LMFDB - Embedded newform 7623.2.a.cm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 2x + 1 $ x^4 - 2x^3 - 4x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.326909$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.05896 q^{2} +2.23931 q^{4} +1.05896 q^{5} -1.00000 q^{7} -0.492737 q^{8} +O(q^{10})$	$q-2.05896 q^{2} +2.23931 q^{4} +1.05896 q^{5} -1.00000 q^{7} -0.492737 q^{8} -2.18035 q^{10} +3.80554 q^{13} +2.05896 q^{14} -3.46410 q^{16} +2.56622 q^{17} -0.180354 q^{19} +2.37134 q^{20} -0.433778 q^{23} -3.87861 q^{25} -7.83544 q^{26} -2.23931 q^{28} +9.03032 q^{29} -0.492737 q^{31} +8.11792 q^{32} -5.28375 q^{34} -1.05896 q^{35} -0.775212 q^{37} +0.371342 q^{38} -0.521789 q^{40} +3.09276 q^{41} -1.61413 q^{43} +0.893131 q^{46} -0.502098 q^{47} +1.00000 q^{49} +7.98589 q^{50} +8.52179 q^{52} +9.94273 q^{53} +0.492737 q^{56} -18.5931 q^{58} -13.4513 q^{59} +5.93756 q^{61} +1.01453 q^{62} -9.78626 q^{64} +4.02991 q^{65} -14.4068 q^{67} +5.74658 q^{68} +2.18035 q^{70} +12.4990 q^{71} -4.27383 q^{73} +1.59613 q^{74} -0.403870 q^{76} +1.37176 q^{79} -3.66834 q^{80} -6.36787 q^{82} +7.47346 q^{83} +2.71753 q^{85} +3.32343 q^{86} +16.1341 q^{89} -3.80554 q^{91} -0.971364 q^{92} +1.03380 q^{94} -0.190988 q^{95} +14.6158 q^{97} -2.05896 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 4 * q^7	$ 4 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 4 q^{7} - 14 q^{10} + 2 q^{13} - 2 q^{14} + 2 q^{17} - 6 q^{19} - 8 q^{20} - 10 q^{23} - 4 q^{28} + 14 q^{29} + 12 q^{32} - 2 q^{34} + 6 q^{35} - 12 q^{37} - 16 q^{38} - 8 q^{40} + 16 q^{41} - 20 q^{43} - 8 q^{46} - 2 q^{47} + 4 q^{49} + 24 q^{50} + 40 q^{52} + 16 q^{53} - 8 q^{58} - 2 q^{59} - 2 q^{61} + 8 q^{62} - 16 q^{64} - 2 q^{65} - 20 q^{67} + 20 q^{68} + 14 q^{70} + 10 q^{71} + 10 q^{73} - 20 q^{74} - 28 q^{76} - 16 q^{79} - 12 q^{80} + 28 q^{82} + 18 q^{83} - 26 q^{86} + 14 q^{89} - 2 q^{91} + 8 q^{92} + 18 q^{94} + 22 q^{95} + 38 q^{97} + 2 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 4 * q^7 - 14 * q^10 + 2 * q^13 - 2 * q^14 + 2 * q^17 - 6 * q^19 - 8 * q^20 - 10 * q^23 - 4 * q^28 + 14 * q^29 + 12 * q^32 - 2 * q^34 + 6 * q^35 - 12 * q^37 - 16 * q^38 - 8 * q^40 + 16 * q^41 - 20 * q^43 - 8 * q^46 - 2 * q^47 + 4 * q^49 + 24 * q^50 + 40 * q^52 + 16 * q^53 - 8 * q^58 - 2 * q^59 - 2 * q^61 + 8 * q^62 - 16 * q^64 - 2 * q^65 - 20 * q^67 + 20 * q^68 + 14 * q^70 + 10 * q^71 + 10 * q^73 - 20 * q^74 - 28 * q^76 - 16 * q^79 - 12 * q^80 + 28 * q^82 + 18 * q^83 - 26 * q^86 + 14 * q^89 - 2 * q^91 + 8 * q^92 + 18 * q^94 + 22 * q^95 + 38 * q^97 + 2 * q^98

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 $

$2$	−2.05896	−1.45590	−0.727952	−	0.685628i	$-0.759528\pi$
$2$	−2.05896	−1.45590	−0.727952	+	0.685628i	$0.759528\pi$
$3$	0	0
$3$	0	0
$4$	2.23931	1.11966
$5$	1.05896	0.473581	0.236791	−	0.971561i	$-0.423904\pi$
$5$	1.05896	0.473581	0.236791	+	0.971561i	$0.423904\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.492737	−0.174209
$9$	0	0
$10$	−2.18035	−0.689489
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.80554	1.05547	0.527733	−	0.849410i	$-0.323042\pi$
$13$	3.80554	1.05547	0.527733	+	0.849410i	$0.323042\pi$
$14$	2.05896	0.550280
$15$	0	0
$16$	−3.46410	−0.866025
$17$	2.56622	0.622400	0.311200	−	0.950344i	$-0.399269\pi$
$17$	2.56622	0.622400	0.311200	+	0.950344i	$0.399269\pi$
$18$	0	0
$19$	−0.180354	−0.0413761	−0.0206881	−	0.999786i	$-0.506586\pi$
$19$	−0.180354	−0.0413761	−0.0206881	+	0.999786i	$0.506586\pi$
$20$	2.37134	0.530248
$21$	0	0
$22$	0	0
$23$	−0.433778	−0.0904489	−0.0452245	−	0.998977i	$-0.514400\pi$
$23$	−0.433778	−0.0904489	−0.0452245	+	0.998977i	$0.514400\pi$
$24$	0	0
$25$	−3.87861	−0.775721
$26$	−7.83544	−1.53666
$27$	0	0
$28$	−2.23931	−0.423191
$29$	9.03032	1.67689	0.838445	−	0.544987i	$-0.183465\pi$
$29$	9.03032	1.67689	0.838445	+	0.544987i	$0.183465\pi$
$30$	0	0
$31$	−0.492737	−0.0884982	−0.0442491	−	0.999021i	$-0.514090\pi$
$31$	−0.492737	−0.0884982	−0.0442491	+	0.999021i	$0.514090\pi$
$32$	8.11792	1.43506
$33$	0	0
$34$	−5.28375	−0.906155
$35$	−1.05896	−0.178997
$36$	0	0
$37$	−0.775212	−0.127444	−0.0637220	−	0.997968i	$-0.520297\pi$
$37$	−0.775212	−0.127444	−0.0637220	+	0.997968i	$0.520297\pi$
$38$	0.371342	0.0602397
$39$	0	0
$40$	−0.521789	−0.0825020
$41$	3.09276	0.483008	0.241504	−	0.970400i	$-0.422359\pi$
$41$	3.09276	0.483008	0.241504	+	0.970400i	$0.422359\pi$
$42$	0	0
$43$	−1.61413	−0.246153	−0.123076	−	0.992397i	$-0.539276\pi$
$43$	−1.61413	−0.246153	−0.123076	+	0.992397i	$0.539276\pi$
