^55 + 19 * q^57 + 22 * q^59 - 3 * q^61 - 6 * q^63 + 5 * q^65 + 9 * q^67 + 7 * q^69 - 16 * q^71 + 3 * q^73 + 2 * q^75 - 4 * q^77 + 27 * q^79 - 8 * q^81 - 6 * q^83 + 3 * q^85 - 4 * q^87 - 4 * q^89 + 29 * q^91 - 2 * q^93 - q^95 + 41 * q^97 - 21 * 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
$3$	1.29496	0.747647	0.373824	−	0.927500i	$-0.378047\pi$
$3$	1.29496	0.747647	0.373824	+	0.927500i	$0.378047\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.19059	1.20593	0.602964	−	0.797768i	$-0.293986\pi$
$7$	3.19059	1.20593	0.602964	+	0.797768i	$0.293986\pi$
$8$	0	0
$9$	−1.32307	−0.441024
$10$	0	0
$11$	2.70504	0.815599	0.407800	−	0.913071i	$-0.366296\pi$
$11$	2.70504	0.815599	0.407800	+	0.913071i	$0.366296\pi$
$12$	0	0
$13$	2.03640	0.564795	0.282398	−	0.959297i	$-0.408870\pi$
$13$	2.03640	0.564795	0.282398	+	0.959297i	$0.408870\pi$
$14$	0	0
$15$	−1.29496	−0.334358
$16$	0	0
$17$	2.67180	0.648008	0.324004	−	0.946056i	$-0.394971\pi$
$17$	2.67180	0.648008	0.324004	+	0.946056i	$0.394971\pi$
$18$	0	0
$19$	−1.45989	−0.334921	−0.167461	−	0.985879i	$-0.553557\pi$
$19$	−1.45989	−0.334921	−0.167461	+	0.985879i	$0.553557\pi$
$20$	0	0
$21$	4.13169	0.901609
$22$	0	0
$23$	5.62632	1.17317	0.586585	−	0.809888i	$-0.300472\pi$
$23$	5.62632	1.17317	0.586585	+	0.809888i	$0.300472\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.59822	−1.07738
$28$	0	0
$29$	3.44007	0.638805	0.319403	−	0.947619i	$-0.396518\pi$
$29$	3.44007	0.638805	0.319403	+	0.947619i	$0.396518\pi$
$30$	0	0
$31$	8.62237	1.54862	0.774312	−	0.632805i	$-0.218096\pi$
$31$	8.62237	1.54862	0.774312	+	0.632805i	$0.218096\pi$
$32$	0	0
$33$	3.50292	0.609781
$34$	0	0
$35$	−3.19059	−0.539308
$36$	0	0
$37$	9.46573	1.55616	0.778079	−	0.628167i	$-0.216195\pi$
$37$	9.46573	1.55616	0.778079	+	0.628167i	$0.216195\pi$
$38$	0	0
$39$	2.63706	0.422268
$40$	0	0
$41$	−10.1119	−1.57921	−0.789605	−	0.613616i	$-0.789714\pi$
$41$	−10.1119	−1.57921	−0.789605	+	0.613616i	$0.789714\pi$
$42$	0	0
$43$	−2.51878	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$43$	−2.51878	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$44$	0	0
$45$	1.32307	0.197232
$46$	0	0
$47$	−1.05810	−0.154340	−0.0771702	−	0.997018i	$-0.524588\pi$
$47$	−1.05810	−0.154340	−0.0771702	+	0.997018i	$0.524588\pi$
$48$	0	0
$49$	3.17985	0.454265
$50$	0	0
$51$	3.45989	0.484481
$52$	0	0
$53$	−1.72802	−0.237362	−0.118681	−	0.992932i	$-0.537867\pi$
$53$	−1.72802	−0.237362	−0.118681	+	0.992932i	$0.537867\pi$
$54$	0	0
$55$	−2.70504	−0.364747
$56$	0	0
$57$	−1.89050	−0.250403
$58$	0	0
$59$	3.39934	0.442556	0.221278	−	0.975211i	$-0.428977\pi$
$59$	3.39934	0.442556	0.221278	+	0.975211i	$0.428977\pi$
$60$	0	0
$61$	−14.9071	−1.90866	−0.954328	−	0.298760i	$-0.903427\pi$
$61$	−14.9071	−1.90866	−0.954328	+	0.298760i	$0.903427\pi$
$62$	0	0
$63$	−4.22137	−0.531843
$64$	0	0
$65$	−2.03640	−0.252584
$66$	0	0
$67$	2.16888	0.264971	0.132486	−	0.991185i	$-0.457704\pi$
$67$	2.16888	0.264971	0.132486	+	0.991185i	$0.457704\pi$
$68$	0	0
$69$	7.28588	0.877117
$70$	0	0
$71$	−6.88221	−0.816768	−0.408384	−	0.912810i	$-0.633908\pi$
$71$	−6.88221	−0.816768	−0.408384	+	0.912810i	$0.633908\pi$
$72$	0	0
$73$	−8.79954	−1.02991	−0.514954	−	0.857218i	$-0.672191\pi$
$73$	−8.79954	−1.02991	−0.514954	+	0.857218i	$0.672191\pi$
$74$	0	0
$75$	1.29496	0.149529
$76$	0	0
$77$	8.63066	0.983555
$78$	0	0
$79$	−5.97939	−0.672734	−0.336367	−	0.941731i	$-0.609198\pi$
$79$	−5.97939	−0.672734	−0.336367	+	0.941731i	$0.609198\pi$
$80$	0	0
$81$	−3.28027	−0.364474
$82$	0	0
$83$	7.29685	0.800933	0.400467	−	0.916311i	$-0.368848\pi$
$83$	7.29685	0.800933	0.400467	+	0.916311i	$0.368848\pi$
$84$	0	0
$85$	−2.67180	−0.289798
$86$	0	0
$87$	4.45476	0.477601
$88$	0	0
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	6.49731	0.681103
$92$	0	0
$93$	11.1656	1.15782
$94$	0	0
$95$	1.45989	0.149781
$96$	0	0
$97$	15.6505	1.58907	0.794533	−	0.607222i	$-0.207716\pi$
$97$	15.6505	1.58907	0.794533	+	0.607222i	$0.207716\pi$
$98$	0	0
$99$	−3.57896	−0.359699
$100$	0	0
$101$	17.0561	1.69715	0.848575	−	0.529075i	$-0.177461\pi$
$101$	17.0561	1.69715	0.848575	+	0.529075i	$0.177461\pi$
$102$	0	0
$103$	−6.69162	−0.659345	−0.329673	−	0.944095i	$-0.606938\pi$
$103$	−6.69162	−0.659345	−0.329673	+	0.944095i	$0.606938\pi$
$104$	0	0
$105$	−4.13169	−0.403212
$106$	0	0
$107$	15.9256	1.53959	0.769794	−	0.638292i	$-0.220359\pi$
$107$	15.9256	1.53959	0.769794	+	0.638292i	$0.220359\pi$
$108$	0	0
$109$	−1.65443	−0.158466	−0.0792330	−	0.996856i	$-0.525247\pi$
