LMFDB - Embedded newform 4376.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4376,2,Mod(1,4376)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4376, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4376.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4376 = 2^{3} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4376.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:	$34.9425359245$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	4376.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.24040 q^{3} -2.27349 q^{5} +0.587186 q^{7} +7.50019 q^{9} +O(q^{10})$	$q-3.24040 q^{3} -2.27349 q^{5} +0.587186 q^{7} +7.50019 q^{9} -0.562791 q^{11} +1.76770 q^{13} +7.36701 q^{15} -3.97692 q^{17} -5.97060 q^{19} -1.90272 q^{21} +0.0297800 q^{23} +0.168753 q^{25} -14.5824 q^{27} +2.02198 q^{29} +2.64436 q^{31} +1.82367 q^{33} -1.33496 q^{35} +6.81586 q^{37} -5.72804 q^{39} +8.90757 q^{41} -5.63983 q^{43} -17.0516 q^{45} +8.87979 q^{47} -6.65521 q^{49} +12.8868 q^{51} +11.8378 q^{53} +1.27950 q^{55} +19.3471 q^{57} -11.2148 q^{59} -7.50492 q^{61} +4.40400 q^{63} -4.01884 q^{65} +3.77078 q^{67} -0.0964993 q^{69} +7.23903 q^{71} +4.75020 q^{73} -0.546828 q^{75} -0.330463 q^{77} +14.7615 q^{79} +24.7523 q^{81} -6.66318 q^{83} +9.04149 q^{85} -6.55202 q^{87} -4.60576 q^{89} +1.03797 q^{91} -8.56877 q^{93} +13.5741 q^{95} +0.593574 q^{97} -4.22104 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 8 q^{3} - 11 q^{5} - 10 q^{7} + 39 q^{9}+O(q^{10}) $ 31 * q - 8 * q^3 - 11 * q^5 - 10 * q^7 + 39 * q^9	$ 31 q - 8 q^{3} - 11 q^{5} - 10 q^{7} + 39 q^{9} - 11 q^{11} - 20 q^{13} - 11 q^{15} + 26 q^{17} - 32 q^{19} - 23 q^{21} - 29 q^{23} + 26 q^{25} - 53 q^{27} - 6 q^{29} - 53 q^{31} + 8 q^{33} - 13 q^{35} - 26 q^{37} - 24 q^{39} - 6 q^{41} - 11 q^{43} - 12 q^{45} - 41 q^{47} + 29 q^{49} - 35 q^{51} - 17 q^{53} - 10 q^{55} + 9 q^{57} - 27 q^{59} - 62 q^{61} - 34 q^{63} + 4 q^{65} - 28 q^{67} - 20 q^{69} - 39 q^{71} + 16 q^{73} - 36 q^{75} - 19 q^{77} - 39 q^{79} + 35 q^{81} - 32 q^{83} - 58 q^{85} - 38 q^{87} + 61 q^{89} - 25 q^{91} - q^{93} - 12 q^{95} - 85 q^{99}+O(q^{100}) $ 31 * q - 8 * q^3 - 11 * q^5 - 10 * q^7 + 39 * q^9 - 11 * q^11 - 20 * q^13 - 11 * q^15 + 26 * q^17 - 32 * q^19 - 23 * q^21 - 29 * q^23 + 26 * q^25 - 53 * q^27 - 6 * q^29 - 53 * q^31 + 8 * q^33 - 13 * q^35 - 26 * q^37 - 24 * q^39 - 6 * q^41 - 11 * q^43 - 12 * q^45 - 41 * q^47 + 29 * q^49 - 35 * q^51 - 17 * q^53 - 10 * q^55 + 9 * q^57 - 27 * q^59 - 62 * q^61 - 34 * q^63 + 4 * q^65 - 28 * q^67 - 20 * q^69 - 39 * q^71 + 16 * q^73 - 36 * q^75 - 19 * q^77 - 39 * q^79 + 35 * q^81 - 32 * q^83 - 58 * q^85 - 38 * q^87 + 61 * q^89 - 25 * q^91 - q^93 - 12 * q^95 - 85 * 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
$3$	−3.24040	−1.87085	−0.935423	−	0.353531i	$-0.884981\pi$
$3$	−3.24040	−1.87085	−0.935423	+	0.353531i	$0.884981\pi$
$4$	0	0
$5$	−2.27349	−1.01674	−0.508368	−	0.861140i	$-0.669751\pi$
$5$	−2.27349	−1.01674	−0.508368	+	0.861140i	$0.669751\pi$
$6$	0	0
$7$	0.587186	0.221935	0.110968	−	0.993824i	$-0.464605\pi$
$7$	0.587186	0.221935	0.110968	+	0.993824i	$0.464605\pi$
$8$	0	0
$9$	7.50019	2.50006
$10$	0	0
$11$	−0.562791	−0.169688	−0.0848440	−	0.996394i	$-0.527039\pi$
$11$	−0.562791	−0.169688	−0.0848440	+	0.996394i	$0.527039\pi$
$12$	0	0
$13$	1.76770	0.490271	0.245135	−	0.969489i	$-0.421168\pi$
$13$	1.76770	0.490271	0.245135	+	0.969489i	$0.421168\pi$
$14$	0	0
$15$	7.36701	1.90215
$16$	0	0
$17$	−3.97692	−0.964545	−0.482273	−	0.876021i	$-0.660189\pi$
$17$	−3.97692	−0.964545	−0.482273	+	0.876021i	$0.660189\pi$
$18$	0	0
$19$	−5.97060	−1.36975	−0.684875	−	0.728660i	$-0.740143\pi$
$19$	−5.97060	−1.36975	−0.684875	+	0.728660i	$0.740143\pi$
$20$	0	0
$21$	−1.90272	−0.415207
$22$	0	0
$23$	0.0297800	0.00620957	0.00310478	−	0.999995i	$-0.499012\pi$
$23$	0.0297800	0.00620957	0.00310478	+	0.999995i	$0.499012\pi$
$24$	0	0
$25$	0.168753	0.0337506
$26$	0	0
$27$	−14.5824	−2.80639
$28$	0	0
$29$	2.02198	0.375472	0.187736	−	0.982220i	$-0.439885\pi$
$29$	2.02198	0.375472	0.187736	+	0.982220i	$0.439885\pi$
$30$	0	0
$31$	2.64436	0.474941	0.237470	−	0.971395i	$-0.423682\pi$
$31$	2.64436	0.474941	0.237470	+	0.971395i	$0.423682\pi$
$32$	0	0
$33$	1.82367	0.317460
$34$	0	0
$35$	−1.33496	−0.225649
$36$	0	0
$37$	6.81586	1.12052	0.560260	−	0.828316i	$-0.310701\pi$
$37$	6.81586	1.12052	0.560260	+	0.828316i	$0.310701\pi$
$38$	0	0
$39$	−5.72804	−0.917221
$40$	0	0
$41$	8.90757	1.39113	0.695564	−	0.718464i	$-0.255155\pi$
$41$	8.90757	1.39113	0.695564	+	0.718464i	$0.255155\pi$
$42$	0	0
$43$	−5.63983	−0.860066	−0.430033	−	0.902813i	$-0.641498\pi$
$43$	−5.63983	−0.860066	−0.430033	+	0.902813i	$0.641498\pi$
$44$	0	0
$45$	−17.0516	−2.54190
$46$	0	0
$47$	8.87979	1.29525	0.647625	−	0.761959i	$-0.275762\pi$
$47$	8.87979	1.29525	0.647625	+	0.761959i	$0.275762\pi$