$44$	0	0
$45$	0	0
$46$	0.893131	0.131685
$47$	−0.502098	−0.0732386	−0.0366193	−	0.999329i	$-0.511659\pi$
$47$	−0.502098	−0.0732386	−0.0366193	+	0.999329i	$0.511659\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.98589	1.12938
$51$	0	0
$52$	8.52179	1.18176
$53$	9.94273	1.36574	0.682869	−	0.730540i	$-0.260732\pi$
$53$	9.94273	1.36574	0.682869	+	0.730540i	$0.260732\pi$
$54$	0	0
$55$	0	0
$56$	0.492737	0.0658448
$57$	0	0
$58$	−18.5931	−2.44139
$59$	−13.4513	−1.75121	−0.875603	−	0.483032i	$-0.839535\pi$
$59$	−13.4513	−1.75121	−0.875603	+	0.483032i	$0.839535\pi$
$60$	0	0
$61$	5.93756	0.760227	0.380114	−	0.924940i	$-0.375885\pi$
$61$	5.93756	0.760227	0.380114	+	0.924940i	$0.375885\pi$
$62$	1.01453	0.128845
$63$	0	0
$64$	−9.78626	−1.22328
$65$	4.02991	0.499849
$66$	0	0
$67$	−14.4068	−1.76007	−0.880037	−	0.474905i	$-0.842483\pi$
$67$	−14.4068	−1.76007	−0.880037	+	0.474905i	$0.842483\pi$
$68$	5.74658	0.696875
$69$	0	0
$70$	2.18035	0.260602
$71$	12.4990	1.48336	0.741681	−	0.670752i	$-0.234029\pi$
$71$	12.4990	1.48336	0.741681	+	0.670752i	$0.234029\pi$
$72$	0	0
$73$	−4.27383	−0.500214	−0.250107	−	0.968218i	$-0.580466\pi$
$73$	−4.27383	−0.500214	−0.250107	+	0.968218i	$0.580466\pi$
$74$	1.59613	0.185546
$75$	0	0
$76$	−0.403870	−0.0463271
$77$	0	0
$78$	0	0
$79$	1.37176	0.154335	0.0771674	−	0.997018i	$-0.475412\pi$
$79$	1.37176	0.154335	0.0771674	+	0.997018i	$0.475412\pi$
$80$	−3.66834	−0.410133
$81$	0	0
$82$	−6.36787	−0.703213
$83$	7.47346	0.820319	0.410160	−	0.912014i	$-0.365473\pi$
$83$	7.47346	0.820319	0.410160	+	0.912014i	$0.365473\pi$
$84$	0	0
$85$	2.71753	0.294757
$86$	3.32343	0.358375
$87$	0	0
$88$	0	0
$89$	16.1341	1.71021	0.855107	−	0.518451i	$-0.173491\pi$
$89$	16.1341	1.71021	0.855107	+	0.518451i	$0.173491\pi$
$90$	0	0
$91$	−3.80554	−0.398929
$92$	−0.971364	−0.101272
$93$	0	0
$94$	1.03380	0.106628
$95$	−0.190988	−0.0195949
$96$	0	0
$97$	14.6158	1.48401	0.742006	−	0.670393i	$-0.233875\pi$
$97$	14.6158	1.48401	0.742006	+	0.670393i	$0.233875\pi$
$98$	−2.05896	−0.207986
$99$	0	0
$100$	−8.68541	−0.868541
$101$	6.10856	0.607824	0.303912	−	0.952700i	$-0.401707\pi$
$101$	6.10856	0.607824	0.303912	+	0.952700i	$0.401707\pi$
$102$	0	0
$103$	−9.56275	−0.942245	−0.471123	−	0.882068i	$-0.656151\pi$
$103$	−9.56275	−0.942245	−0.471123	+	0.882068i	$0.656151\pi$
$104$	−1.87513	−0.183872
$105$	0	0
$106$	−20.4717	−1.98838
$107$	4.59486	0.444202	0.222101	−	0.975024i	$-0.428709\pi$
$107$	4.59486	0.444202	0.222101	+	0.975024i	$0.428709\pi$
$108$	0	0
$109$	−1.99009	−0.190616	−0.0953079	−	0.995448i	$-0.530384\pi$
$109$	−1.99009	−0.190616	−0.0953079	+	0.995448i	$0.530384\pi$
$110$	0	0
$111$	0	0
$112$	3.46410	0.327327
$113$	−8.61527	−0.810456	−0.405228	−	0.914216i	$-0.632808\pi$
$113$	−8.61527	−0.810456	−0.405228	+	0.914216i	$0.632808\pi$
$114$	0	0
$115$	−0.459353	−0.0428349
$116$	20.2217	1.87754
$117$	0	0
$118$	27.6956	2.54959
$119$	−2.56622	−0.235245
$120$	0	0
$121$	0	0
$122$	−12.2252	−1.10682
$123$	0	0
$124$	−1.10339	−0.0990876
$125$	−9.40208	−0.840948
$126$	0	0
$127$	19.1294	1.69746	0.848729	−	0.528828i	$-0.177368\pi$
$127$	19.1294	1.69746	0.848729	+	0.528828i	$0.177368\pi$
$128$	3.91368	0.345923
$129$	0	0
$130$	−8.29742	−0.727732
$131$	−6.91998	−0.604601	−0.302301	−	0.953213i	$-0.597755\pi$
$131$	−6.91998	−0.604601	−0.302301	+	0.953213i	$0.597755\pi$
$132$	0	0
$133$	0.180354	0.0156387
$134$	29.6631	2.56250
$135$	0	0
$136$	−1.26447	−0.108428
$137$	−6.81197	−0.581986	−0.290993	−	0.956725i	$-0.593986\pi$
$137$	−6.81197	−0.581986	−0.290993	+	0.956725i	$0.593986\pi$
$138$	0	0
$139$	−19.6013	−1.66256	−0.831280	−	0.555854i	$-0.812391\pi$
$139$	−19.6013	−1.66256	−0.831280	+	0.555854i	$0.812391\pi$
$140$	−2.37134	−0.200415
$141$	0	0
$142$	−25.7350	−2.15963
$143$	0	0
$144$	0	0
$145$	9.56275	0.794143
$146$	8.79965	0.728264
$147$	0	0
$148$	−1.73594	−0.142694
$149$	−8.18721	−0.670722	−0.335361	−	0.942090i	$-0.608858\pi$
$149$	−8.18721	−0.670722	−0.335361	+	0.942090i	$0.608858\pi$
$150$	0	0
$151$	−17.1576	−1.39627	−0.698133	−	0.715968i	$-0.745986\pi$
$151$	−17.1576	−1.39627	−0.698133	+	0.715968i	$0.745986\pi$
$152$	0.0888673	0.00720809
$153$	0	0
$154$	0	0
$155$	−0.521789	−0.0419111
$156$	0	0
$157$	5.02864	0.401329	0.200664	−	0.979660i	$-0.435690\pi$
$157$	5.02864	0.401329	0.200664	+	0.979660i	$0.435690\pi$
$158$	−2.82439	−0.224697
$159$	0	0
$160$	8.59655	0.679617
$161$	0.433778	0.0341865
$162$	0	0
$163$	15.2469	1.19423	0.597114	−	0.802156i	$-0.296314\pi$
$163$	15.2469	1.19423	0.597114	+	0.802156i	$0.296314\pi$