$109$	−1.65443	−0.158466	−0.0792330	+	0.996856i	$0.525247\pi$
$110$	0	0
$111$	12.2578	1.16346
$112$	0	0
$113$	−2.74388	−0.258123	−0.129061	−	0.991637i	$-0.541196\pi$
$113$	−2.74388	−0.258123	−0.129061	+	0.991637i	$0.541196\pi$
$114$	0	0
$115$	−5.62632	−0.524657
$116$	0	0
$117$	−2.69430	−0.249088
$118$	0	0
$119$	8.52463	0.781451
$120$	0	0
$121$	−3.68277	−0.334798
$122$	0	0
$123$	−13.0945	−1.18069
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.0122478	0.00108681	0.000543407	−	1.00000i	$-0.499827\pi$
$127$	0.0122478	0.00108681	0.000543407		1.00000i	$0.499827\pi$
$128$	0	0
$129$	−3.26173	−0.287179
$130$	0	0
$131$	18.4349	1.61067	0.805334	−	0.592821i	$-0.201986\pi$
$131$	18.4349	1.61067	0.805334	+	0.592821i	$0.201986\pi$
$132$	0	0
$133$	−4.65790	−0.403891
$134$	0	0
$135$	5.59822	0.481818
$136$	0	0
$137$	17.8348	1.52373	0.761864	−	0.647737i	$-0.224285\pi$
$137$	17.8348	1.52373	0.761864	+	0.647737i	$0.224285\pi$
$138$	0	0
$139$	7.69547	0.652721	0.326361	−	0.945245i	$-0.394178\pi$
$139$	7.69547	0.652721	0.326361	+	0.945245i	$0.394178\pi$
$140$	0	0
$141$	−1.37021	−0.115392
$142$	0	0
$143$	5.50854	0.460647
$144$	0	0
$145$	−3.44007	−0.285682
$146$	0	0
$147$	4.11779	0.339630
$148$	0	0
$149$	−16.0825	−1.31753	−0.658764	−	0.752349i	$-0.728921\pi$
$149$	−16.0825	−1.31753	−0.658764	+	0.752349i	$0.728921\pi$
$150$	0	0
$151$	3.01304	0.245198	0.122599	−	0.992456i	$-0.460877\pi$
$151$	3.01304	0.245198	0.122599	+	0.992456i	$0.460877\pi$
$152$	0	0
$153$	−3.53499	−0.285787
$154$	0	0
$155$	−8.62237	−0.692565
$156$	0	0
$157$	−10.7128	−0.854972	−0.427486	−	0.904022i	$-0.640601\pi$
$157$	−10.7128	−0.854972	−0.427486	+	0.904022i	$0.640601\pi$
$158$	0	0
$159$	−2.23772	−0.177463
$160$	0	0
$161$	17.9513	1.41476
$162$	0	0
$163$	1.55250	0.121601	0.0608007	−	0.998150i	$-0.480635\pi$
$163$	1.55250	0.121601	0.0608007	+	0.998150i	$0.480635\pi$
$164$	0	0
$165$	−3.50292	−0.272702
$166$	0	0
$167$	12.8252	0.992444	0.496222	−	0.868196i	$-0.334720\pi$
$167$	12.8252	0.992444	0.496222	+	0.868196i	$0.334720\pi$
$168$	0	0
$169$	−8.85308	−0.681006
$170$	0	0
$171$	1.93154	0.147708
$172$	0	0
$173$	−14.9094	−1.13354	−0.566770	−	0.823876i	$-0.691807\pi$
$173$	−14.9094	−1.13354	−0.566770	+	0.823876i	$0.691807\pi$
$174$	0	0
$175$	3.19059	0.241186
$176$	0	0
$177$	4.40202	0.330876
$178$	0	0
$179$	20.1349	1.50495	0.752475	−	0.658621i	$-0.228860\pi$
$179$	20.1349	1.50495	0.752475	+	0.658621i	$0.228860\pi$
$180$	0	0
$181$	−4.51916	−0.335907	−0.167953	−	0.985795i	$-0.553716\pi$
$181$	−4.51916	−0.335907	−0.167953	+	0.985795i	$0.553716\pi$
$182$	0	0
$183$	−19.3041	−1.42700
$184$	0	0
$185$	−9.46573	−0.695935
$186$	0	0
$187$	7.22733	0.528515
$188$	0	0
$189$	−17.8616	−1.29924
$190$	0	0
$191$	−9.55425	−0.691321	−0.345661	−	0.938360i	$-0.612345\pi$
$191$	−9.55425	−0.691321	−0.345661	+	0.938360i	$0.612345\pi$
$192$	0	0
$193$	24.9743	1.79769	0.898844	−	0.438270i	$-0.144408\pi$
$193$	24.9743	1.79769	0.898844	+	0.438270i	$0.144408\pi$
$194$	0	0
$195$	−2.63706	−0.188844
$196$	0	0
$197$	11.1011	0.790923	0.395462	−	0.918482i	$-0.370585\pi$
$197$	11.1011	0.790923	0.395462	+	0.918482i	$0.370585\pi$
$198$	0	0
$199$	7.82995	0.555050	0.277525	−	0.960718i	$-0.410486\pi$
$199$	7.82995	0.555050	0.277525	+	0.960718i	$0.410486\pi$
$200$	0	0
$201$	2.80862	0.198105
$202$	0	0
$203$	10.9758	0.770354
$204$	0	0
$205$	10.1119	0.706244
$206$	0	0
$207$	−7.44403	−0.517396
$208$	0	0
$209$	−3.94905	−0.273162
$210$	0	0
$211$	−20.5375	−1.41386	−0.706929	−	0.707284i	$-0.749920\pi$
$211$	−20.5375	−1.41386	−0.706929	+	0.707284i	$0.749920\pi$
$212$	0	0
$213$	−8.91221	−0.610654
$214$	0	0
$215$	2.51878	0.171780
$216$	0	0
$217$	27.5104	1.86753
$218$	0	0
$219$	−11.3951	−0.770008
$220$	0	0
$221$	5.44086	0.365992
$222$	0	0
$223$	4.16530	0.278929	0.139465	−	0.990227i	$-0.455462\pi$
$223$	4.16530	0.278929	0.139465	+	0.990227i	$0.455462\pi$
$224$	0	0
$225$	−1.32307	−0.0882047
$226$	0	0
$227$	7.69980	0.511054	0.255527	−	0.966802i	$-0.417751\pi$
$227$	7.69980	0.511054	0.255527	+	0.966802i	$0.417751\pi$
$228$	0	0
$229$	0.206071	0.0136176	0.00680879	−	0.999977i	$-0.497833\pi$
$229$	0.206071	0.0136176	0.00680879	+	0.999977i	$0.497833\pi$
$230$	0	0
$231$	11.1764	0.735352
$232$	0	0
$233$	15.9391	1.04421	0.522104	−	0.852882i	$-0.325147\pi$
$233$	15.9391	1.04421	0.522104	+	0.852882i	$0.325147\pi$
$234$	0	0
$235$	1.05810	0.0690231
$236$	0	0
$237$	−7.74309	−0.502968
$238$	0	0
$239$	−16.9216	−1.09457	−0.547284	−	0.836947i	$-0.684338\pi$