$48$	0	0
$49$	−6.65521	−0.950745
$50$	0	0
$51$	12.8868	1.80452
$52$	0	0
$53$	11.8378	1.62604	0.813021	−	0.582235i	$-0.197821\pi$
$53$	11.8378	1.62604	0.813021	+	0.582235i	$0.197821\pi$
$54$	0	0
$55$	1.27950	0.172528
$56$	0	0
$57$	19.3471	2.56259
$58$	0	0
$59$	−11.2148	−1.46004	−0.730019	−	0.683427i	$-0.760489\pi$
$59$	−11.2148	−1.46004	−0.730019	+	0.683427i	$0.760489\pi$
$60$	0	0
$61$	−7.50492	−0.960906	−0.480453	−	0.877020i	$-0.659528\pi$
$61$	−7.50492	−0.960906	−0.480453	+	0.877020i	$0.659528\pi$
$62$	0	0
$63$	4.40400	0.554852
$64$	0	0
$65$	−4.01884	−0.498475
$66$	0	0
$67$	3.77078	0.460673	0.230337	−	0.973111i	$-0.426017\pi$
$67$	3.77078	0.460673	0.230337	+	0.973111i	$0.426017\pi$
$68$	0	0
$69$	−0.0964993	−0.0116171
$70$	0	0
$71$	7.23903	0.859115	0.429557	−	0.903040i	$-0.358670\pi$
$71$	7.23903	0.859115	0.429557	+	0.903040i	$0.358670\pi$
$72$	0	0
$73$	4.75020	0.555969	0.277984	−	0.960586i	$-0.410334\pi$
$73$	4.75020	0.555969	0.277984	+	0.960586i	$0.410334\pi$
$74$	0	0
$75$	−0.546828	−0.0631422
$76$	0	0
$77$	−0.330463	−0.0376597
$78$	0	0
$79$	14.7615	1.66080	0.830398	−	0.557171i	$-0.188113\pi$
$79$	14.7615	1.66080	0.830398	+	0.557171i	$0.188113\pi$
$80$	0	0
$81$	24.7523	2.75025
$82$	0	0
$83$	−6.66318	−0.731379	−0.365689	−	0.930737i	$-0.619167\pi$
$83$	−6.66318	−0.731379	−0.365689	+	0.930737i	$0.619167\pi$
$84$	0	0
$85$	9.04149	0.980687
$86$	0	0
$87$	−6.55202	−0.702450
$88$	0	0
$89$	−4.60576	−0.488209	−0.244105	−	0.969749i	$-0.578494\pi$
$89$	−4.60576	−0.488209	−0.244105	+	0.969749i	$0.578494\pi$
$90$	0	0
$91$	1.03797	0.108808
$92$	0	0
$93$	−8.56877	−0.888540
$94$	0	0
$95$	13.5741	1.39267
$96$	0	0
$97$	0.593574	0.0602683	0.0301341	−	0.999546i	$-0.490407\pi$
$97$	0.593574	0.0602683	0.0301341	+	0.999546i	$0.490407\pi$
$98$	0	0
$99$	−4.22104	−0.424231
$100$	0	0
$101$	6.41705	0.638520	0.319260	−	0.947667i	$-0.396566\pi$
$101$	6.41705	0.638520	0.319260	+	0.947667i	$0.396566\pi$
$102$	0	0
$103$	9.73359	0.959079	0.479540	−	0.877520i	$-0.340804\pi$
$103$	9.73359	0.959079	0.479540	+	0.877520i	$0.340804\pi$
$104$	0	0
$105$	4.32580	0.422155
$106$	0	0
$107$	1.13664	0.109883	0.0549415	−	0.998490i	$-0.482503\pi$
$107$	1.13664	0.109883	0.0549415	+	0.998490i	$0.482503\pi$
$108$	0	0
$109$	13.3043	1.27432	0.637162	−	0.770730i	$-0.280108\pi$
$109$	13.3043	1.27432	0.637162	+	0.770730i	$0.280108\pi$
$110$	0	0
$111$	−22.0861	−2.09632
$112$	0	0
$113$	2.93632	0.276226	0.138113	−	0.990416i	$-0.455896\pi$
$113$	2.93632	0.276226	0.138113	+	0.990416i	$0.455896\pi$
$114$	0	0
$115$	−0.0677046	−0.00631349
$116$	0	0
$117$	13.2581	1.22571
$118$	0	0
$119$	−2.33519	−0.214067
$120$	0	0
$121$	−10.6833	−0.971206
$122$	0	0
$123$	−28.8641	−2.60259
$124$	0	0
$125$	10.9838	0.982420
$126$	0	0
$127$	−10.1544	−0.901062	−0.450531	−	0.892761i	$-0.648765\pi$
$127$	−10.1544	−0.901062	−0.450531	+	0.892761i	$0.648765\pi$
$128$	0	0
$129$	18.2753	1.60905
$130$	0	0
$131$	−1.08434	−0.0947397	−0.0473698	−	0.998877i	$-0.515084\pi$
$131$	−1.08434	−0.0947397	−0.0473698	+	0.998877i	$0.515084\pi$
$132$	0	0
$133$	−3.50585	−0.303996
$134$	0	0
$135$	33.1530	2.85335
$136$	0	0
$137$	11.8922	1.01602	0.508008	−	0.861352i	$-0.330382\pi$
$137$	11.8922	1.01602	0.508008	+	0.861352i	$0.330382\pi$
$138$	0	0
$139$	−19.1230	−1.62199	−0.810994	−	0.585054i	$-0.801073\pi$
$139$	−19.1230	−1.62199	−0.810994	+	0.585054i	$0.801073\pi$
$140$	0	0
$141$	−28.7741	−2.42321
$142$	0	0
$143$	−0.994844	−0.0831930
$144$	0	0
$145$	−4.59695	−0.381756
$146$	0	0
$147$	21.5656	1.77870
$148$	0	0
$149$	17.9709	1.47223	0.736116	−	0.676855i	$-0.236658\pi$
$149$	17.9709	1.47223	0.736116	+	0.676855i	$0.236658\pi$
$150$	0	0
$151$	−2.15574	−0.175432	−0.0877158	−	0.996146i	$-0.527957\pi$
$151$	−2.15574	−0.175432	−0.0877158	+	0.996146i	$0.527957\pi$
$152$	0	0
$153$	−29.8277	−2.41142
$154$	0	0
$155$	−6.01192	−0.482889
$156$	0	0
$157$	−4.13075	−0.329670	−0.164835	−	0.986321i	$-0.552709\pi$
$157$	−4.13075	−0.329670	−0.164835	+	0.986321i	$0.552709\pi$
$158$	0	0
$159$	−38.3591	−3.04207
$160$	0	0
$161$	0.0174864	0.00137812
$162$	0	0
$163$	−15.1626	−1.18762	−0.593812	−	0.804604i	$-0.702378\pi$
$163$	−15.1626	−1.18762	−0.593812	+	0.804604i	$0.702378\pi$
$164$	0	0
$165$	−4.14609	−0.322773
$166$	0	0
$167$	−18.4168	−1.42513	−0.712567	−	0.701604i	$-0.752467\pi$
$167$	−18.4168	−1.42513	−0.712567	+	0.701604i	$0.752467\pi$
$168$	0	0
$169$	−9.87525	−0.759635
$170$	0	0
$171$	−44.7807	−3.42446
$172$	0	0
$173$	6.16631	0.468816	0.234408	−	0.972138i	$-0.424685\pi$
$173$	6.16631	0.468816	0.234408	+	0.972138i	$0.424685\pi$
$174$	0	0
$175$	0.0990894	0.00749046
$176$	0	0
$177$	36.3403	2.73150
$178$	0	0
$179$	−6.84166	−0.511370	−0.255685	−	0.966760i	$-0.582301\pi$
$179$	−6.84166	−0.511370	−0.255685	+	0.966760i	$0.582301\pi$
$180$	0	0
$181$	−4.24699	−0.315676	−0.157838	−	0.987465i	$-0.550452\pi$