$164$	6.92566	0.540803
$165$	0	0
$166$	−15.3876	−1.19431
$167$	−4.43072	−0.342859	−0.171430	−	0.985196i	$-0.554839\pi$
$167$	−4.43072	−0.342859	−0.171430	+	0.985196i	$0.554839\pi$
$168$	0	0
$169$	1.48210	0.114008
$170$	−5.59527	−0.429138
$171$	0	0
$172$	−3.61455	−0.275607
$173$	10.7124	0.814446	0.407223	−	0.913329i	$-0.366497\pi$
$173$	10.7124	0.814446	0.407223	+	0.913329i	$0.366497\pi$
$174$	0	0
$175$	3.87861	0.293195
$176$	0	0
$177$	0	0
$178$	−33.2195	−2.48991
$179$	−11.4546	−0.856157	−0.428079	−	0.903741i	$-0.640809\pi$
$179$	−11.4546	−0.856157	−0.428079	+	0.903741i	$0.640809\pi$
$180$	0	0
$181$	16.5252	1.22831	0.614153	−	0.789187i	$-0.289498\pi$
$181$	16.5252	1.22831	0.614153	+	0.789187i	$0.289498\pi$
$182$	7.83544	0.580802
$183$	0	0
$184$	0.213738	0.0157570
$185$	−0.820918	−0.0603551
$186$	0	0
$187$	0	0
$188$	−1.12436	−0.0820021
$189$	0	0
$190$	0.393236	0.0285284
$191$	2.19794	0.159037	0.0795187	−	0.996833i	$-0.474662\pi$
$191$	2.19794	0.159037	0.0795187	+	0.996833i	$0.474662\pi$
$192$	0	0
$193$	25.1481	1.81020	0.905100	−	0.425198i	$-0.139796\pi$
$193$	25.1481	1.81020	0.905100	+	0.425198i	$0.139796\pi$
$194$	−30.0934	−2.16058
$195$	0	0
$196$	2.23931	0.159951
$197$	−6.13203	−0.436889	−0.218444	−	0.975849i	$-0.570098\pi$
$197$	−6.13203	−0.436889	−0.218444	+	0.975849i	$0.570098\pi$
$198$	0	0
$199$	20.4956	1.45289	0.726446	−	0.687224i	$-0.241171\pi$
$199$	20.4956	1.45289	0.726446	+	0.687224i	$0.241171\pi$
$200$	1.91113	0.135137
$201$	0	0
$202$	−12.5773	−0.884934
$203$	−9.03032	−0.633804
$204$	0	0
$205$	3.27511	0.228743
$206$	19.6893	1.37182
$207$	0	0
$208$	−13.1828	−0.914060
$209$	0	0
$210$	0	0
$211$	−8.82942	−0.607843	−0.303921	−	0.952697i	$-0.598296\pi$
$211$	−8.82942	−0.607843	−0.303921	+	0.952697i	$0.598296\pi$
$212$	22.2649	1.52916
$213$	0	0
$214$	−9.46063	−0.646715
$215$	−1.70930	−0.116573
$216$	0	0
$217$	0.492737	0.0334492
$218$	4.09751	0.277518
$219$	0	0
$220$	0	0
$221$	9.76585	0.656922
$222$	0	0
$223$	−20.6012	−1.37956	−0.689778	−	0.724021i	$-0.742292\pi$
$223$	−20.6012	−1.37956	−0.689778	+	0.724021i	$0.742292\pi$
$224$	−8.11792	−0.542401
$225$	0	0
$226$	17.7385	1.17995
$227$	17.4349	1.15720	0.578598	−	0.815613i	$-0.303600\pi$
$227$	17.4349	1.15720	0.578598	+	0.815613i	$0.303600\pi$
$228$	0	0
$229$	23.4477	1.54946	0.774732	−	0.632289i	$-0.217885\pi$
$229$	23.4477	1.54946	0.774732	+	0.632289i	$0.217885\pi$
$230$	0.945789	0.0623635
$231$	0	0
$232$	−4.44958	−0.292129
$233$	−14.8692	−0.974117	−0.487058	−	0.873369i	$-0.661930\pi$
$233$	−14.8692	−0.974117	−0.487058	+	0.873369i	$0.661930\pi$
$234$	0	0
$235$	−0.531702	−0.0346844
$236$	−30.1216	−1.96075
$237$	0	0
$238$	5.28375	0.342494
$239$	22.5721	1.46007	0.730034	−	0.683411i	$-0.239504\pi$
$239$	22.5721	1.46007	0.730034	+	0.683411i	$0.239504\pi$
$240$	0	0
$241$	−2.20245	−0.141872	−0.0709362	−	0.997481i	$-0.522599\pi$
$241$	−2.20245	−0.141872	−0.0709362	+	0.997481i	$0.522599\pi$
$242$	0	0
$243$	0	0
$244$	13.2961	0.851194
$245$	1.05896	0.0676544
$246$	0	0
$247$	−0.686345	−0.0436711
$248$	0.242790	0.0154172
$249$	0	0
$250$	19.3585	1.22434
$251$	−7.39733	−0.466916	−0.233458	−	0.972367i	$-0.575004\pi$
$251$	−7.39733	−0.466916	−0.233458	+	0.972367i	$0.575004\pi$
$252$	0	0
$253$	0	0
$254$	−39.3866	−2.47134
$255$	0	0
$256$	11.5144	0.719651
$257$	−6.29786	−0.392850	−0.196425	−	0.980519i	$-0.562933\pi$
$257$	−6.29786	−0.392850	−0.196425	+	0.980519i	$0.562933\pi$
$258$	0	0
$259$	0.775212	0.0481693
$260$	9.02423	0.559659
$261$	0	0
$262$	14.2480	0.880242
$263$	32.2131	1.98634	0.993172	−	0.116659i	$-0.0372185\pi$
$263$	32.2131	1.98634	0.993172	+	0.116659i	$0.0372185\pi$
$264$	0	0
$265$	10.5289	0.646788
$266$	−0.371342	−0.0227685
$267$	0	0
$268$	−32.2614	−1.97068
$269$	−27.6136	−1.68363	−0.841816	−	0.539765i	$-0.818513\pi$
$269$	−27.6136	−1.68363	−0.841816	+	0.539765i	$0.818513\pi$
$270$	0	0
$271$	30.9678	1.88116	0.940580	−	0.339573i	$-0.110283\pi$
$271$	30.9678	1.88116	0.940580	+	0.339573i	$0.110283\pi$
$272$	−8.88965	−0.539014
$273$	0	0
$274$	14.0256	0.847316
$275$	0	0
$276$	0	0
$277$	−0.420940	−0.0252919	−0.0126459	−	0.999920i	$-0.504025\pi$
$277$	−0.420940	−0.0252919	−0.0126459	+	0.999920i	$0.504025\pi$
$278$	40.3583	2.42053
$279$	0	0
$280$	0.521789	0.0311828
$281$	19.8996	1.18711	0.593554	−	0.804794i	$-0.297724\pi$
$281$	19.8996	1.18711	0.593554	+	0.804794i	$0.297724\pi$
$282$	0	0
$283$	−24.3641	−1.44829	−0.724147	−	0.689645i	$-0.757766\pi$
$283$	−24.3641	−1.44829	−0.724147	+	0.689645i	$0.757766\pi$
$284$	27.9893	1.66086
$285$	0	0
$286$	0	0