$239$	−16.9216	−1.09457	−0.547284	+	0.836947i	$0.684338\pi$
$240$	0	0
$241$	3.84197	0.247483	0.123741	−	0.992315i	$-0.460511\pi$
$241$	3.84197	0.247483	0.123741	+	0.992315i	$0.460511\pi$
$242$	0	0
$243$	12.5468	0.804879
$244$	0	0
$245$	−3.17985	−0.203153
$246$	0	0
$247$	−2.97291	−0.189162
$248$	0	0
$249$	9.44915	0.598816
$250$	0	0
$251$	1.25626	0.0792946	0.0396473	−	0.999214i	$-0.487377\pi$
$251$	1.25626	0.0792946	0.0396473	+	0.999214i	$0.487377\pi$
$252$	0	0
$253$	15.2194	0.956837
$254$	0	0
$255$	−3.45989	−0.216667
$256$	0	0
$257$	6.84313	0.426863	0.213431	−	0.976958i	$-0.431536\pi$
$257$	6.84313	0.426863	0.213431	+	0.976958i	$0.431536\pi$
$258$	0	0
$259$	30.2013	1.87661
$260$	0	0
$261$	−4.55146	−0.281728
$262$	0	0
$263$	4.38743	0.270541	0.135270	−	0.990809i	$-0.456810\pi$
$263$	4.38743	0.270541	0.135270	+	0.990809i	$0.456810\pi$
$264$	0	0
$265$	1.72802	0.106152
$266$	0	0
$267$	−1.29496	−0.0792504
$268$	0	0
$269$	−25.5807	−1.55968	−0.779840	−	0.625978i	$-0.784700\pi$
$269$	−25.5807	−1.55968	−0.779840	+	0.625978i	$0.784700\pi$
$270$	0	0
$271$	−14.2568	−0.866036	−0.433018	−	0.901385i	$-0.642551\pi$
$271$	−14.2568	−0.866036	−0.433018	+	0.901385i	$0.642551\pi$
$272$	0	0
$273$	8.41378	0.509225
$274$	0	0
$275$	2.70504	0.163120
$276$	0	0
$277$	9.43597	0.566952	0.283476	−	0.958979i	$-0.408512\pi$
$277$	9.43597	0.566952	0.283476	+	0.958979i	$0.408512\pi$
$278$	0	0
$279$	−11.4080	−0.682979
$280$	0	0
$281$	−20.3543	−1.21423	−0.607117	−	0.794612i	$-0.707674\pi$
$281$	−20.3543	−1.21423	−0.607117	+	0.794612i	$0.707674\pi$
$282$	0	0
$283$	−4.77106	−0.283610	−0.141805	−	0.989895i	$-0.545291\pi$
$283$	−4.77106	−0.283610	−0.141805	+	0.989895i	$0.545291\pi$
$284$	0	0
$285$	1.89050	0.111984
$286$	0	0
$287$	−32.2628	−1.90441
$288$	0	0
$289$	−9.86146	−0.580086
$290$	0	0
$291$	20.2668	1.18806
$292$	0	0
$293$	12.5065	0.730636	0.365318	−	0.930883i	$-0.380960\pi$
$293$	12.5065	0.730636	0.365318	+	0.930883i	$0.380960\pi$
$294$	0	0
$295$	−3.39934	−0.197917
$296$	0	0
$297$	−15.1434	−0.878708
$298$	0	0
$299$	11.4574	0.662601
$300$	0	0
$301$	−8.03640	−0.463210
$302$	0	0
$303$	22.0871	1.26887
$304$	0	0
$305$	14.9071	0.853577
$306$	0	0
$307$	−15.9679	−0.911334	−0.455667	−	0.890150i	$-0.650599\pi$
$307$	−15.9679	−0.911334	−0.455667	+	0.890150i	$0.650599\pi$
$308$	0	0
$309$	−8.66540	−0.492958
$310$	0	0
$311$	17.7230	1.00498	0.502490	−	0.864583i	$-0.332417\pi$
$311$	17.7230	1.00498	0.502490	+	0.864583i	$0.332417\pi$
$312$	0	0
$313$	19.8006	1.11920	0.559599	−	0.828763i	$-0.310955\pi$
$313$	19.8006	1.11920	0.559599	+	0.828763i	$0.310955\pi$
$314$	0	0
$315$	4.22137	0.237848
$316$	0	0
$317$	−1.15759	−0.0650167	−0.0325083	−	0.999471i	$-0.510350\pi$
$317$	−1.15759	−0.0650167	−0.0325083	+	0.999471i	$0.510350\pi$
$318$	0	0
$319$	9.30552	0.521009
$320$	0	0
$321$	20.6231	1.15107
$322$	0	0
$323$	−3.90054	−0.217032
$324$	0	0
$325$	2.03640	0.112959
$326$	0	0
$327$	−2.14243	−0.118477
$328$	0	0
$329$	−3.37598	−0.186124
$330$	0	0
$331$	1.88134	0.103408	0.0517040	−	0.998662i	$-0.483535\pi$
$331$	1.88134	0.103408	0.0517040	+	0.998662i	$0.483535\pi$
$332$	0	0
$333$	−12.5238	−0.686302
$334$	0	0
$335$	−2.16888	−0.118499
$336$	0	0
$337$	1.86868	0.101794	0.0508969	−	0.998704i	$-0.483792\pi$
$337$	1.86868	0.101794	0.0508969	+	0.998704i	$0.483792\pi$
$338$	0	0
$339$	−3.55322	−0.192985
$340$	0	0
$341$	23.3238	1.26306
$342$	0	0
$343$	−12.1885	−0.658118
$344$	0	0
$345$	−7.28588	−0.392259
$346$	0	0
$347$	9.65267	0.518182	0.259091	−	0.965853i	$-0.416577\pi$
$347$	9.65267	0.518182	0.259091	+	0.965853i	$0.416577\pi$
$348$	0	0
$349$	34.0970	1.82517	0.912585	−	0.408886i	$-0.134083\pi$
$349$	34.0970	1.82517	0.912585	+	0.408886i	$0.134083\pi$
$350$	0	0
$351$	−11.4002	−0.608498
$352$	0	0
$353$	−36.9671	−1.96756	−0.983779	−	0.179382i	$-0.942590\pi$
$353$	−36.9671	−1.96756	−0.983779	+	0.179382i	$0.942590\pi$
$354$	0	0
$355$	6.88221	0.365270
$356$	0	0
$357$	11.0391	0.584250
$358$	0	0
$359$	−1.00561	−0.0530742	−0.0265371	−	0.999648i	$-0.508448\pi$
$359$	−1.00561	−0.0530742	−0.0265371	+	0.999648i	$0.508448\pi$
$360$	0	0
$361$	−16.8687	−0.887828
$362$	0	0
$363$	−4.76906	−0.250311
$364$	0	0
$365$	8.79954	0.460589
$366$	0	0
$367$	−4.09416	−0.213713	−0.106857	−	0.994274i	$-0.534079\pi$
$367$	−4.09416	−0.213713	−0.106857	+	0.994274i	$0.534079\pi$
$368$	0	0
$369$	13.3787	0.696469
$370$	0	0
$371$	−5.51340	−0.286242
$372$	0	0
$373$	6.86807	0.355615	0.177808	−	0.984065i	$-0.443099\pi$
$373$	6.86807	0.355615	0.177808	+	0.984065i	$0.443099\pi$