$181$	−4.24699	−0.315676	−0.157838	+	0.987465i	$0.550452\pi$
$182$	0	0
$183$	24.3189	1.79771
$184$	0	0
$185$	−15.4958	−1.13927
$186$	0	0
$187$	2.23818	0.163672
$188$	0	0
$189$	−8.56258	−0.622836
$190$	0	0
$191$	−18.0427	−1.30553	−0.652763	−	0.757562i	$-0.726390\pi$
$191$	−18.0427	−1.30553	−0.652763	+	0.757562i	$0.726390\pi$
$192$	0	0
$193$	3.33626	0.240149	0.120075	−	0.992765i	$-0.461687\pi$
$193$	3.33626	0.240149	0.120075	+	0.992765i	$0.461687\pi$
$194$	0	0
$195$	13.0226	0.932571
$196$	0	0
$197$	−16.0137	−1.14093	−0.570465	−	0.821322i	$-0.693237\pi$
$197$	−16.0137	−1.14093	−0.570465	+	0.821322i	$0.693237\pi$
$198$	0	0
$199$	−14.7712	−1.04710	−0.523552	−	0.851993i	$-0.675394\pi$
$199$	−14.7712	−1.04710	−0.523552	+	0.851993i	$0.675394\pi$
$200$	0	0
$201$	−12.2188	−0.861849
$202$	0	0
$203$	1.18728	0.0833305
$204$	0	0
$205$	−20.2513	−1.41441
$206$	0	0
$207$	0.223356	0.0155243
$208$	0	0
$209$	3.36020	0.232430
$210$	0	0
$211$	−9.34263	−0.643173	−0.321586	−	0.946880i	$-0.604216\pi$
$211$	−9.34263	−0.643173	−0.321586	+	0.946880i	$0.604216\pi$
$212$	0	0
$213$	−23.4574	−1.60727
$214$	0	0
$215$	12.8221	0.874460
$216$	0	0
$217$	1.55273	0.105406
$218$	0	0
$219$	−15.3925	−1.04013
$220$	0	0
$221$	−7.02999	−0.472888
$222$	0	0
$223$	−4.97674	−0.333267	−0.166634	−	0.986019i	$-0.553290\pi$
$223$	−4.97674	−0.333267	−0.166634	+	0.986019i	$0.553290\pi$
$224$	0	0
$225$	1.26568	0.0843787
$226$	0	0
$227$	13.0354	0.865193	0.432596	−	0.901588i	$-0.357598\pi$
$227$	13.0354	0.865193	0.432596	+	0.901588i	$0.357598\pi$
$228$	0	0
$229$	−15.8286	−1.04598	−0.522992	−	0.852337i	$-0.675184\pi$
$229$	−15.8286	−1.04598	−0.522992	+	0.852337i	$0.675184\pi$
$230$	0	0
$231$	1.07083	0.0704555
$232$	0	0
$233$	11.7920	0.772521	0.386261	−	0.922390i	$-0.373767\pi$
$233$	11.7920	0.772521	0.386261	+	0.922390i	$0.373767\pi$
$234$	0	0
$235$	−20.1881	−1.31693
$236$	0	0
$237$	−47.8331	−3.10709
$238$	0	0
$239$	−2.94791	−0.190684	−0.0953421	−	0.995445i	$-0.530394\pi$
$239$	−2.94791	−0.190684	−0.0953421	+	0.995445i	$0.530394\pi$
$240$	0	0
$241$	−11.6939	−0.753267	−0.376634	−	0.926362i	$-0.622918\pi$
$241$	−11.6939	−0.753267	−0.376634	+	0.926362i	$0.622918\pi$
$242$	0	0
$243$	−36.4601	−2.33891
$244$	0	0
$245$	15.1306	0.966656
$246$	0	0
$247$	−10.5542	−0.671548
$248$	0	0
$249$	21.5914	1.36830
$250$	0	0
$251$	−19.8117	−1.25050	−0.625252	−	0.780423i	$-0.715004\pi$
$251$	−19.8117	−1.25050	−0.625252	+	0.780423i	$0.715004\pi$
$252$	0	0
$253$	−0.0167599	−0.00105369
$254$	0	0
$255$	−29.2980	−1.83471
$256$	0	0
$257$	17.0221	1.06181	0.530905	−	0.847431i	$-0.321852\pi$
$257$	17.0221	1.06181	0.530905	+	0.847431i	$0.321852\pi$
$258$	0	0
$259$	4.00218	0.248683
$260$	0	0
$261$	15.1652	0.938704
$262$	0	0
$263$	3.72903	0.229942	0.114971	−	0.993369i	$-0.463323\pi$
$263$	3.72903	0.229942	0.114971	+	0.993369i	$0.463323\pi$
$264$	0	0
$265$	−26.9130	−1.65325
$266$	0	0
$267$	14.9245	0.913364
$268$	0	0
$269$	8.78605	0.535695	0.267847	−	0.963461i	$-0.413688\pi$
$269$	8.78605	0.535695	0.267847	+	0.963461i	$0.413688\pi$
$270$	0	0
$271$	6.66712	0.404999	0.202499	−	0.979282i	$-0.435094\pi$
$271$	6.66712	0.404999	0.202499	+	0.979282i	$0.435094\pi$
$272$	0	0
$273$	−3.36342	−0.203564
$274$	0	0
$275$	−0.0949728	−0.00572707
$276$	0	0
$277$	−32.2600	−1.93831	−0.969157	−	0.246444i	$-0.920738\pi$
$277$	−32.2600	−1.93831	−0.969157	+	0.246444i	$0.920738\pi$
$278$	0	0
$279$	19.8332	1.18738
$280$	0	0
$281$	11.8821	0.708826	0.354413	−	0.935089i	$-0.384681\pi$
$281$	11.8821	0.708826	0.354413	+	0.935089i	$0.384681\pi$
$282$	0	0
$283$	22.8141	1.35616	0.678078	−	0.734990i	$-0.262813\pi$
$283$	22.8141	1.35616	0.678078	+	0.734990i	$0.262813\pi$
$284$	0	0
$285$	−43.9855	−2.60548
$286$	0	0
$287$	5.23039	0.308740
$288$	0	0
$289$	−1.18410	−0.0696527
$290$	0	0
$291$	−1.92342	−0.112753
$292$	0	0
$293$	−1.20021	−0.0701173	−0.0350586	−	0.999385i	$-0.511162\pi$
$293$	−1.20021	−0.0701173	−0.0350586	+	0.999385i	$0.511162\pi$
$294$	0	0
$295$	25.4966	1.48447
$296$	0	0
$297$	8.20685	0.476210
$298$	0	0
$299$	0.0526421	0.00304437
$300$	0	0
$301$	−3.31163	−0.190879
$302$	0	0
$303$	−20.7938	−1.19457
$304$	0	0
$305$	17.0624	0.976987
$306$	0	0
$307$	12.7967	0.730347	0.365174	−	0.930939i	$-0.381010\pi$
$307$	12.7967	0.730347	0.365174	+	0.930939i	$0.381010\pi$
$308$	0	0
$309$	−31.5407	−1.79429
$310$	0	0
$311$	17.3789	0.985467	0.492733	−	0.870180i	$-0.335998\pi$
$311$	17.3789	0.985467	0.492733	+	0.870180i	$0.335998\pi$
$312$	0	0
$313$	−5.91950	−0.334590	−0.167295	−	0.985907i	$-0.553503\pi$
$313$	−5.91950	−0.334590	−0.167295	+	0.985907i	$0.553503\pi$
$314$	0	0
$315$	−10.0125	−0.564138
$316$	0	0
$317$	−16.7646	−0.941592	−0.470796	−	0.882242i	$-0.656033\pi$
$317$	−16.7646	−0.941592	−0.470796	+	0.882242i	$0.656033\pi$
$318$	0	0
$319$	−1.13795	−0.0637131
$320$	0	0
$321$	−3.68316	−0.205574