$287$	−3.09276	−0.182560
$288$	0	0
$289$	−10.4145	−0.612618
$290$	−19.6893	−1.15620
$291$	0	0
$292$	−9.57046	−0.560069
$293$	−17.4662	−1.02039	−0.510193	−	0.860060i	$-0.670426\pi$
$293$	−17.4662	−1.02039	−0.510193	+	0.860060i	$0.670426\pi$
$294$	0	0
$295$	−14.2443	−0.829338
$296$	0.381976	0.0222019
$297$	0	0
$298$	16.8571	0.976507
$299$	−1.65076	−0.0954657
$300$	0	0
$301$	1.61413	0.0930370
$302$	35.3268	2.03283
$303$	0	0
$304$	0.624766	0.0358328
$305$	6.28764	0.360029
$306$	0	0
$307$	−10.9239	−0.623458	−0.311729	−	0.950171i	$-0.600908\pi$
$307$	−10.9239	−0.623458	−0.311729	+	0.950171i	$0.600908\pi$
$308$	0	0
$309$	0	0
$310$	1.07434	0.0610185
$311$	−3.51662	−0.199409	−0.0997047	−	0.995017i	$-0.531790\pi$
$311$	−3.51662	−0.199409	−0.0997047	+	0.995017i	$0.531790\pi$
$312$	0	0
$313$	25.9789	1.46842	0.734208	−	0.678924i	$-0.237554\pi$
$313$	25.9789	1.46842	0.734208	+	0.678924i	$0.237554\pi$
$314$	−10.3538	−0.584296
$315$	0	0
$316$	3.07180	0.172802
$317$	3.72269	0.209087	0.104544	−	0.994520i	$-0.466662\pi$
$317$	3.72269	0.209087	0.104544	+	0.994520i	$0.466662\pi$
$318$	0	0
$319$	0	0
$320$	−10.3633	−0.579323
$321$	0	0
$322$	−0.893131	−0.0497722
$323$	−0.462829	−0.0257525
$324$	0	0
$325$	−14.7602	−0.818747
$326$	−31.3927	−1.73868
$327$	0	0
$328$	−1.52392	−0.0841443
$329$	0.502098	0.0276816
$330$	0	0
$331$	17.7683	0.976632	0.488316	−	0.872667i	$-0.337611\pi$
$331$	17.7683	0.976632	0.488316	+	0.872667i	$0.337611\pi$
$332$	16.7354	0.918476
$333$	0	0
$334$	9.12267	0.499170
$335$	−15.2562	−0.833538
$336$	0	0
$337$	−26.1178	−1.42273	−0.711364	−	0.702824i	$-0.751922\pi$
$337$	−26.1178	−1.42273	−0.711364	+	0.702824i	$0.751922\pi$
$338$	−3.05159	−0.165985
$339$	0	0
$340$	6.08539	0.330027
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0.795343	0.0428820
$345$	0	0
$346$	−22.0563	−1.18575
$347$	0.794486	0.0426503	0.0213251	−	0.999773i	$-0.493211\pi$
$347$	0.794486	0.0426503	0.0213251	+	0.999773i	$0.493211\pi$
$348$	0	0
$349$	−11.7816	−0.630657	−0.315329	−	0.948983i	$-0.602115\pi$
$349$	−11.7816	−0.630657	−0.315329	+	0.948983i	$0.602115\pi$
$350$	−7.98589	−0.426864
$351$	0	0
$352$	0	0
$353$	−6.45041	−0.343321	−0.171660	−	0.985156i	$-0.554913\pi$
$353$	−6.45041	−0.343321	−0.171660	+	0.985156i	$0.554913\pi$
$354$	0	0
$355$	13.2360	0.702493
$356$	36.1294	1.91485
$357$	0	0
$358$	23.5846	1.24648
$359$	5.87916	0.310290	0.155145	−	0.987892i	$-0.450416\pi$
$359$	5.87916	0.310290	0.155145	+	0.987892i	$0.450416\pi$
$360$	0	0
$361$	−18.9675	−0.998288
$362$	−34.0246	−1.78830
$363$	0	0
$364$	−8.52179	−0.446663
$365$	−4.52582	−0.236892
$366$	0	0
$367$	−29.3478	−1.53194	−0.765971	−	0.642876i	$-0.777741\pi$
$367$	−29.3478	−1.53194	−0.765971	+	0.642876i	$0.777741\pi$
$368$	1.50265	0.0783311
$369$	0	0
$370$	1.69024	0.0878712
$371$	−9.94273	−0.516201
$372$	0	0
$373$	8.83207	0.457307	0.228654	−	0.973508i	$-0.426568\pi$
$373$	8.83207	0.457307	0.228654	+	0.973508i	$0.426568\pi$
$374$	0	0
$375$	0	0
$376$	0.247402	0.0127588
$377$	34.3652	1.76990
$378$	0	0
$379$	−11.7171	−0.601867	−0.300934	−	0.953645i	$-0.597298\pi$
$379$	−11.7171	−0.601867	−0.300934	+	0.953645i	$0.597298\pi$
$380$	−0.427682	−0.0219396
$381$	0	0
$382$	−4.52547	−0.231543
$383$	−30.6516	−1.56622	−0.783112	−	0.621881i	$-0.786369\pi$
$383$	−30.6516	−1.56622	−0.783112	+	0.621881i	$0.786369\pi$
$384$	0	0
$385$	0	0
$386$	−51.7789	−2.63548
$387$	0	0
$388$	32.7294	1.66158
$389$	3.61107	0.183089	0.0915443	−	0.995801i	$-0.470820\pi$
$389$	3.61107	0.183089	0.0915443	+	0.995801i	$0.470820\pi$
$390$	0	0
$391$	−1.11317	−0.0562954
$392$	−0.492737	−0.0248870
$393$	0	0
$394$	12.6256	0.636068
$395$	1.45264	0.0730901
$396$	0	0
$397$	3.10339	0.155755	0.0778774	−	0.996963i	$-0.475186\pi$
$397$	3.10339	0.155755	0.0778774	+	0.996963i	$0.475186\pi$
$398$	−42.1995	−2.11527
$399$	0	0
$400$	13.4359	0.671794
$401$	−29.8141	−1.48884	−0.744422	−	0.667709i	$-0.767275\pi$
$401$	−29.8141	−1.48884	−0.744422	+	0.667709i	$0.767275\pi$
$402$	0	0
$403$	−1.87513	−0.0934068
$404$	13.6790	0.680555
$405$	0	0
$406$	18.5931	0.922759
$407$	0	0
$408$	0	0
$409$	−19.0348	−0.941212	−0.470606	−	0.882343i	$-0.655965\pi$
$409$	−19.0348	−0.941212	−0.470606	+	0.882343i	$0.655965\pi$
$410$	−6.74331	−0.333028
$411$	0	0
$412$	−21.4140	−1.05499
$413$	13.4513	0.661893
$414$	0	0
$415$	7.91409	0.388488
$416$	30.8930	1.51466
$417$	0	0
$418$	0	0
$419$	−6.05476	−0.295795	−0.147897	−	0.989003i	$-0.547250\pi$
$419$	−6.05476	−0.295795	−0.147897	+	0.989003i	$0.547250\pi$
$420$	0	0
$421$	29.2007	1.42315	0.711577	−	0.702608i	$-0.247981\pi$