$374$	0	0
$375$	−1.29496	−0.0668716
$376$	0	0
$377$	7.00536	0.360794
$378$	0	0
$379$	1.27864	0.0656791	0.0328396	−	0.999461i	$-0.489545\pi$
$379$	1.27864	0.0656791	0.0328396	+	0.999461i	$0.489545\pi$
$380$	0	0
$381$	0.0158604	0.000812554 0
$382$	0	0
$383$	19.8336	1.01345	0.506726	−	0.862107i	$-0.330856\pi$
$383$	19.8336	1.01345	0.506726	+	0.862107i	$0.330856\pi$
$384$	0	0
$385$	−8.63066	−0.439859
$386$	0	0
$387$	3.33253	0.169402
$388$	0	0
$389$	13.9074	0.705132	0.352566	−	0.935787i	$-0.385309\pi$
$389$	13.9074	0.705132	0.352566	+	0.935787i	$0.385309\pi$
$390$	0	0
$391$	15.0324	0.760223
$392$	0	0
$393$	23.8726	1.20421
$394$	0	0
$395$	5.97939	0.300856
$396$	0	0
$397$	−2.27223	−0.114040	−0.0570201	−	0.998373i	$-0.518160\pi$
$397$	−2.27223	−0.114040	−0.0570201	+	0.998373i	$0.518160\pi$
$398$	0	0
$399$	−6.03181	−0.301968
$400$	0	0
$401$	−20.7709	−1.03725	−0.518624	−	0.855002i	$-0.673556\pi$
$401$	−20.7709	−1.03725	−0.518624	+	0.855002i	$0.673556\pi$
$402$	0	0
$403$	17.5586	0.874655
$404$	0	0
$405$	3.28027	0.162998
$406$	0	0
$407$	25.6052	1.26920
$408$	0	0
$409$	37.1089	1.83492	0.917458	−	0.397833i	$-0.130238\pi$
$409$	37.1089	1.83492	0.917458	+	0.397833i	$0.130238\pi$
$410$	0	0
$411$	23.0954	1.13921
$412$	0	0
$413$	10.8459	0.533691
$414$	0	0
$415$	−7.29685	−0.358188
$416$	0	0
$417$	9.96535	0.488005
$418$	0	0
$419$	−28.9496	−1.41428	−0.707139	−	0.707074i	$-0.750015\pi$
$419$	−28.9496	−1.41428	−0.707139	+	0.707074i	$0.750015\pi$
$420$	0	0
$421$	34.0480	1.65940	0.829699	−	0.558211i	$-0.188512\pi$
$421$	34.0480	1.65940	0.829699	+	0.558211i	$0.188512\pi$
$422$	0	0
$423$	1.39995	0.0680678
$424$	0	0
$425$	2.67180	0.129602
$426$	0	0
$427$	−47.5624	−2.30170
$428$	0	0
$429$	7.13335	0.344401
$430$	0	0
$431$	−15.7133	−0.756881	−0.378441	−	0.925626i	$-0.623540\pi$
$431$	−15.7133	−0.756881	−0.378441	+	0.925626i	$0.623540\pi$
$432$	0	0
$433$	−22.0260	−1.05850	−0.529251	−	0.848465i	$-0.677527\pi$
$433$	−22.0260	−1.05850	−0.529251	+	0.848465i	$0.677527\pi$
$434$	0	0
$435$	−4.45476	−0.213590
$436$	0	0
$437$	−8.21381	−0.392920
$438$	0	0
$439$	−32.5888	−1.55538	−0.777690	−	0.628648i	$-0.783609\pi$
$439$	−32.5888	−1.55538	−0.777690	+	0.628648i	$0.783609\pi$
$440$	0	0
$441$	−4.20717	−0.200341
$442$	0	0
$443$	−33.3935	−1.58657	−0.793287	−	0.608849i	$-0.791632\pi$
$443$	−33.3935	−1.58657	−0.793287	+	0.608849i	$0.791632\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	0	0
$447$	−20.8262	−0.985047
$448$	0	0
$449$	−2.89521	−0.136634	−0.0683168	−	0.997664i	$-0.521763\pi$
$449$	−2.89521	−0.136634	−0.0683168	+	0.997664i	$0.521763\pi$
$450$	0	0
$451$	−27.3530	−1.28800
$452$	0	0
$453$	3.90177	0.183321
$454$	0	0
$455$	−6.49731	−0.304599
$456$	0	0
$457$	18.5490	0.867686	0.433843	−	0.900988i	$-0.357157\pi$
$457$	18.5490	0.867686	0.433843	+	0.900988i	$0.357157\pi$
$458$	0	0
$459$	−14.9573	−0.698149
$460$	0	0
$461$	20.3168	0.946248	0.473124	−	0.880996i	$-0.343126\pi$
$461$	20.3168	0.946248	0.473124	+	0.880996i	$0.343126\pi$
$462$	0	0
$463$	−20.0650	−0.932499	−0.466250	−	0.884653i	$-0.654395\pi$
$463$	−20.0650	−0.932499	−0.466250	+	0.884653i	$0.654395\pi$
$464$	0	0
$465$	−11.1656	−0.517794
$466$	0	0
$467$	35.1654	1.62726	0.813631	−	0.581382i	$-0.197488\pi$
$467$	35.1654	1.62726	0.813631	+	0.581382i	$0.197488\pi$
$468$	0	0
$469$	6.92001	0.319536
$470$	0	0
$471$	−13.8726	−0.639218
$472$	0	0
$473$	−6.81340	−0.313281
$474$	0	0
$475$	−1.45989	−0.0669843
$476$	0	0
$477$	2.28629	0.104682
$478$	0	0
$479$	36.9126	1.68658	0.843290	−	0.537458i	$-0.180615\pi$
$479$	36.9126	1.68658	0.843290	+	0.537458i	$0.180615\pi$
$480$	0	0
$481$	19.2760	0.878910
$482$	0	0
$483$	23.2462	1.05774
$484$	0	0
$485$	−15.6505	−0.710652
$486$	0	0
$487$	13.4262	0.608400	0.304200	−	0.952608i	$-0.401611\pi$
$487$	13.4262	0.608400	0.304200	+	0.952608i	$0.401611\pi$
$488$	0	0
$489$	2.01043	0.0909150
$490$	0	0
$491$	29.1214	1.31423	0.657115	−	0.753791i	$-0.271777\pi$
$491$	29.1214	1.31423	0.657115	+	0.753791i	$0.271777\pi$
$492$	0	0
$493$	9.19120	0.413951
$494$	0	0
$495$	3.57896	0.160862
$496$	0	0
$497$	−21.9583	−0.984964
$498$	0	0
$499$	−6.11813	−0.273885	−0.136943	−	0.990579i	$-0.543728\pi$
$499$	−6.11813	−0.273885	−0.136943	+	0.990579i	$0.543728\pi$
$500$	0	0
$501$	16.6082	0.741998
$502$	0	0
$503$	16.6291	0.741456	0.370728	−	0.928742i	$-0.379108\pi$
$503$	16.6291	0.741456	0.370728	+	0.928742i	$0.379108\pi$
$504$	0	0
$505$	−17.0561	−0.758989
$506$	0	0
$507$	−11.4644	−0.509152
$508$	0	0