$322$	0	0
$323$	23.7446	1.32119
$324$	0	0
$325$	0.298304	0.0165469
$326$	0	0
$327$	−43.1114	−2.38406
$328$	0	0
$329$	5.21408	0.287462
$330$	0	0
$331$	28.8384	1.58510	0.792550	−	0.609807i	$-0.208753\pi$
$331$	28.8384	1.58510	0.792550	+	0.609807i	$0.208753\pi$
$332$	0	0
$333$	51.1203	2.80137
$334$	0	0
$335$	−8.57282	−0.468383
$336$	0	0
$337$	−23.7748	−1.29509	−0.647547	−	0.762025i	$-0.724205\pi$
$337$	−23.7748	−1.29509	−0.647547	+	0.762025i	$0.724205\pi$
$338$	0	0
$339$	−9.51486	−0.516776
$340$	0	0
$341$	−1.48822	−0.0805917
$342$	0	0
$343$	−8.01814	−0.432939
$344$	0	0
$345$	0.219390	0.0118116
$346$	0	0
$347$	−17.1041	−0.918197	−0.459099	−	0.888385i	$-0.651828\pi$
$347$	−17.1041	−0.918197	−0.459099	+	0.888385i	$0.651828\pi$
$348$	0	0
$349$	−9.57720	−0.512656	−0.256328	−	0.966590i	$-0.582513\pi$
$349$	−9.57720	−0.512656	−0.256328	+	0.966590i	$0.582513\pi$
$350$	0	0
$351$	−25.7773	−1.37589
$352$	0	0
$353$	5.28253	0.281161	0.140580	−	0.990069i	$-0.455103\pi$
$353$	5.28253	0.281161	0.140580	+	0.990069i	$0.455103\pi$
$354$	0	0
$355$	−16.4579	−0.873492
$356$	0	0
$357$	7.56695	0.400486
$358$	0	0
$359$	−7.10365	−0.374916	−0.187458	−	0.982273i	$-0.560025\pi$
$359$	−7.10365	−0.374916	−0.187458	+	0.982273i	$0.560025\pi$
$360$	0	0
$361$	16.6481	0.876216
$362$	0	0
$363$	34.6181	1.81698
$364$	0	0
$365$	−10.7995	−0.565273
$366$	0	0
$367$	−23.4431	−1.22372	−0.611859	−	0.790967i	$-0.709578\pi$
$367$	−23.4431	−1.22372	−0.611859	+	0.790967i	$0.709578\pi$
$368$	0	0
$369$	66.8084	3.47791
$370$	0	0
$371$	6.95096	0.360876
$372$	0	0
$373$	25.6748	1.32939	0.664696	−	0.747114i	$-0.268561\pi$
$373$	25.6748	1.32939	0.664696	+	0.747114i	$0.268561\pi$
$374$	0	0
$375$	−35.5919	−1.83796
$376$	0	0
$377$	3.57424	0.184083
$378$	0	0
$379$	−24.4131	−1.25402	−0.627009	−	0.779012i	$-0.715721\pi$
$379$	−24.4131	−1.25402	−0.627009	+	0.779012i	$0.715721\pi$
$380$	0	0
$381$	32.9045	1.68575
$382$	0	0
$383$	21.7677	1.11228	0.556139	−	0.831090i	$-0.312282\pi$
$383$	21.7677	1.11228	0.556139	+	0.831090i	$0.312282\pi$
$384$	0	0
$385$	0.751304	0.0382900
$386$	0	0
$387$	−42.2998	−2.15022
$388$	0	0
$389$	12.9005	0.654080	0.327040	−	0.945011i	$-0.393949\pi$
$389$	12.9005	0.654080	0.327040	+	0.945011i	$0.393949\pi$
$390$	0	0
$391$	−0.118433	−0.00598941
$392$	0	0
$393$	3.51371	0.177243
$394$	0	0
$395$	−33.5601	−1.68859
$396$	0	0
$397$	−8.75062	−0.439181	−0.219591	−	0.975592i	$-0.570472\pi$
$397$	−8.75062	−0.439181	−0.219591	+	0.975592i	$0.570472\pi$
$398$	0	0
$399$	11.3604	0.568729
$400$	0	0
$401$	−31.1301	−1.55456	−0.777281	−	0.629154i	$-0.783401\pi$
$401$	−31.1301	−1.55456	−0.777281	+	0.629154i	$0.783401\pi$
$402$	0	0
$403$	4.67442	0.232849
$404$	0	0
$405$	−56.2741	−2.79628
$406$	0	0
$407$	−3.83591	−0.190139
$408$	0	0
$409$	22.9962	1.13709	0.568545	−	0.822652i	$-0.307506\pi$
$409$	22.9962	1.13709	0.568545	+	0.822652i	$0.307506\pi$
$410$	0	0
$411$	−38.5354	−1.90081
$412$	0	0
$413$	−6.58514	−0.324034
$414$	0	0
$415$	15.1487	0.743619
$416$	0	0
$417$	61.9661	3.03449
$418$	0	0
$419$	−37.0817	−1.81156	−0.905781	−	0.423747i	$-0.860715\pi$
$419$	−37.0817	−1.81156	−0.905781	+	0.423747i	$0.860715\pi$
$420$	0	0
$421$	−31.8735	−1.55342	−0.776710	−	0.629858i	$-0.783113\pi$
$421$	−31.8735	−1.55342	−0.776710	+	0.629858i	$0.783113\pi$
$422$	0	0
$423$	66.6001	3.23821
$424$	0	0
$425$	−0.671118	−0.0325540
$426$	0	0
$427$	−4.40678	−0.213259
$428$	0	0
$429$	3.22369	0.155641
$430$	0	0
$431$	−30.5841	−1.47318	−0.736592	−	0.676337i	$-0.763566\pi$
$431$	−30.5841	−1.47318	−0.736592	+	0.676337i	$0.763566\pi$
$432$	0	0
$433$	−21.8714	−1.05107	−0.525537	−	0.850771i	$-0.676136\pi$
$433$	−21.8714	−1.05107	−0.525537	+	0.850771i	$0.676136\pi$
$434$	0	0
$435$	14.8959	0.714206
$436$	0	0
$437$	−0.177805	−0.00850556
$438$	0	0
$439$	5.10090	0.243452	0.121726	−	0.992564i	$-0.461157\pi$
$439$	5.10090	0.243452	0.121726	+	0.992564i	$0.461157\pi$
$440$	0	0
$441$	−49.9154	−2.37692
$442$	0	0
$443$	14.0318	0.666671	0.333336	−	0.942808i	$-0.391826\pi$
$443$	14.0318	0.666671	0.333336	+	0.942808i	$0.391826\pi$
$444$	0	0
$445$	10.4711	0.496379
$446$	0	0
$447$	−58.2329	−2.75432
$448$	0	0
$449$	−35.6533	−1.68258	−0.841291	−	0.540583i	$-0.818204\pi$
$449$	−35.6533	−1.68258	−0.841291	+	0.540583i	$0.818204\pi$
$450$	0	0
$451$	−5.01310	−0.236058
$452$	0	0
$453$	6.98546	0.328205
$454$	0	0
$455$	−2.35980	−0.110629
$456$	0	0
$457$	34.9710	1.63588	0.817938	−	0.575306i	$-0.195117\pi$
$457$	34.9710	1.63588	0.817938	+	0.575306i	$0.195117\pi$
$458$	0	0
$459$	57.9931	2.70689
$460$	0	0
$461$	−12.0084	−0.559286	−0.279643	−	0.960104i	$-0.590216\pi$
$461$	−12.0084	−0.559286	−0.279643	+	0.960104i	$0.590216\pi$
$462$	0	0
$463$	15.5326	0.721862	0.360931	−	0.932593i	$-0.382459\pi$
$463$	15.5326	0.721862	0.360931	+	0.932593i	$0.382459\pi$
$464$	0	0
$465$	19.4810	0.903410
$466$	0	0