$421$	29.2007	1.42315	0.711577	+	0.702608i	$0.247981\pi$
$422$	18.1794	0.884960
$423$	0	0
$424$	−4.89915	−0.237924
$425$	−9.95336	−0.482809
$426$	0	0
$427$	−5.93756	−0.287339
$428$	10.2893	0.497353
$429$	0	0
$430$	3.51938	0.169720
$431$	15.3401	0.738905	0.369452	−	0.929250i	$-0.379545\pi$
$431$	15.3401	0.738905	0.369452	+	0.929250i	$0.379545\pi$
$432$	0	0
$433$	4.47808	0.215203	0.107601	−	0.994194i	$-0.465683\pi$
$433$	4.47808	0.215203	0.107601	+	0.994194i	$0.465683\pi$
$434$	−1.01453	−0.0486988
$435$	0	0
$436$	−4.45643	−0.213424
$437$	0.0782337	0.00374242
$438$	0	0
$439$	−26.4717	−1.26342	−0.631712	−	0.775203i	$-0.717647\pi$
$439$	−26.4717	−1.26342	−0.631712	+	0.775203i	$0.717647\pi$
$440$	0	0
$441$	0	0
$442$	−20.1075	−0.956416
$443$	3.86545	0.183653	0.0918266	−	0.995775i	$-0.470729\pi$
$443$	3.86545	0.183653	0.0918266	+	0.995775i	$0.470729\pi$
$444$	0	0
$445$	17.0854	0.809925
$446$	42.4169	2.00850
$447$	0	0
$448$	9.78626	0.462357
$449$	7.26530	0.342871	0.171435	−	0.985195i	$-0.445159\pi$
$449$	7.26530	0.342871	0.171435	+	0.985195i	$0.445159\pi$
$450$	0	0
$451$	0	0
$452$	−19.2923	−0.907433
$453$	0	0
$454$	−35.8978	−1.68477
$455$	−4.02991	−0.188925
$456$	0	0
$457$	25.7449	1.20430	0.602148	−	0.798384i	$-0.294312\pi$
$457$	25.7449	1.20430	0.602148	+	0.798384i	$0.294312\pi$
$458$	−48.2778	−2.25587
$459$	0	0
$460$	−1.02864	−0.0479604
$461$	22.9619	1.06944	0.534721	−	0.845029i	$-0.320417\pi$
$461$	22.9619	1.06944	0.534721	+	0.845029i	$0.320417\pi$
$462$	0	0
$463$	4.68972	0.217950	0.108975	−	0.994044i	$-0.465243\pi$
$463$	4.68972	0.217950	0.108975	+	0.994044i	$0.465243\pi$
$464$	−31.2820	−1.45223
$465$	0	0
$466$	30.6152	1.41822
$467$	32.8849	1.52173	0.760867	−	0.648908i	$-0.224774\pi$
$467$	32.8849	1.52173	0.760867	+	0.648908i	$0.224774\pi$
$468$	0	0
$469$	14.4068	0.665246
$470$	1.09475	0.0504972
$471$	0	0
$472$	6.62794	0.305076
$473$	0	0
$474$	0	0
$475$	0.699523	0.0320963
$476$	−5.74658	−0.263394
$477$	0	0
$478$	−46.4751	−2.12572
$479$	10.3796	0.474255	0.237127	−	0.971479i	$-0.423794\pi$
$479$	10.3796	0.474255	0.237127	+	0.971479i	$0.423794\pi$
$480$	0	0
$481$	−2.95010	−0.134513
$482$	4.53476	0.206553
$483$	0	0
$484$	0	0
$485$	15.4776	0.702800
$486$	0	0
$487$	28.4384	1.28867	0.644334	−	0.764744i	$-0.277135\pi$
$487$	28.4384	1.28867	0.644334	+	0.764744i	$0.277135\pi$
$488$	−2.92566	−0.132438
$489$	0	0
$490$	−2.18035	−0.0984984
$491$	30.9474	1.39664	0.698318	−	0.715788i	$-0.253932\pi$
$491$	30.9474	1.39664	0.698318	+	0.715788i	$0.253932\pi$
$492$	0	0
$493$	23.1738	1.04370
$494$	1.41316	0.0635809
$495$	0	0
$496$	1.70689	0.0766417
$497$	−12.4990	−0.560658
$498$	0	0
$499$	−34.2751	−1.53436	−0.767182	−	0.641429i	$-0.778342\pi$
$499$	−34.2751	−1.53436	−0.767182	+	0.641429i	$0.778342\pi$
$500$	−21.0542	−0.941573
$501$	0	0
$502$	15.2308	0.679784
$503$	13.0742	0.582950	0.291475	−	0.956578i	$-0.405854\pi$
$503$	13.0742	0.582950	0.291475	+	0.956578i	$0.405854\pi$
$504$	0	0
$505$	6.46871	0.287854
$506$	0	0
$507$	0	0
$508$	42.8367	1.90057
$509$	32.5765	1.44393	0.721963	−	0.691932i	$-0.243240\pi$
$509$	32.5765	1.44393	0.721963	+	0.691932i	$0.243240\pi$
$510$	0	0
$511$	4.27383	0.189063
$512$	−31.5351	−1.39367
$513$	0	0
$514$	12.9670	0.571951
$515$	−10.1266	−0.446230
$516$	0	0
$517$	0	0
$518$	−1.59613	−0.0701299
$519$	0	0
$520$	−1.98569	−0.0870781
$521$	−11.0109	−0.482397	−0.241198	−	0.970476i	$-0.577540\pi$
$521$	−11.0109	−0.482397	−0.241198	+	0.970476i	$0.577540\pi$
$522$	0	0
$523$	−9.20204	−0.402377	−0.201189	−	0.979553i	$-0.564480\pi$
$523$	−9.20204	−0.402377	−0.201189	+	0.979553i	$0.564480\pi$
$524$	−15.4960	−0.676946
$525$	0	0
$526$	−66.3254	−2.89193
$527$	−1.26447	−0.0550813
$528$	0	0
$529$	−22.8118	−0.991819
$530$	−21.6787	−0.941661
$531$	0	0
$532$	0.403870	0.0175100
$533$	11.7696	0.509798
$534$	0	0
$535$	4.86577	0.210365
$536$	7.09878	0.306621
$537$	0	0
$538$	56.8553	2.45121
$539$	0	0
$540$	0	0
$541$	−9.28585	−0.399230	−0.199615	−	0.979874i	$-0.563969\pi$
$541$	−9.28585	−0.399230	−0.199615	+	0.979874i	$0.563969\pi$
$542$	−63.7614	−2.73879
$543$	0	0
$544$	20.8324	0.893181
$545$	−2.10742	−0.0902720
$546$	0	0
$547$	−5.40335	−0.231031	−0.115515	−	0.993306i	$-0.536852\pi$
$547$	−5.40335	−0.231031	−0.115515	+	0.993306i	$0.536852\pi$
$548$	−15.2541	−0.651625
$549$	0	0
$550$	0	0
$551$	−1.62866	−0.0693832
$552$	0	0
$553$	−1.37176	−0.0583331
$554$	0.866699	0.0368225
$555$	0	0
$556$	−43.8934	−1.86150
$557$	23.3870	0.990939	0.495470	−	0.868625i	$-0.334996\pi$
$557$	23.3870	0.990939	0.495470	+	0.868625i	$0.334996\pi$