$509$	−6.47859	−0.287159	−0.143579	−	0.989639i	$-0.545861\pi$
$509$	−6.47859	−0.287159	−0.143579	+	0.989639i	$0.545861\pi$
$510$	0	0
$511$	−28.0757	−1.24200
$512$	0	0
$513$	8.17277	0.360837
$514$	0	0
$515$	6.69162	0.294868
$516$	0	0
$517$	−2.86221	−0.125880
$518$	0	0
$519$	−19.3071	−0.847488
$520$	0	0
$521$	22.4933	0.985449	0.492724	−	0.870185i	$-0.336001\pi$
$521$	22.4933	0.985449	0.492724	+	0.870185i	$0.336001\pi$
$522$	0	0
$523$	−14.1820	−0.620134	−0.310067	−	0.950715i	$-0.600352\pi$
$523$	−14.1820	−0.620134	−0.310067	+	0.950715i	$0.600352\pi$
$524$	0	0
$525$	4.13169	0.180322
$526$	0	0
$527$	23.0373	1.00352
$528$	0	0
$529$	8.65553	0.376327
$530$	0	0
$531$	−4.49757	−0.195178
$532$	0	0
$533$	−20.5918	−0.891930
$534$	0	0
$535$	−15.9256	−0.688525
$536$	0	0
$537$	26.0739	1.12517
$538$	0	0
$539$	8.60162	0.370498
$540$	0	0
$541$	−35.2225	−1.51433	−0.757166	−	0.653222i	$-0.773417\pi$
$541$	−35.2225	−1.51433	−0.757166	+	0.653222i	$0.773417\pi$
$542$	0	0
$543$	−5.85214	−0.251140
$544$	0	0
$545$	1.65443	0.0708681
$546$	0	0
$547$	1.43480	0.0613477	0.0306739	−	0.999529i	$-0.490235\pi$
$547$	1.43480	0.0613477	0.0306739	+	0.999529i	$0.490235\pi$
$548$	0	0
$549$	19.7231	0.841763
$550$	0	0
$551$	−5.02212	−0.213949
$552$	0	0
$553$	−19.0778	−0.811270
$554$	0	0
$555$	−12.2578	−0.520314
$556$	0	0
$557$	−0.533965	−0.0226248	−0.0113124	−	0.999936i	$-0.503601\pi$
$557$	−0.533965	−0.0226248	−0.0113124	+	0.999936i	$0.503601\pi$
$558$	0	0
$559$	−5.12925	−0.216944
$560$	0	0
$561$	9.35913	0.395143
$562$	0	0
$563$	36.0788	1.52054	0.760270	−	0.649607i	$-0.225067\pi$
$563$	36.0788	1.52054	0.760270	+	0.649607i	$0.225067\pi$
$564$	0	0
$565$	2.74388	0.115436
$566$	0	0
$567$	−10.4660	−0.439530
$568$	0	0
$569$	−39.9398	−1.67436	−0.837181	−	0.546925i	$-0.815798\pi$
$569$	−39.9398	−1.67436	−0.837181	+	0.546925i	$0.815798\pi$
$570$	0	0
$571$	34.5619	1.44637	0.723184	−	0.690655i	$-0.242678\pi$
$571$	34.5619	1.44637	0.723184	+	0.690655i	$0.242678\pi$
$572$	0	0
$573$	−12.3724	−0.516864
$574$	0	0
$575$	5.62632	0.234634
$576$	0	0
$577$	46.3225	1.92843	0.964215	−	0.265122i	$-0.0854121\pi$
$577$	46.3225	1.92843	0.964215	+	0.265122i	$0.0854121\pi$
$578$	0	0
$579$	32.3408	1.34404
$580$	0	0
$581$	23.2812	0.965869
$582$	0	0
$583$	−4.67436	−0.193592
$584$	0	0
$585$	2.69430	0.111396
$586$	0	0
$587$	−14.1115	−0.582445	−0.291223	−	0.956655i	$-0.594062\pi$
$587$	−14.1115	−0.582445	−0.291223	+	0.956655i	$0.594062\pi$
$588$	0	0
$589$	−12.5877	−0.518667
$590$	0	0
$591$	14.3756	0.591332
$592$	0	0
$593$	−46.5842	−1.91298	−0.956491	−	0.291760i	$-0.905759\pi$
$593$	−46.5842	−1.91298	−0.956491	+	0.291760i	$0.905759\pi$
$594$	0	0
$595$	−8.52463	−0.349476
$596$	0	0
$597$	10.1395	0.414982
$598$	0	0
$599$	4.68560	0.191448	0.0957241	−	0.995408i	$-0.469483\pi$
$599$	4.68560	0.191448	0.0957241	+	0.995408i	$0.469483\pi$
$600$	0	0
$601$	−29.7661	−1.21419	−0.607093	−	0.794631i	$-0.707664\pi$
$601$	−29.7661	−1.21419	−0.607093	+	0.794631i	$0.707664\pi$
$602$	0	0
$603$	−2.86959	−0.116859
$604$	0	0
$605$	3.68277	0.149726
$606$	0	0
$607$	−8.61468	−0.349659	−0.174829	−	0.984599i	$-0.555937\pi$
$607$	−8.61468	−0.349659	−0.174829	+	0.984599i	$0.555937\pi$
$608$	0	0
$609$	14.2133	0.575953
$610$	0	0
$611$	−2.15472	−0.0871708
$612$	0	0
$613$	−16.8722	−0.681463	−0.340731	−	0.940161i	$-0.610675\pi$
$613$	−16.8722	−0.681463	−0.340731	+	0.940161i	$0.610675\pi$
$614$	0	0
$615$	13.0945	0.528021
$616$	0	0
$617$	0.826593	0.0332774	0.0166387	−	0.999862i	$-0.494703\pi$
$617$	0.826593	0.0332774	0.0166387	+	0.999862i	$0.494703\pi$
$618$	0	0
$619$	7.92336	0.318467	0.159233	−	0.987241i	$-0.449098\pi$
$619$	7.92336	0.318467	0.159233	+	0.987241i	$0.449098\pi$
$620$	0	0
$621$	−31.4974	−1.26395
$622$	0	0
$623$	−3.19059	−0.127828
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.11388	−0.204229
$628$	0	0
$629$	25.2906	1.00840
$630$	0	0
$631$	−36.1596	−1.43949	−0.719745	−	0.694238i	$-0.755741\pi$
$631$	−36.1596	−1.43949	−0.719745	+	0.694238i	$0.755741\pi$
$632$	0	0
$633$	−26.5953	−1.05707
$634$	0	0
$635$	−0.0122478	−0.000486038 0
$636$	0	0
$637$	6.47545	0.256567
$638$	0	0
$639$	9.10565	0.360214
$640$	0	0
$641$	−17.4184	−0.687984	−0.343992	−	0.938973i	$-0.611779\pi$
$641$	−17.4184	−0.687984	−0.343992	+	0.938973i	$0.611779\pi$
$642$	0	0
$643$	42.1096	1.66064	0.830321	−	0.557286i	$-0.188157\pi$
$643$	42.1096	1.66064	0.830321	+	0.557286i	$0.188157\pi$
$644$	0	0
$645$	3.26173	0.128431
$646$	0	0
$647$	21.0971	0.829413	0.414707	−	0.909955i	$-0.363884\pi$