$467$	−1.36267	−0.0630569	−0.0315284	−	0.999503i	$-0.510037\pi$
$467$	−1.36267	−0.0630569	−0.0315284	+	0.999503i	$0.510037\pi$
$468$	0	0
$469$	2.21414	0.102240
$470$	0	0
$471$	13.3853	0.616761
$472$	0	0
$473$	3.17405	0.145943
$474$	0	0
$475$	−1.00756	−0.0462299
$476$	0	0
$477$	88.7855	4.06521
$478$	0	0
$479$	−6.33484	−0.289446	−0.144723	−	0.989472i	$-0.546229\pi$
$479$	−6.33484	−0.289446	−0.144723	+	0.989472i	$0.546229\pi$
$480$	0	0
$481$	12.0484	0.549359
$482$	0	0
$483$	−0.0566630	−0.00257825
$484$	0	0
$485$	−1.34948	−0.0612769
$486$	0	0
$487$	21.2210	0.961617	0.480809	−	0.876825i	$-0.340343\pi$
$487$	21.2210	0.961617	0.480809	+	0.876825i	$0.340343\pi$
$488$	0	0
$489$	49.1328	2.22186
$490$	0	0
$491$	37.0589	1.67244	0.836222	−	0.548390i	$-0.184759\pi$
$491$	37.0589	1.67244	0.836222	+	0.548390i	$0.184759\pi$
$492$	0	0
$493$	−8.04125	−0.362160
$494$	0	0
$495$	9.59649	0.431330
$496$	0	0
$497$	4.25065	0.190668
$498$	0	0
$499$	7.12480	0.318950	0.159475	−	0.987202i	$-0.449020\pi$
$499$	7.12480	0.318950	0.159475	+	0.987202i	$0.449020\pi$
$500$	0	0
$501$	59.6777	2.66620
$502$	0	0
$503$	−4.88102	−0.217634	−0.108817	−	0.994062i	$-0.534706\pi$
$503$	−4.88102	−0.217634	−0.108817	+	0.994062i	$0.534706\pi$
$504$	0	0
$505$	−14.5891	−0.649206
$506$	0	0
$507$	31.9998	1.42116
$508$	0	0
$509$	−0.254610	−0.0112854	−0.00564270	−	0.999984i	$-0.501796\pi$
$509$	−0.254610	−0.0112854	−0.00564270	+	0.999984i	$0.501796\pi$
$510$	0	0
$511$	2.78925	0.123389
$512$	0	0
$513$	87.0658	3.84405
$514$	0	0
$515$	−22.1292	−0.975130
$516$	0	0
$517$	−4.99746	−0.219788
$518$	0	0
$519$	−19.9813	−0.877082
$520$	0	0
$521$	26.4908	1.16058	0.580291	−	0.814409i	$-0.302939\pi$
$521$	26.4908	1.16058	0.580291	+	0.814409i	$0.302939\pi$
$522$	0	0
$523$	10.3065	0.450673	0.225336	−	0.974281i	$-0.427652\pi$
$523$	10.3065	0.450673	0.225336	+	0.974281i	$0.427652\pi$
$524$	0	0
$525$	−0.321089	−0.0140135
$526$	0	0
$527$	−10.5164	−0.458102
$528$	0	0
$529$	−22.9991	−0.999961
$530$	0	0
$531$	−84.1128	−3.65019
$532$	0	0
$533$	15.7459	0.682029
$534$	0	0
$535$	−2.58414	−0.111722
$536$	0	0
$537$	22.1697	0.956694
$538$	0	0
$539$	3.74550	0.161330
$540$	0	0
$541$	−13.9736	−0.600772	−0.300386	−	0.953818i	$-0.597115\pi$
$541$	−13.9736	−0.600772	−0.300386	+	0.953818i	$0.597115\pi$
$542$	0	0
$543$	13.7619	0.590582
$544$	0	0
$545$	−30.2473	−1.29565
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	0	0
$549$	−56.2883	−2.40233
$550$	0	0
$551$	−12.0724	−0.514303
$552$	0	0
$553$	8.66773	0.368589
$554$	0	0
$555$	50.2126	2.13140
$556$	0	0
$557$	−7.56610	−0.320586	−0.160293	−	0.987069i	$-0.551244\pi$
$557$	−7.56610	−0.320586	−0.160293	+	0.987069i	$0.551244\pi$
$558$	0	0
$559$	−9.96951	−0.421665
$560$	0	0
$561$	−7.25259	−0.306204
$562$	0	0
$563$	4.92242	0.207455	0.103728	−	0.994606i	$-0.466923\pi$
$563$	4.92242	0.207455	0.103728	+	0.994606i	$0.466923\pi$
$564$	0	0
$565$	−6.67570	−0.280849
$566$	0	0
$567$	14.5342	0.610378
$568$	0	0
$569$	6.39281	0.268000	0.134000	−	0.990981i	$-0.457218\pi$
$569$	6.39281	0.268000	0.134000	+	0.990981i	$0.457218\pi$
$570$	0	0
$571$	−17.6163	−0.737220	−0.368610	−	0.929584i	$-0.620166\pi$
$571$	−17.6163	−0.737220	−0.368610	+	0.929584i	$0.620166\pi$
$572$	0	0
$573$	58.4656	2.44244
$574$	0	0
$575$	0.00502548	0.000209577 0
$576$	0	0
$577$	−19.7554	−0.822426	−0.411213	−	0.911539i	$-0.634895\pi$
$577$	−19.7554	−0.822426	−0.411213	+	0.911539i	$0.634895\pi$
$578$	0	0
$579$	−10.8108	−0.449282
$580$	0	0
$581$	−3.91252	−0.162319
$582$	0	0
$583$	−6.66219	−0.275920
$584$	0	0
$585$	−30.1420	−1.24622
$586$	0	0
$587$	−23.1482	−0.955428	−0.477714	−	0.878515i	$-0.658535\pi$
$587$	−23.1482	−0.955428	−0.477714	+	0.878515i	$0.658535\pi$
$588$	0	0
$589$	−15.7884	−0.650550
$590$	0	0
$591$	51.8908	2.13450
$592$	0	0
$593$	3.99710	0.164141	0.0820706	−	0.996627i	$-0.473847\pi$
$593$	3.99710	0.164141	0.0820706	+	0.996627i	$0.473847\pi$
$594$	0	0
$595$	5.30903	0.217649
$596$	0	0
$597$	47.8647	1.95897
$598$	0	0
$599$	−29.3903	−1.20086	−0.600428	−	0.799679i	$-0.705003\pi$
$599$	−29.3903	−1.20086	−0.600428	+	0.799679i	$0.705003\pi$
$600$	0	0
$601$	−34.0838	−1.39031	−0.695154	−	0.718861i	$-0.744664\pi$
$601$	−34.0838	−1.39031	−0.695154	+	0.718861i	$0.744664\pi$
$602$	0	0
$603$	28.2815	1.15171
$604$	0	0
$605$	24.2883	0.987459
$606$	0	0
$607$	−16.0013	−0.649473	−0.324737	−	0.945804i	$-0.605276\pi$
$607$	−16.0013	−0.649473	−0.324737	+	0.945804i	$0.605276\pi$
$608$	0	0
$609$	−3.84725	−0.155898
$610$	0	0
$611$	15.6968	0.635023
$612$	0	0
$613$	−48.6288	−1.96410	−0.982049	−	0.188623i	$-0.939597\pi$
$613$	−48.6288	−1.96410	−0.982049	+	0.188623i	$0.939597\pi$
$614$	0	0
$615$	65.6222	2.64614
$616$	0	0
$617$	−13.1578	−0.529712	−0.264856	−	0.964288i	$-0.585324\pi$
$617$	−13.1578	−0.529712	−0.264856	+	0.964288i	$0.585324\pi$
$618$	0	0
$619$	−35.6049	−1.43108	−0.715540	−	0.698572i	$-0.753819\pi$