$558$	0	0
$559$	−6.14264	−0.259806
$560$	3.66834	0.155016
$561$	0	0
$562$	−40.9724	−1.72832
$563$	−24.5311	−1.03386	−0.516930	−	0.856028i	$-0.672925\pi$
$563$	−24.5311	−1.03386	−0.516930	+	0.856028i	$0.672925\pi$
$564$	0	0
$565$	−9.12322	−0.383817
$566$	50.1647	2.10858
$567$	0	0
$568$	−6.15874	−0.258415
$569$	−39.4165	−1.65243	−0.826213	−	0.563358i	$-0.809509\pi$
$569$	−39.4165	−1.65243	−0.826213	+	0.563358i	$0.809509\pi$
$570$	0	0
$571$	41.0006	1.71582	0.857911	−	0.513798i	$-0.171762\pi$
$571$	41.0006	1.71582	0.857911	+	0.513798i	$0.171762\pi$
$572$	0	0
$573$	0	0
$574$	6.36787	0.265790
$575$	1.68245	0.0701631
$576$	0	0
$577$	−44.1963	−1.83992	−0.919959	−	0.392015i	$-0.871778\pi$
$577$	−44.1963	−1.83992	−0.919959	+	0.392015i	$0.871778\pi$
$578$	21.4430	0.891913
$579$	0	0
$580$	21.4140	0.889167
$581$	−7.47346	−0.310051
$582$	0	0
$583$	0	0
$584$	2.10588	0.0871418
$585$	0	0
$586$	35.9622	1.48559
$587$	2.75552	0.113733	0.0568663	−	0.998382i	$-0.481889\pi$
$587$	2.75552	0.113733	0.0568663	+	0.998382i	$0.481889\pi$
$588$	0	0
$589$	0.0888673	0.00366171
$590$	29.3285	1.20744
$591$	0	0
$592$	2.68541	0.110370
$593$	−30.8619	−1.26735	−0.633673	−	0.773601i	$-0.718453\pi$
$593$	−30.8619	−1.26735	−0.633673	+	0.773601i	$0.718453\pi$
$594$	0	0
$595$	−2.71753	−0.111408
$596$	−18.3337	−0.750979
$597$	0	0
$598$	3.39884	0.138989
$599$	13.3297	0.544638	0.272319	−	0.962207i	$-0.412209\pi$
$599$	13.3297	0.544638	0.272319	+	0.962207i	$0.412209\pi$
$600$	0	0
$601$	16.4395	0.670582	0.335291	−	0.942115i	$-0.391165\pi$
$601$	16.4395	0.670582	0.335291	+	0.942115i	$0.391165\pi$
$602$	−3.32343	−0.135453
$603$	0	0
$604$	−38.4213	−1.56334
$605$	0	0
$606$	0	0
$607$	24.2669	0.984962	0.492481	−	0.870323i	$-0.336090\pi$
$607$	24.2669	0.984962	0.492481	+	0.870323i	$0.336090\pi$
$608$	−1.46410	−0.0593772
$609$	0	0
$610$	−12.9460	−0.524168
$611$	−1.91075	−0.0773008
$612$	0	0
$613$	24.5773	0.992668	0.496334	−	0.868132i	$-0.334679\pi$
$613$	24.5773	0.992668	0.496334	+	0.868132i	$0.334679\pi$
$614$	22.4918	0.907695
$615$	0	0
$616$	0	0
$617$	15.4094	0.620358	0.310179	−	0.950678i	$-0.399611\pi$
$617$	15.4094	0.620358	0.310179	+	0.950678i	$0.399611\pi$
$618$	0	0
$619$	35.6568	1.43317	0.716583	−	0.697501i	$-0.245705\pi$
$619$	35.6568	1.43317	0.716583	+	0.697501i	$0.245705\pi$
$620$	−1.16845	−0.0469260
$621$	0	0
$622$	7.24059	0.290321
$623$	−16.1341	−0.646400
$624$	0	0
$625$	9.43660	0.377464
$626$	−53.4896	−2.13787
$627$	0	0
$628$	11.2607	0.449351
$629$	−1.98937	−0.0793212
$630$	0	0
$631$	11.6305	0.463003	0.231502	−	0.972834i	$-0.425636\pi$
$631$	11.6305	0.463003	0.231502	+	0.972834i	$0.425636\pi$
$632$	−0.675916	−0.0268865
$633$	0	0
$634$	−7.66487	−0.304411
$635$	20.2572	0.803884
$636$	0	0
$637$	3.80554	0.150781
$638$	0	0
$639$	0	0
$640$	4.14443	0.163823
$641$	−36.8237	−1.45445	−0.727225	−	0.686399i	$-0.759190\pi$
$641$	−36.8237	−1.45445	−0.727225	+	0.686399i	$0.759190\pi$
$642$	0	0
$643$	48.4049	1.90890	0.954452	−	0.298366i	$-0.0964415\pi$
$643$	48.4049	1.90890	0.954452	+	0.298366i	$0.0964415\pi$
$644$	0.971364	0.0382771
$645$	0	0
$646$	0.952947	0.0374932
$647$	26.2987	1.03391	0.516954	−	0.856013i	$-0.327066\pi$
$647$	26.2987	1.03391	0.516954	+	0.856013i	$0.327066\pi$
$648$	0	0
$649$	0	0
$650$	30.3906	1.19202
$651$	0	0
$652$	34.1426	1.33713
$653$	23.9772	0.938302	0.469151	−	0.883118i	$-0.344560\pi$
$653$	23.9772	0.938302	0.469151	+	0.883118i	$0.344560\pi$
$654$	0	0
$655$	−7.32798	−0.286328
$656$	−10.7136	−0.418297
$657$	0	0
$658$	−1.03380	−0.0403017
$659$	5.02206	0.195632	0.0978159	−	0.995205i	$-0.468814\pi$
$659$	5.02206	0.195632	0.0978159	+	0.995205i	$0.468814\pi$
$660$	0	0
$661$	−13.2645	−0.515928	−0.257964	−	0.966155i	$-0.583052\pi$
$661$	−13.2645	−0.515928	−0.257964	+	0.966155i	$0.583052\pi$
$662$	−36.5841	−1.42188
$663$	0	0
$664$	−3.68245	−0.142907
$665$	0.190988	0.00740619
$666$	0	0
$667$	−3.91715	−0.151673
$668$	−9.92177	−0.383885
$669$	0	0
$670$	31.4120	1.21355
$671$	0	0
$672$	0	0
$673$	36.2097	1.39578	0.697892	−	0.716203i	$-0.254122\pi$
$673$	36.2097	1.39578	0.697892	+	0.716203i	$0.254122\pi$
$674$	53.7755	2.07136
$675$	0	0
$676$	3.31889	0.127650
$677$	−8.90851	−0.342382	−0.171191	−	0.985238i	$-0.554762\pi$
$677$	−8.90851	−0.342382	−0.171191	+	0.985238i	$0.554762\pi$
$678$	0	0
$679$	−14.6158	−0.560904
$680$	−1.33903	−0.0513493
$681$	0	0
$682$	0	0
$683$	23.2377	0.889166	0.444583	−	0.895738i	$-0.353352\pi$
$683$	23.2377	0.889166	0.444583	+	0.895738i	$0.353352\pi$
$684$	0	0
$685$	−7.21360	−0.275618
$686$	2.05896	0.0786114