$647$	21.0971	0.829413	0.414707	+	0.909955i	$0.363884\pi$
$648$	0	0
$649$	9.19533	0.360948
$650$	0	0
$651$	35.6250	1.39625
$652$	0	0
$653$	−28.2222	−1.10442	−0.552210	−	0.833705i	$-0.686215\pi$
$653$	−28.2222	−1.10442	−0.552210	+	0.833705i	$0.686215\pi$
$654$	0	0
$655$	−18.4349	−0.720313
$656$	0	0
$657$	11.6424	0.454214
$658$	0	0
$659$	−33.8928	−1.32028	−0.660138	−	0.751144i	$-0.729502\pi$
$659$	−33.8928	−1.32028	−0.660138	+	0.751144i	$0.729502\pi$
$660$	0	0
$661$	−5.46911	−0.212724	−0.106362	−	0.994327i	$-0.533920\pi$
$661$	−5.46911	−0.212724	−0.106362	+	0.994327i	$0.533920\pi$
$662$	0	0
$663$	7.04571	0.273633
$664$	0	0
$665$	4.65790	0.180626
$666$	0	0
$667$	19.3550	0.749427
$668$	0	0
$669$	5.39391	0.208541
$670$	0	0
$671$	−40.3242	−1.55670
$672$	0	0
$673$	−40.9495	−1.57849	−0.789244	−	0.614080i	$-0.789527\pi$
$673$	−40.9495	−1.57849	−0.789244	+	0.614080i	$0.789527\pi$
$674$	0	0
$675$	−5.59822	−0.215475
$676$	0	0
$677$	12.1363	0.466437	0.233219	−	0.972424i	$-0.425074\pi$
$677$	12.1363	0.466437	0.233219	+	0.972424i	$0.425074\pi$
$678$	0	0
$679$	49.9342	1.91630
$680$	0	0
$681$	9.97096	0.382088
$682$	0	0
$683$	12.9361	0.494986	0.247493	−	0.968890i	$-0.420393\pi$
$683$	12.9361	0.494986	0.247493	+	0.968890i	$0.420393\pi$
$684$	0	0
$685$	−17.8348	−0.681432
$686$	0	0
$687$	0.266855	0.0101811
$688$	0	0
$689$	−3.51894	−0.134061
$690$	0	0
$691$	6.09541	0.231880	0.115940	−	0.993256i	$-0.463012\pi$
$691$	6.09541	0.231880	0.115940	+	0.993256i	$0.463012\pi$
$692$	0	0
$693$	−11.4190	−0.433771
$694$	0	0
$695$	−7.69547	−0.291906
$696$	0	0
$697$	−27.0170	−1.02334
$698$	0	0
$699$	20.6406	0.780699
$700$	0	0
$701$	24.9250	0.941403	0.470701	−	0.882293i	$-0.344001\pi$
$701$	24.9250	0.941403	0.470701	+	0.882293i	$0.344001\pi$
$702$	0	0
$703$	−13.8189	−0.521190
$704$	0	0
$705$	1.37021	0.0516050
$706$	0	0
$707$	54.4191	2.04664
$708$	0	0
$709$	−48.1319	−1.80763	−0.903816	−	0.427921i	$-0.859246\pi$
$709$	−48.1319	−1.80763	−0.903816	+	0.427921i	$0.859246\pi$
$710$	0	0
$711$	7.91116	0.296692
$712$	0	0
$713$	48.5122	1.81680
$714$	0	0
$715$	−5.50854	−0.206008
$716$	0	0
$717$	−21.9128	−0.818350
$718$	0	0
$719$	−16.3018	−0.607954	−0.303977	−	0.952679i	$-0.598315\pi$
$719$	−16.3018	−0.607954	−0.303977	+	0.952679i	$0.598315\pi$
$720$	0	0
$721$	−21.3502	−0.795123
$722$	0	0
$723$	4.97520	0.185030
$724$	0	0
$725$	3.44007	0.127761
$726$	0	0
$727$	−44.8912	−1.66492	−0.832462	−	0.554083i	$-0.813069\pi$
$727$	−44.8912	−1.66492	−0.832462	+	0.554083i	$0.813069\pi$
$728$	0	0
$729$	26.0885	0.966240
$730$	0	0
$731$	−6.72970	−0.248907
$732$	0	0
$733$	7.45088	0.275204	0.137602	−	0.990488i	$-0.456060\pi$
$733$	7.45088	0.275204	0.137602	+	0.990488i	$0.456060\pi$
$734$	0	0
$735$	−4.11779	−0.151887
$736$	0	0
$737$	5.86691	0.216110
$738$	0	0
$739$	−11.2444	−0.413633	−0.206817	−	0.978380i	$-0.566310\pi$
$739$	−11.2444	−0.413633	−0.206817	+	0.978380i	$0.566310\pi$
$740$	0	0
$741$	−3.84981	−0.141426
$742$	0	0
$743$	−2.94251	−0.107950	−0.0539750	−	0.998542i	$-0.517189\pi$
$743$	−2.94251	−0.107950	−0.0539750	+	0.998542i	$0.517189\pi$
$744$	0	0
$745$	16.0825	0.589217
$746$	0	0
$747$	−9.65425	−0.353231
$748$	0	0
$749$	50.8121	1.85663
$750$	0	0
$751$	−13.0559	−0.476416	−0.238208	−	0.971214i	$-0.576560\pi$
$751$	−13.0559	−0.476416	−0.238208	+	0.971214i	$0.576560\pi$
$752$	0	0
$753$	1.62681	0.0592844
$754$	0	0
$755$	−3.01304	−0.109656
$756$	0	0
$757$	−16.1988	−0.588754	−0.294377	−	0.955689i	$-0.595112\pi$
$757$	−16.1988	−0.588754	−0.294377	+	0.955689i	$0.595112\pi$
$758$	0	0
$759$	19.7086	0.715376
$760$	0	0
$761$	23.7871	0.862282	0.431141	−	0.902285i	$-0.358111\pi$
$761$	23.7871	0.862282	0.431141	+	0.902285i	$0.358111\pi$
$762$	0	0
$763$	−5.27861	−0.191099
$764$	0	0
$765$	3.53499	0.127808
$766$	0	0
$767$	6.92241	0.249954
$768$	0	0
$769$	−20.9770	−0.756451	−0.378225	−	0.925714i	$-0.623466\pi$
$769$	−20.9770	−0.756451	−0.378225	+	0.925714i	$0.623466\pi$
$770$	0	0
$771$	8.86160	0.319143
$772$	0	0
$773$	−16.6061	−0.597280	−0.298640	−	0.954366i	$-0.596533\pi$
$773$	−16.6061	−0.597280	−0.298640	+	0.954366i	$0.596533\pi$
$774$	0	0
$775$	8.62237	0.309725
$776$	0	0
$777$	39.1095	1.40305
$778$	0	0
$779$	14.7622	0.528911
$780$	0	0
$781$	−18.6166	−0.666156
$782$	0	0
$783$	−19.2583	−0.688234
$784$	0	0
$785$	10.7128	0.382355
$786$	0	0
$787$	50.6611	1.80587	0.902937	−	0.429773i	$-0.141407\pi$
$787$	50.6611	1.80587	0.902937	+	0.429773i	$0.141407\pi$
$788$	0	0
$789$	5.68156	0.202269
$790$	0	0
$791$	−8.75460	−0.311278