$619$	−35.6049	−1.43108	−0.715540	+	0.698572i	$0.753819\pi$
$620$	0	0
$621$	−0.434265	−0.0174265
$622$	0	0
$623$	−2.70443	−0.108351
$624$	0	0
$625$	−25.8153	−1.03261
$626$	0	0
$627$	−10.8884	−0.434841
$628$	0	0
$629$	−27.1062	−1.08079
$630$	0	0
$631$	−3.98773	−0.158749	−0.0793745	−	0.996845i	$-0.525292\pi$
$631$	−3.98773	−0.158749	−0.0793745	+	0.996845i	$0.525292\pi$
$632$	0	0
$633$	30.2738	1.20328
$634$	0	0
$635$	23.0860	0.916141
$636$	0	0
$637$	−11.7644	−0.466122
$638$	0	0
$639$	54.2941	2.14784
$640$	0	0
$641$	−30.9516	−1.22252	−0.611258	−	0.791432i	$-0.709336\pi$
$641$	−30.9516	−1.22252	−0.611258	+	0.791432i	$0.709336\pi$
$642$	0	0
$643$	−13.8181	−0.544934	−0.272467	−	0.962165i	$-0.587840\pi$
$643$	−13.8181	−0.544934	−0.272467	+	0.962165i	$0.587840\pi$
$644$	0	0
$645$	−41.5487	−1.63598
$646$	0	0
$647$	30.7237	1.20788	0.603938	−	0.797032i	$-0.293598\pi$
$647$	30.7237	1.20788	0.603938	+	0.797032i	$0.293598\pi$
$648$	0	0
$649$	6.31157	0.247751
$650$	0	0
$651$	−5.03146	−0.197198
$652$	0	0
$653$	27.9296	1.09297	0.546484	−	0.837469i	$-0.315966\pi$
$653$	27.9296	1.09297	0.546484	+	0.837469i	$0.315966\pi$
$654$	0	0
$655$	2.46525	0.0963252
$656$	0	0
$657$	35.6274	1.38996
$658$	0	0
$659$	5.40191	0.210429	0.105214	−	0.994450i	$-0.466447\pi$
$659$	5.40191	0.210429	0.105214	+	0.994450i	$0.466447\pi$
$660$	0	0
$661$	−1.04994	−0.0408377	−0.0204189	−	0.999792i	$-0.506500\pi$
$661$	−1.04994	−0.0408377	−0.0204189	+	0.999792i	$0.506500\pi$
$662$	0	0
$663$	22.7800	0.884701
$664$	0	0
$665$	7.97052	0.309083
$666$	0	0
$667$	0.0602146	0.00233152
$668$	0	0
$669$	16.1266	0.623492
$670$	0	0
$671$	4.22370	0.163054
$672$	0	0
$673$	26.9770	1.03989	0.519943	−	0.854201i	$-0.325953\pi$
$673$	26.9770	1.03989	0.519943	+	0.854201i	$0.325953\pi$
$674$	0	0
$675$	−2.46083	−0.0947174
$676$	0	0
$677$	11.7287	0.450772	0.225386	−	0.974270i	$-0.427636\pi$
$677$	11.7287	0.450772	0.225386	+	0.974270i	$0.427636\pi$
$678$	0	0
$679$	0.348538	0.0133757
$680$	0	0
$681$	−42.2400	−1.61864
$682$	0	0
$683$	19.2853	0.737931	0.368965	−	0.929443i	$-0.379712\pi$
$683$	19.2853	0.737931	0.368965	+	0.929443i	$0.379712\pi$
$684$	0	0
$685$	−27.0367	−1.03302
$686$	0	0
$687$	51.2911	1.95688
$688$	0	0
$689$	20.9256	0.797201
$690$	0	0
$691$	−4.50485	−0.171373	−0.0856863	−	0.996322i	$-0.527308\pi$
$691$	−4.50485	−0.171373	−0.0856863	+	0.996322i	$0.527308\pi$
$692$	0	0
$693$	−2.47853	−0.0941517
$694$	0	0
$695$	43.4759	1.64913
$696$	0	0
$697$	−35.4247	−1.34181
$698$	0	0
$699$	−38.2109	−1.44527
$700$	0	0
$701$	23.3201	0.880787	0.440394	−	0.897805i	$-0.354839\pi$
$701$	23.3201	0.880787	0.440394	+	0.897805i	$0.354839\pi$
$702$	0	0
$703$	−40.6948	−1.53483
$704$	0	0
$705$	65.4175	2.46377
$706$	0	0
$707$	3.76800	0.141710
$708$	0	0
$709$	−15.8879	−0.596683	−0.298341	−	0.954459i	$-0.596433\pi$
$709$	−15.8879	−0.596683	−0.298341	+	0.954459i	$0.596433\pi$
$710$	0	0
$711$	110.714	4.15210
$712$	0	0
$713$	0.0787491	0.00294918
$714$	0	0
$715$	2.26177	0.0845853
$716$	0	0
$717$	9.55239	0.356741
$718$	0	0
$719$	−8.94825	−0.333714	−0.166857	−	0.985981i	$-0.553362\pi$
$719$	−8.94825	−0.333714	−0.166857	+	0.985981i	$0.553362\pi$
$720$	0	0
$721$	5.71542	0.212853
$722$	0	0
$723$	37.8928	1.40925
$724$	0	0
$725$	0.341215	0.0126724
$726$	0	0
$727$	−19.1142	−0.708905	−0.354452	−	0.935074i	$-0.615333\pi$
$727$	−19.1142	−0.708905	−0.354452	+	0.935074i	$0.615333\pi$
$728$	0	0
$729$	43.8883	1.62549
$730$	0	0
$731$	22.4292	0.829573
$732$	0	0
$733$	8.46101	0.312514	0.156257	−	0.987716i	$-0.450057\pi$
$733$	8.46101	0.312514	0.156257	+	0.987716i	$0.450057\pi$
$734$	0	0
$735$	−49.0290	−1.80846
$736$	0	0
$737$	−2.12216	−0.0781707
$738$	0	0
$739$	0.611238	0.0224848	0.0112424	−	0.999937i	$-0.496421\pi$
$739$	0.611238	0.0224848	0.0112424	+	0.999937i	$0.496421\pi$
$740$	0	0
$741$	34.1999	1.25636
$742$	0	0
$743$	46.7180	1.71392	0.856958	−	0.515387i	$-0.172352\pi$
$743$	46.7180	1.71392	0.856958	+	0.515387i	$0.172352\pi$
$744$	0	0
$745$	−40.8566	−1.49687
$746$	0	0
$747$	−49.9751	−1.82849
$748$	0	0
$749$	0.667418	0.0243869
$750$	0	0
$751$	13.2159	0.482257	0.241128	−	0.970493i	$-0.422483\pi$
$751$	13.2159	0.482257	0.241128	+	0.970493i	$0.422483\pi$
$752$	0	0
$753$	64.1979	2.33950
$754$	0	0
$755$	4.90105	0.178367
$756$	0	0
$757$	50.1147	1.82145	0.910725	−	0.413013i	$-0.135524\pi$
$757$	50.1147	1.82145	0.910725	+	0.413013i	$0.135524\pi$
$758$	0	0
$759$	0.0543089	0.00197129
$760$	0	0
$761$	−44.6658	−1.61913	−0.809567	−	0.587027i	$-0.800298\pi$
$761$	−44.6658	−1.61913	−0.809567	+	0.587027i	$0.800298\pi$
$762$	0	0
$763$	7.81211	0.282817
$764$	0	0
$765$	67.8129	2.45178
$766$	0	0
$767$	−19.8243	−0.715813
$768$	0	0
$769$	2.34044	0.0843983	0.0421992	−	0.999109i	$-0.486564\pi$
$769$	2.34044	0.0843983	0.0421992	+	0.999109i	$0.486564\pi$
$770$	0	0
$771$	−55.1585	−1.98648
$772$	0	0