$687$	0	0
$688$	5.59152	0.213175
$689$	37.8374	1.44149
$690$	0	0
$691$	29.2034	1.11095	0.555475	−	0.831533i	$-0.312536\pi$
$691$	29.2034	1.11095	0.555475	+	0.831533i	$0.312536\pi$
$692$	23.9883	0.911900
$693$	0	0
$694$	−1.63582	−0.0620947
$695$	−20.7570	−0.787357
$696$	0	0
$697$	7.93671	0.300624
$698$	24.2579	0.918176
$699$	0	0
$700$	8.68541	0.328278
$701$	45.7062	1.72630	0.863150	−	0.504947i	$-0.168488\pi$
$701$	45.7062	1.72630	0.863150	+	0.504947i	$0.168488\pi$
$702$	0	0
$703$	0.139813	0.00527314
$704$	0	0
$705$	0	0
$706$	13.2811	0.499842
$707$	−6.10856	−0.229736
$708$	0	0
$709$	−5.12487	−0.192469	−0.0962343	−	0.995359i	$-0.530680\pi$
$709$	−5.12487	−0.192469	−0.0962343	+	0.995359i	$0.530680\pi$
$710$	−27.2523	−1.02276
$711$	0	0
$712$	−7.94989	−0.297935
$713$	0.213738	0.00800457
$714$	0	0
$715$	0	0
$716$	−25.6505	−0.958602
$717$	0	0
$718$	−12.1049	−0.451753
$719$	−2.68899	−0.100282	−0.0501412	−	0.998742i	$-0.515967\pi$
$719$	−2.68899	−0.100282	−0.0501412	+	0.998742i	$0.515967\pi$
$720$	0	0
$721$	9.56275	0.356135
$722$	39.0533	1.45341
$723$	0	0
$724$	37.0050	1.37528
$725$	−35.0251	−1.30080
$726$	0	0
$727$	1.17656	0.0436363	0.0218181	−	0.999762i	$-0.493055\pi$
$727$	1.17656	0.0436363	0.0218181	+	0.999762i	$0.493055\pi$
$728$	1.87513	0.0694969
$729$	0	0
$730$	9.31847	0.344892
$731$	−4.14222	−0.153206
$732$	0	0
$733$	16.2128	0.598833	0.299416	−	0.954123i	$-0.403208\pi$
$733$	16.2128	0.598833	0.299416	+	0.954123i	$0.403208\pi$
$734$	60.4259	2.23036
$735$	0	0
$736$	−3.52137	−0.129800
$737$	0	0
$738$	0	0
$739$	−37.2564	−1.37050	−0.685249	−	0.728309i	$-0.740307\pi$
$739$	−37.2564	−1.37050	−0.685249	+	0.728309i	$0.740307\pi$
$740$	−1.83829	−0.0675770
$741$	0	0
$742$	20.4717	0.751539
$743$	31.9013	1.17034	0.585172	−	0.810909i	$-0.301027\pi$
$743$	31.9013	1.17034	0.585172	+	0.810909i	$0.301027\pi$
$744$	0	0
$745$	−8.66992	−0.317641
$746$	−18.1849	−0.665795
$747$	0	0
$748$	0	0
$749$	−4.59486	−0.167892
$750$	0	0
$751$	22.2186	0.810767	0.405384	−	0.914147i	$-0.367138\pi$
$751$	22.2186	0.810767	0.405384	+	0.914147i	$0.367138\pi$
$752$	1.73932	0.0634265
$753$	0	0
$754$	−70.7566	−2.57680
$755$	−18.1692	−0.661245
$756$	0	0
$757$	−38.8014	−1.41026	−0.705131	−	0.709077i	$-0.749112\pi$
$757$	−38.8014	−1.41026	−0.705131	+	0.709077i	$0.749112\pi$
$758$	24.1251	0.876261
$759$	0	0
$760$	0.0941068	0.00341361
$761$	49.4020	1.79082	0.895410	−	0.445242i	$-0.146882\pi$
$761$	49.4020	1.79082	0.895410	+	0.445242i	$0.146882\pi$
$762$	0	0
$763$	1.99009	0.0720460
$764$	4.92188	0.178067
$765$	0	0
$766$	63.1104	2.28027
$767$	−51.1893	−1.84834
$768$	0	0
$769$	16.2939	0.587575	0.293787	−	0.955871i	$-0.405084\pi$
$769$	16.2939	0.587575	0.293787	+	0.955871i	$0.405084\pi$
$770$	0	0
$771$	0	0
$772$	56.3145	2.02680
$773$	47.5351	1.70972	0.854860	−	0.518858i	$-0.173643\pi$
$773$	47.5351	1.70972	0.854860	+	0.518858i	$0.173643\pi$
$774$	0	0
$775$	1.91113	0.0686499
$776$	−7.20176	−0.258528
$777$	0	0
$778$	−7.43505	−0.266559
$779$	−0.557792	−0.0199850
$780$	0	0
$781$	0	0
$782$	2.29197	0.0819608
$783$	0	0
$784$	−3.46410	−0.123718
$785$	5.32512	0.190062
$786$	0	0
$787$	−46.8753	−1.67093	−0.835463	−	0.549547i	$-0.814800\pi$
$787$	−46.8753	−1.67093	−0.835463	+	0.549547i	$0.814800\pi$
$788$	−13.7315	−0.489166
$789$	0	0
$790$	−2.99092	−0.106412
$791$	8.61527	0.306324
$792$	0	0
$793$	22.5956	0.802394
$794$	−6.38976	−0.226764
$795$	0	0
$796$	45.8960	1.62674
$797$	22.5463	0.798631	0.399315	−	0.916814i	$-0.369248\pi$
$797$	22.5463	0.798631	0.399315	+	0.916814i	$0.369248\pi$
$798$	0	0
$799$	−1.28850	−0.0455837
$800$	−31.4862	−1.11321
$801$	0	0
$802$	61.3860	2.16761
$803$	0	0
$804$	0	0
$805$	0.459353	0.0161901
$806$	3.86081	0.135991
$807$	0	0
$808$	−3.00991	−0.105888
$809$	12.8111	0.450416	0.225208	−	0.974311i	$-0.427694\pi$
$809$	12.8111	0.450416	0.225208	+	0.974311i	$0.427694\pi$
$810$	0	0
$811$	−4.65327	−0.163398	−0.0816991	−	0.996657i	$-0.526035\pi$
$811$	−4.65327	−0.163398	−0.0816991	+	0.996657i	$0.526035\pi$
$812$	−20.2217	−0.709644
$813$	0	0
$814$	0	0
$815$	16.1458	0.565564
$816$	0	0
$817$	0.291116	0.0101848
$818$	39.1920	1.37031
$819$	0	0
$820$	7.33399	0.256114
$821$	45.9133	1.60238	0.801192	−	0.598407i	$-0.204199\pi$
$821$	45.9133	1.60238	0.801192	+	0.598407i	$0.204199\pi$
$822$	0	0
$823$	39.0742	1.36204	0.681021	−	0.732264i	$-0.261536\pi$
$823$	39.0742	1.36204	0.681021	+	0.732264i	$0.261536\pi$
$824$	4.71192	0.164148
$825$	0	0
$826$	−27.6956	−0.963653
$827$	43.9735	1.52911	0.764554	−	0.644560i	$-0.222959\pi$