$792$	0	0
$793$	−30.3568	−1.07800
$794$	0	0
$795$	2.23772	0.0793639
$796$	0	0
$797$	−47.4446	−1.68057	−0.840287	−	0.542142i	$-0.817614\pi$
$797$	−47.4446	−1.68057	−0.840287	+	0.542142i	$0.817614\pi$
$798$	0	0
$799$	−2.82705	−0.100014
$800$	0	0
$801$	1.32307	0.0467484
$802$	0	0
$803$	−23.8031	−0.839993
$804$	0	0
$805$	−17.9513	−0.632700
$806$	0	0
$807$	−33.1260	−1.16609
$808$	0	0
$809$	27.5792	0.969631	0.484816	−	0.874616i	$-0.338887\pi$
$809$	27.5792	0.969631	0.484816	+	0.874616i	$0.338887\pi$
$810$	0	0
$811$	−11.4096	−0.400645	−0.200322	−	0.979730i	$-0.564199\pi$
$811$	−11.4096	−0.400645	−0.200322	+	0.979730i	$0.564199\pi$
$812$	0	0
$813$	−18.4620	−0.647489
$814$	0	0
$815$	−1.55250	−0.0543818
$816$	0	0
$817$	3.67714	0.128647
$818$	0	0
$819$	−8.59640	−0.300383
$820$	0	0
$821$	28.7824	1.00451	0.502257	−	0.864719i	$-0.332503\pi$
$821$	28.7824	1.00451	0.502257	+	0.864719i	$0.332503\pi$
$822$	0	0
$823$	−52.5379	−1.83136	−0.915679	−	0.401911i	$-0.868346\pi$
$823$	−52.5379	−1.83136	−0.915679	+	0.401911i	$0.868346\pi$
$824$	0	0
$825$	3.50292	0.121956
$826$	0	0
$827$	−51.6561	−1.79626	−0.898129	−	0.439732i	$-0.855073\pi$
$827$	−51.6561	−1.79626	−0.898129	+	0.439732i	$0.855073\pi$
$828$	0	0
$829$	−44.4506	−1.54383	−0.771917	−	0.635723i	$-0.780702\pi$
$829$	−44.4506	−1.54383	−0.771917	+	0.635723i	$0.780702\pi$
$830$	0	0
$831$	12.2192	0.423880
$832$	0	0
$833$	8.49594	0.294367
$834$	0	0
$835$	−12.8252	−0.443834
$836$	0	0
$837$	−48.2699	−1.66845
$838$	0	0
$839$	24.5071	0.846078	0.423039	−	0.906111i	$-0.360963\pi$
$839$	24.5071	0.846078	0.423039	+	0.906111i	$0.360963\pi$
$840$	0	0
$841$	−17.1659	−0.591928
$842$	0	0
$843$	−26.3580	−0.907819
$844$	0	0
$845$	8.85308	0.304555
$846$	0	0
$847$	−11.7502	−0.403742
$848$	0	0
$849$	−6.17834	−0.212040
$850$	0	0
$851$	53.2573	1.82564
$852$	0	0
$853$	−7.76300	−0.265800	−0.132900	−	0.991129i	$-0.542429\pi$
$853$	−7.76300	−0.265800	−0.132900	+	0.991129i	$0.542429\pi$
$854$	0	0
$855$	−1.93154	−0.0660571
$856$	0	0
$857$	−6.72226	−0.229628	−0.114814	−	0.993387i	$-0.536627\pi$
$857$	−6.72226	−0.229628	−0.114814	+	0.993387i	$0.536627\pi$
$858$	0	0
$859$	−6.13678	−0.209384	−0.104692	−	0.994505i	$-0.533386\pi$
$859$	−6.13678	−0.209384	−0.104692	+	0.994505i	$0.533386\pi$
$860$	0	0
$861$	−41.7792	−1.42383
$862$	0	0
$863$	7.96255	0.271048	0.135524	−	0.990774i	$-0.456728\pi$
$863$	7.96255	0.271048	0.135524	+	0.990774i	$0.456728\pi$
$864$	0	0
$865$	14.9094	0.506934
$866$	0	0
$867$	−12.7702	−0.433700
$868$	0	0
$869$	−16.1745	−0.548682
$870$	0	0
$871$	4.41671	0.149654
$872$	0	0
$873$	−20.7067	−0.700815
$874$	0	0
$875$	−3.19059	−0.107862
$876$	0	0
$877$	3.78873	0.127936	0.0639681	−	0.997952i	$-0.479624\pi$
$877$	3.78873	0.127936	0.0639681	+	0.997952i	$0.479624\pi$
$878$	0	0
$879$	16.1954	0.546258
$880$	0	0
$881$	−35.4998	−1.19602	−0.598009	−	0.801490i	$-0.704041\pi$
$881$	−35.4998	−1.19602	−0.598009	+	0.801490i	$0.704041\pi$
$882$	0	0
$883$	22.4461	0.755371	0.377685	−	0.925934i	$-0.376720\pi$
$883$	22.4461	0.755371	0.377685	+	0.925934i	$0.376720\pi$
$884$	0	0
$885$	−4.40202	−0.147972
$886$	0	0
$887$	25.8592	0.868267	0.434133	−	0.900849i	$-0.357055\pi$
$887$	25.8592	0.868267	0.434133	+	0.900849i	$0.357055\pi$
$888$	0	0
$889$	0.0390776	0.00131062
$890$	0	0
$891$	−8.87325	−0.297265
$892$	0	0
$893$	1.54471	0.0516919
$894$	0	0
$895$	−20.1349	−0.673034
$896$	0	0
$897$	14.8370	0.495392
$898$	0	0
$899$	29.6616	0.989268
$900$	0	0
$901$	−4.61694	−0.153812
$902$	0	0
$903$	−10.4068	−0.346318
$904$	0	0
$905$	4.51916	0.150222
$906$	0	0
$907$	45.8976	1.52400	0.762002	−	0.647575i	$-0.224217\pi$
$907$	45.8976	1.52400	0.762002	+	0.647575i	$0.224217\pi$
$908$	0	0
$909$	−22.5665	−0.748483
$910$	0	0
$911$	−50.0150	−1.65707	−0.828536	−	0.559936i	$-0.810826\pi$
$911$	−50.0150	−1.65707	−0.828536	+	0.559936i	$0.810826\pi$
$912$	0	0
$913$	19.7383	0.653241
$914$	0	0
$915$	19.3041	0.638174
$916$	0	0
$917$	58.8183	1.94235
$918$	0	0
$919$	40.9072	1.34940	0.674702	−	0.738090i	$-0.264272\pi$
$919$	40.9072	1.34940	0.674702	+	0.738090i	$0.264272\pi$
$920$	0	0
$921$	−20.6778	−0.681356
$922$	0	0
$923$	−14.0149	−0.461307
$924$	0	0
$925$	9.46573	0.311231
$926$	0	0
$927$	8.85349	0.290787
$928$	0	0
$929$	22.6729	0.743874	0.371937	−	0.928258i	$-0.378694\pi$
$929$	22.6729	0.743874	0.371937	+	0.928258i	$0.378694\pi$
$930$	0	0
$931$	−4.64223	−0.152143
$932$	0	0
$933$	22.9507	0.751371
$934$	0	0
$935$	−7.22733	−0.236359
$936$	0	0
$937$	−38.5330	−1.25882	−0.629409	−	0.777074i	$-0.716703\pi$