$773$	−37.2217	−1.33877	−0.669386	−	0.742915i	$-0.733443\pi$
$773$	−37.2217	−1.33877	−0.669386	+	0.742915i	$0.733443\pi$
$774$	0	0
$775$	0.446244	0.0160295
$776$	0	0
$777$	−12.9687	−0.465248
$778$	0	0
$779$	−53.1835	−1.90550
$780$	0	0
$781$	−4.07406	−0.145781
$782$	0	0
$783$	−29.4853	−1.05372
$784$	0	0
$785$	9.39122	0.335187
$786$	0	0
$787$	25.2383	0.899647	0.449823	−	0.893117i	$-0.351487\pi$
$787$	25.2383	0.899647	0.449823	+	0.893117i	$0.351487\pi$
$788$	0	0
$789$	−12.0835	−0.430185
$790$	0	0
$791$	1.72417	0.0613043
$792$	0	0
$793$	−13.2664	−0.471104
$794$	0	0
$795$	87.2090	3.09298
$796$	0	0
$797$	−42.7479	−1.51421	−0.757104	−	0.653295i	$-0.773386\pi$
$797$	−42.7479	−1.51421	−0.757104	+	0.653295i	$0.773386\pi$
$798$	0	0
$799$	−35.3142	−1.24933
$800$	0	0
$801$	−34.5440	−1.22055
$802$	0	0
$803$	−2.67337	−0.0943412
$804$	0	0
$805$	−0.0397552	−0.00140119
$806$	0	0
$807$	−28.4703	−1.00220
$808$	0	0
$809$	−55.6137	−1.95527	−0.977637	−	0.210301i	$-0.932555\pi$
$809$	−55.6137	−1.95527	−0.977637	+	0.210301i	$0.932555\pi$
$810$	0	0
$811$	46.9117	1.64729	0.823646	−	0.567104i	$-0.191936\pi$
$811$	46.9117	1.64729	0.823646	+	0.567104i	$0.191936\pi$
$812$	0	0
$813$	−21.6041	−0.757690
$814$	0	0
$815$	34.4719	1.20750
$816$	0	0
$817$	33.6732	1.17808
$818$	0	0
$819$	7.78494	0.272028
$820$	0	0
$821$	−25.3283	−0.883965	−0.441982	−	0.897024i	$-0.645725\pi$
$821$	−25.3283	−0.883965	−0.441982	+	0.897024i	$0.645725\pi$
$822$	0	0
$823$	−46.7507	−1.62963	−0.814813	−	0.579724i	$-0.803160\pi$
$823$	−46.7507	−1.62963	−0.814813	+	0.579724i	$0.803160\pi$
$824$	0	0
$825$	0.307750	0.0107145
$826$	0	0
$827$	5.41638	0.188346	0.0941729	−	0.995556i	$-0.469979\pi$
$827$	5.41638	0.188346	0.0941729	+	0.995556i	$0.469979\pi$
$828$	0	0
$829$	49.3103	1.71262	0.856309	−	0.516463i	$-0.172752\pi$
$829$	49.3103	1.71262	0.856309	+	0.516463i	$0.172752\pi$
$830$	0	0
$831$	104.535	3.62629
$832$	0	0
$833$	26.4673	0.917036
$834$	0	0
$835$	41.8703	1.44898
$836$	0	0
$837$	−38.5611	−1.33287
$838$	0	0
$839$	45.9938	1.58788	0.793941	−	0.607995i	$-0.208026\pi$
$839$	45.9938	1.58788	0.793941	+	0.607995i	$0.208026\pi$
$840$	0	0
$841$	−24.9116	−0.859021
$842$	0	0
$843$	−38.5027	−1.32610
$844$	0	0
$845$	22.4513	0.772347
$846$	0	0
$847$	−6.27306	−0.215545
$848$	0	0
$849$	−73.9268	−2.53716
$850$	0	0
$851$	0.202977	0.00695795
$852$	0	0
$853$	−21.5129	−0.736589	−0.368295	−	0.929709i	$-0.620058\pi$
$853$	−21.5129	−0.736589	−0.368295	+	0.929709i	$0.620058\pi$
$854$	0	0
$855$	101.808	3.48177
$856$	0	0
$857$	−54.1519	−1.84979	−0.924897	−	0.380217i	$-0.875849\pi$
$857$	−54.1519	−1.84979	−0.924897	+	0.380217i	$0.875849\pi$
$858$	0	0
$859$	16.5943	0.566190	0.283095	−	0.959092i	$-0.408639\pi$
$859$	16.5943	0.566190	0.283095	+	0.959092i	$0.408639\pi$
$860$	0	0
$861$	−16.9486	−0.577606
$862$	0	0
$863$	34.3208	1.16830	0.584148	−	0.811648i	$-0.301429\pi$
$863$	34.3208	1.16830	0.584148	+	0.811648i	$0.301429\pi$
$864$	0	0
$865$	−14.0190	−0.476661
$866$	0	0
$867$	3.83695	0.130310
$868$	0	0
$869$	−8.30763	−0.281817
$870$	0	0
$871$	6.66558	0.225855
$872$	0	0
$873$	4.45191	0.150674
$874$	0	0
$875$	6.44952	0.218034
$876$	0	0
$877$	3.56028	0.120222	0.0601110	−	0.998192i	$-0.480855\pi$
$877$	3.56028	0.120222	0.0601110	+	0.998192i	$0.480855\pi$
$878$	0	0
$879$	3.88917	0.131179
$880$	0	0
$881$	25.4637	0.857894	0.428947	−	0.903330i	$-0.358885\pi$
$881$	25.4637	0.857894	0.428947	+	0.903330i	$0.358885\pi$
$882$	0	0
$883$	−43.7365	−1.47185	−0.735925	−	0.677063i	$-0.763252\pi$
$883$	−43.7365	−1.47185	−0.735925	+	0.677063i	$0.763252\pi$
$884$	0	0
$885$	−82.6193	−2.77722
$886$	0	0
$887$	−4.63265	−0.155549	−0.0777745	−	0.996971i	$-0.524781\pi$
$887$	−4.63265	−0.155549	−0.0777745	+	0.996971i	$0.524781\pi$
$888$	0	0
$889$	−5.96255	−0.199977
$890$	0	0
$891$	−13.9304	−0.466685
$892$	0	0
$893$	−53.0177	−1.77417
$894$	0	0
$895$	15.5544	0.519928
$896$	0	0
$897$	−0.170581	−0.00569555
$898$	0	0
$899$	5.34683	0.178327
$900$	0	0
$901$	−47.0779	−1.56839
$902$	0	0
$903$	10.7310	0.357105
$904$	0	0
$905$	9.65548	0.320959
$906$	0	0
$907$	−35.0844	−1.16496	−0.582478	−	0.812846i	$-0.697917\pi$
$907$	−35.0844	−1.16496	−0.582478	+	0.812846i	$0.697917\pi$
$908$	0	0
$909$	48.1291	1.59634
$910$	0	0
$911$	−12.3081	−0.407785	−0.203893	−	0.978993i	$-0.565359\pi$
$911$	−12.3081	−0.407785	−0.203893	+	0.978993i	$0.565359\pi$
$912$	0	0
$913$	3.74998	0.124106
$914$	0	0
$915$	−55.2888	−1.82779
$916$	0	0
$917$	−0.636712	−0.0210261
$918$	0	0
$919$	−14.0514	−0.463515	−0.231757	−	0.972774i	$-0.574448\pi$
$919$	−14.0514	−0.463515	−0.231757	+	0.972774i	$0.574448\pi$
$920$	0	0
$921$	−41.4665	−1.36637
$922$	0	0
$923$	12.7964	0.421199
$924$	0	0
$925$	1.15020	0.0378183
$926$	0	0
$927$	73.0038	2.39776
$928$	0	0
$929$	10.6938	0.350853	0.175427	−	0.984493i	$-0.443870\pi$
$929$	10.6938	0.350853	0.175427	+	0.984493i	$0.443870\pi$
$930$	0	0
$931$	39.7356	1.30228
$932$	0	0
$933$	−56.3146	−1.84366
$934$	0	0
$935$	−5.08847	−0.166411
$936$	0	0
$937$	−1.62113	−0.0529601	−0.0264800	−	0.999649i	$-0.508430\pi$
$937$	−1.62113	−0.0529601	−0.0264800	+	0.999649i	$0.508430\pi$
$938$	0	0
$939$	19.1815	0.625966
$940$	0	0
$941$	21.9839	0.716654	0.358327	−	0.933596i	$-0.383347\pi$
$941$	21.9839	0.716654	0.358327	+	0.933596i	$0.383347\pi$
$942$	0	0
$943$	0.265268	0.00863831
$944$	0	0
$945$	19.4669	0.633260
$946$	0	0
$947$	11.7706	0.382492	0.191246	−	0.981542i	$-0.438747\pi$
$947$	11.7706	0.382492	0.191246	+	0.981542i	$0.438747\pi$
$948$	0	0
$949$	8.39691	0.272575
$950$	0	0
$951$	54.3239	1.76157
$952$	0	0
$953$	10.6036	0.343485	0.171743	−	0.985142i	$-0.445060\pi$
$953$	10.6036	0.343485	0.171743	+	0.985142i	$0.445060\pi$
$954$	0	0
$955$	41.0199	1.32737
$956$	0	0
$957$	3.68742	0.119197
$958$	0	0
$959$	6.98291	0.225490
$960$	0	0
$961$	−24.0074	−0.774431
$962$	0	0
$963$	8.52501	0.274715
$964$	0	0
$965$	−7.58495	−0.244168
$966$	0	0
$967$	−7.34001	−0.236039	−0.118019	−	0.993011i	$-0.537654\pi$
$967$	−7.34001	−0.236039	−0.118019	+	0.993011i	$0.537654\pi$
$968$	0	0
$969$	−76.9421	−2.47173
$970$	0	0
$971$	−46.6664	−1.49760	−0.748799	−	0.662797i	$-0.769369\pi$
$971$	−46.6664	−1.49760	−0.748799	+	0.662797i	$0.769369\pi$
$972$	0	0
$973$	−11.2287	−0.359977
$974$	0	0
$975$	−0.966625	−0.0309568
$976$	0	0
$977$	21.3732	0.683789	0.341895	−	0.939738i	$-0.388931\pi$
$977$	21.3732	0.683789	0.341895	+	0.939738i	$0.388931\pi$
$978$	0	0
$979$	2.59208	0.0828432
$980$	0	0
$981$	99.7850	3.18589
$982$	0	0
$983$	14.4017	0.459344	0.229672	−	0.973268i	$-0.426235\pi$
$983$	14.4017	0.459344	0.229672	+	0.973268i	$0.426235\pi$
$984$	0	0
$985$	36.4070	1.16002
$986$	0	0
$987$	−16.8957	−0.537796
$988$	0	0
$989$	−0.167954	−0.00534064
$990$	0	0
$991$	−6.12762	−0.194650	−0.0973251	−	0.995253i	$-0.531029\pi$
$991$	−6.12762	−0.194650	−0.0973251	+	0.995253i	$0.531029\pi$
$992$	0	0
$993$	−93.4478	−2.96548
$994$	0	0
$995$	33.5822	1.06463
$996$	0	0
$997$	16.8290	0.532980	0.266490	−	0.963838i	$-0.414136\pi$
$997$	16.8290	0.532980	0.266490	+	0.963838i	$0.414136\pi$
$998$	0	0
$999$	−99.3918	−3.14462

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4376.2.a.i.1.3	✓	31
4.3	odd	2		8752.2.a.x.1.29		31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4376.2.a.i.1.3	✓	31	1.1	even	1	trivial
8752.2.a.x.1.29		31	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4376 = 2^{3} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4376.a (trivial)

\(f(q)\)	\(=\)	\(q-3.24040 q^{3} -2.27349 q^{5} +0.587186 q^{7} +7.50019 q^{9} +O(q^{10})\)	\(q-3.24040 q^{3} -2.27349 q^{5} +0.587186 q^{7} +7.50019 q^{9} -0.562791 q^{11} +1.76770 q^{13} +7.36701 q^{15} -3.97692 q^{17} -5.97060 q^{19} -1.90272 q^{21} +0.0297800 q^{23} +0.168753 q^{25} -14.5824 q^{27} +2.02198 q^{29} +2.64436 q^{31} +1.82367 q^{33} -1.33496 q^{35} +6.81586 q^{37} -5.72804 q^{39} +8.90757 q^{41} -5.63983 q^{43} -17.0516 q^{45} +8.87979 q^{47} -6.65521 q^{49} +12.8868 q^{51} +11.8378 q^{53} +1.27950 q^{55} +19.3471 q^{57} -11.2148 q^{59} -7.50492 q^{61} +4.40400 q^{63} -4.01884 q^{65} +3.77078 q^{67} -0.0964993 q^{69} +7.23903 q^{71} +4.75020 q^{73} -0.546828 q^{75} -0.330463 q^{77} +14.7615 q^{79} +24.7523 q^{81} -6.66318 q^{83} +9.04149 q^{85} -6.55202 q^{87} -4.60576 q^{89} +1.03797 q^{91} -8.56877 q^{93} +13.5741 q^{95} +0.593574 q^{97} -4.22104 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 8 q^{3} - 11 q^{5} - 10 q^{7} + 39 q^{9}+O(q^{10}) \) 31 * q - 8 * q^3 - 11 * q^5 - 10 * q^7 + 39 * q^9	\( 31 q - 8 q^{3} - 11 q^{5} - 10 q^{7} + 39 q^{9} - 11 q^{11} - 20 q^{13} - 11 q^{15} + 26 q^{17} - 32 q^{19} - 23 q^{21} - 29 q^{23} + 26 q^{25} - 53 q^{27} - 6 q^{29} - 53 q^{31} + 8 q^{33} - 13 q^{35} - 26 q^{37} - 24 q^{39} - 6 q^{41} - 11 q^{43} - 12 q^{45} - 41 q^{47} + 29 q^{49} - 35 q^{51} - 17 q^{53} - 10 q^{55} + 9 q^{57} - 27 q^{59} - 62 q^{61} - 34 q^{63} + 4 q^{65} - 28 q^{67} - 20 q^{69} - 39 q^{71} + 16 q^{73} - 36 q^{75} - 19 q^{77} - 39 q^{79} + 35 q^{81} - 32 q^{83} - 58 q^{85} - 38 q^{87} + 61 q^{89} - 25 q^{91} - q^{93} - 12 q^{95} - 85 q^{99}+O(q^{100}) \) 31 * q - 8 * q^3 - 11 * q^5 - 10 * q^7 + 39 * q^9 - 11 * q^11 - 20 * q^13 - 11 * q^15 + 26 * q^17 - 32 * q^19 - 23 * q^21 - 29 * q^23 + 26 * q^25 - 53 * q^27 - 6 * q^29 - 53 * q^31 + 8 * q^33 - 13 * q^35 - 26 * q^37 - 24 * q^39 - 6 * q^41 - 11 * q^43 - 12 * q^45 - 41 * q^47 + 29 * q^49 - 35 * q^51 - 17 * q^53 - 10 * q^55 + 9 * q^57 - 27 * q^59 - 62 * q^61 - 34 * q^63 + 4 * q^65 - 28 * q^67 - 20 * q^69 - 39 * q^71 + 16 * q^73 - 36 * q^75 - 19 * q^77 - 39 * q^79 + 35 * q^81 - 32 * q^83 - 58 * q^85 - 38 * q^87 + 61 * q^89 - 25 * q^91 - q^93 - 12 * q^95 - 85 * q^99

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