$937$	−38.5330	−1.25882	−0.629409	+	0.777074i	$0.716703\pi$
$938$	0	0
$939$	25.6411	0.836765
$940$	0	0
$941$	4.39678	0.143331	0.0716655	−	0.997429i	$-0.477169\pi$
$941$	4.39678	0.143331	0.0716655	+	0.997429i	$0.477169\pi$
$942$	0	0
$943$	−56.8927	−1.85268
$944$	0	0
$945$	17.8616	0.581038
$946$	0	0
$947$	−5.70322	−0.185330	−0.0926649	−	0.995697i	$-0.529539\pi$
$947$	−5.70322	−0.185330	−0.0926649	+	0.995697i	$0.529539\pi$
$948$	0	0
$949$	−17.9194	−0.581688
$950$	0	0
$951$	−1.49903	−0.0486095
$952$	0	0
$953$	−24.8332	−0.804426	−0.402213	−	0.915546i	$-0.631759\pi$
$953$	−24.8332	−0.804426	−0.402213	+	0.915546i	$0.631759\pi$
$954$	0	0
$955$	9.55425	0.309168
$956$	0	0
$957$	12.0503	0.389531
$958$	0	0
$959$	56.9034	1.83751
$960$	0	0
$961$	43.3452	1.39823
$962$	0	0
$963$	−21.0707	−0.678995
$964$	0	0
$965$	−24.9743	−0.803950
$966$	0	0
$967$	−43.4590	−1.39755	−0.698774	−	0.715343i	$-0.746270\pi$
$967$	−43.4590	−1.39755	−0.698774	+	0.715343i	$0.746270\pi$
$968$	0	0
$969$	−5.05105	−0.162263
$970$	0	0
$971$	35.4713	1.13833	0.569165	−	0.822223i	$-0.307267\pi$
$971$	35.4713	1.13833	0.569165	+	0.822223i	$0.307267\pi$
$972$	0	0
$973$	24.5531	0.787135
$974$	0	0
$975$	2.63706	0.0844535
$976$	0	0
$977$	7.14406	0.228559	0.114279	−	0.993449i	$-0.463544\pi$
$977$	7.14406	0.228559	0.114279	+	0.993449i	$0.463544\pi$
$978$	0	0
$979$	−2.70504	−0.0864534
$980$	0	0
$981$	2.18893	0.0698872
$982$	0	0
$983$	14.0179	0.447100	0.223550	−	0.974692i	$-0.428235\pi$
$983$	14.0179	0.447100	0.223550	+	0.974692i	$0.428235\pi$
$984$	0	0
$985$	−11.1011	−0.353712
$986$	0	0
$987$	−4.37176	−0.139155
$988$	0	0
$989$	−14.1715	−0.450627
$990$	0	0
$991$	25.6111	0.813565	0.406782	−	0.913525i	$-0.366651\pi$
$991$	25.6111	0.813565	0.406782	+	0.913525i	$0.366651\pi$
$992$	0	0
$993$	2.43627	0.0773127
$994$	0	0
$995$	−7.82995	−0.248226
$996$	0	0
$997$	−27.3942	−0.867582	−0.433791	−	0.901013i	$-0.642824\pi$
$997$	−27.3942	−0.867582	−0.433791	+	0.901013i	$0.642824\pi$
$998$	0	0
$999$	−52.9912	−1.67657

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7120.2.a.bc.1.3		4
4.3	odd	2		445.2.a.d.1.4	✓	4
12.11	even	2		4005.2.a.l.1.1		4
20.19	odd	2		2225.2.a.i.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.d.1.4	✓	4	4.3	odd	2
2225.2.a.i.1.1		4	20.19	odd	2
4005.2.a.l.1.1		4	12.11	even	2
7120.2.a.bc.1.3		4	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)

\(f(q)\)	\(=\)	\(q+1.29496 q^{3} -1.00000 q^{5} +3.19059 q^{7} -1.32307 q^{9} +O(q^{10})\)	\(q+1.29496 q^{3} -1.00000 q^{5} +3.19059 q^{7} -1.32307 q^{9} +2.70504 q^{11} +2.03640 q^{13} -1.29496 q^{15} +2.67180 q^{17} -1.45989 q^{19} +4.13169 q^{21} +5.62632 q^{23} +1.00000 q^{25} -5.59822 q^{27} +3.44007 q^{29} +8.62237 q^{31} +3.50292 q^{33} -3.19059 q^{35} +9.46573 q^{37} +2.63706 q^{39} -10.1119 q^{41} -2.51878 q^{43} +1.32307 q^{45} -1.05810 q^{47} +3.17985 q^{49} +3.45989 q^{51} -1.72802 q^{53} -2.70504 q^{55} -1.89050 q^{57} +3.39934 q^{59} -14.9071 q^{61} -4.22137 q^{63} -2.03640 q^{65} +2.16888 q^{67} +7.28588 q^{69} -6.88221 q^{71} -8.79954 q^{73} +1.29496 q^{75} +8.63066 q^{77} -5.97939 q^{79} -3.28027 q^{81} +7.29685 q^{83} -2.67180 q^{85} +4.45476 q^{87} -1.00000 q^{89} +6.49731 q^{91} +11.1656 q^{93} +1.45989 q^{95} +15.6505 q^{97} -3.57896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 4 * q^9	\( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9} + 14 q^{11} - 5 q^{13} - 2 q^{15} - 3 q^{17} + q^{19} - 4 q^{21} + 3 q^{23} + 4 q^{25} - q^{27} - 10 q^{29} + 11 q^{31} + 2 q^{35} + 3 q^{37} - 11 q^{39} - 3 q^{41} - 9 q^{43} + 4 q^{45} + 24 q^{47} + 7 q^{51} + 3 q^{53} - 14 q^{55} + 19 q^{57} + 22 q^{59} - 3 q^{61} - 6 q^{63} + 5 q^{65} + 9 q^{67} + 7 q^{69} - 16 q^{71} + 3 q^{73} + 2 q^{75} - 4 q^{77} + 27 q^{79} - 8 q^{81} - 6 q^{83} + 3 q^{85} - 4 q^{87} - 4 q^{89} + 29 q^{91} - 2 q^{93} - q^{95} + 41 q^{97} - 21 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 4 * q^9 + 14 * q^11 - 5 * q^13 - 2 * q^15 - 3 * q^17 + q^19 - 4 * q^21 + 3 * q^23 + 4 * q^25 - q^27 - 10 * q^29 + 11 * q^31 + 2 * q^35 + 3 * q^37 - 11 * q^39 - 3 * q^41 - 9 * q^43 + 4 * q^45 + 24 * q^47 + 7 * q^51 + 3 * q^53 - 14 * q^55 + 19 * q^57 + 22 * q^59 - 3 * q^61 - 6 * q^63 + 5 * q^65 + 9 * q^67 + 7 * q^69 - 16 * q^71 + 3 * q^73 + 2 * q^75 - 4 * q^77 + 27 * q^79 - 8 * q^81 - 6 * q^83 + 3 * q^85 - 4 * q^87 - 4 * q^89 + 29 * q^91 - 2 * q^93 - q^95 + 41 * q^97 - 21 * q^99

Introduction

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