$827$	43.9735	1.52911	0.764554	+	0.644560i	$0.222959\pi$
$828$	0	0
$829$	16.9035	0.587082	0.293541	−	0.955946i	$-0.405166\pi$
$829$	16.9035	0.587082	0.293541	+	0.955946i	$0.405166\pi$
$830$	−16.2948	−0.565601
$831$	0	0
$832$	−37.2420	−1.29113
$833$	2.56622	0.0889143
$834$	0	0
$835$	−4.69195	−0.162372
$836$	0	0
$837$	0	0
$838$	12.4665	0.430648
$839$	16.2627	0.561450	0.280725	−	0.959788i	$-0.409425\pi$
$839$	16.2627	0.561450	0.280725	+	0.959788i	$0.409425\pi$
$840$	0	0
$841$	52.5467	1.81196
$842$	−60.1230	−2.07197
$843$	0	0
$844$	−19.7718	−0.680575
$845$	1.56949	0.0539920
$846$	0	0
$847$	0	0
$848$	−34.4426	−1.18276
$849$	0	0
$850$	20.4936	0.702924
$851$	0.336270	0.0115272
$852$	0	0
$853$	−4.70771	−0.161189	−0.0805945	−	0.996747i	$-0.525682\pi$
$853$	−4.70771	−0.161189	−0.0805945	+	0.996747i	$0.525682\pi$
$854$	12.2252	0.418338
$855$	0	0
$856$	−2.26406	−0.0773839
$857$	27.6135	0.943259	0.471629	−	0.881797i	$-0.343666\pi$
$857$	27.6135	0.943259	0.471629	+	0.881797i	$0.343666\pi$
$858$	0	0
$859$	18.1645	0.619763	0.309882	−	0.950775i	$-0.399711\pi$
$859$	18.1645	0.619763	0.309882	+	0.950775i	$0.399711\pi$
$860$	−3.82766	−0.130522
$861$	0	0
$862$	−31.5846	−1.07577
$863$	31.2285	1.06303	0.531515	−	0.847049i	$-0.321623\pi$
$863$	31.2285	1.06303	0.531515	+	0.847049i	$0.321623\pi$
$864$	0	0
$865$	11.3440	0.385706
$866$	−9.22018	−0.313314
$867$	0	0
$868$	1.10339	0.0374516
$869$	0	0
$870$	0	0
$871$	−54.8257	−1.85770
$872$	0.980590	0.0332070
$873$	0	0
$874$	−0.161080	−0.00544861
$875$	9.40208	0.317848
$876$	0	0
$877$	−15.4419	−0.521436	−0.260718	−	0.965415i	$-0.583959\pi$
$877$	−15.4419	−0.521436	−0.260718	+	0.965415i	$0.583959\pi$
$878$	54.5041	1.83942
$879$	0	0
$880$	0	0
$881$	47.5879	1.60328	0.801639	−	0.597808i	$-0.203962\pi$
$881$	47.5879	1.60328	0.801639	+	0.597808i	$0.203962\pi$
$882$	0	0
$883$	−22.2786	−0.749734	−0.374867	−	0.927079i	$-0.622312\pi$
$883$	−22.2786	−0.749734	−0.374867	+	0.927079i	$0.622312\pi$
$884$	21.8688	0.735527
$885$	0	0
$886$	−7.95881	−0.267381
$887$	25.5643	0.858365	0.429183	−	0.903218i	$-0.358802\pi$
$887$	25.5643	0.858365	0.429183	+	0.903218i	$0.358802\pi$
$888$	0	0
$889$	−19.1294	−0.641579
$890$	−35.1781	−1.17917
$891$	0	0
$892$	−46.1325	−1.54463
$893$	0.0905556	0.00303033
$894$	0	0
$895$	−12.1300	−0.405460
$896$	−3.91368	−0.130747
$897$	0	0
$898$	−14.9590	−0.499187
$899$	−4.44958	−0.148402
$900$	0	0
$901$	25.5153	0.850036
$902$	0	0
$903$	0	0
$904$	4.24506	0.141189
$905$	17.4995	0.581702
$906$	0	0
$907$	3.08704	0.102504	0.0512518	−	0.998686i	$-0.483679\pi$
$907$	3.08704	0.102504	0.0512518	+	0.998686i	$0.483679\pi$
$908$	39.0422	1.29566
$909$	0	0
$910$	8.29742	0.275057
$911$	5.06946	0.167959	0.0839793	−	0.996467i	$-0.473237\pi$
$911$	5.06946	0.167959	0.0839793	+	0.996467i	$0.473237\pi$
$912$	0	0
$913$	0	0
$914$	−53.0078	−1.75334
$915$	0	0
$916$	52.5066	1.73487
$917$	6.91998	0.228518
$918$	0	0
$919$	−32.7870	−1.08154	−0.540771	−	0.841170i	$-0.681868\pi$
$919$	−32.7870	−1.08154	−0.540771	+	0.841170i	$0.681868\pi$
$920$	0.226340	0.00746222
$921$	0	0
$922$	−47.2776	−1.55701
$923$	47.5655	1.56564
$924$	0	0
$925$	3.00674	0.0988610
$926$	−9.65595	−0.317314
$927$	0	0
$928$	73.3074	2.40643
$929$	−22.3901	−0.734595	−0.367298	−	0.930103i	$-0.619717\pi$
$929$	−22.3901	−0.734595	−0.367298	+	0.930103i	$0.619717\pi$
$930$	0	0
$931$	−0.180354	−0.00591087
$932$	−33.2969	−1.09068
$933$	0	0
$934$	−67.7088	−2.21550
$935$	0	0
$936$	0	0
$937$	−6.67075	−0.217924	−0.108962	−	0.994046i	$-0.534753\pi$
$937$	−6.67075	−0.217924	−0.108962	+	0.994046i	$0.534753\pi$
$938$	−29.6631	−0.968534
$939$	0	0
$940$	−1.19065	−0.0388346
$941$	8.84967	0.288491	0.144245	−	0.989542i	$-0.453925\pi$
$941$	8.84967	0.288491	0.144245	+	0.989542i	$0.453925\pi$
$942$	0	0
$943$	−1.34157	−0.0436875
$944$	46.5965	1.51659
$945$	0	0
$946$	0	0
$947$	−34.9296	−1.13506	−0.567530	−	0.823352i	$-0.692101\pi$
$947$	−34.9296	−1.13506	−0.567530	+	0.823352i	$0.692101\pi$
$948$	0	0
$949$	−16.2642	−0.527959
$950$	−1.44029	−0.0467292
$951$	0	0
$952$	1.26447	0.0409818
$953$	−3.36257	−0.108924	−0.0544621	−	0.998516i	$-0.517344\pi$
$953$	−3.36257	−0.108924	−0.0544621	+	0.998516i	$0.517344\pi$
$954$	0	0
$955$	2.32753	0.0753171
$956$	50.5460	1.63478
$957$	0	0
$958$	−21.3711	−0.690469
$959$	6.81197	0.219970
$960$	0	0
$961$	−30.7572	−0.992168
$962$	6.07413	0.195838
$963$	0	0
$964$	−4.93198	−0.158849
$965$	26.6308	0.857277
$966$	0	0
$967$	6.65340	0.213959	0.106979	−	0.994261i	$-0.465882\pi$
$967$	6.65340	0.213959	0.106979	+	0.994261i	$0.465882\pi$
$968$	0	0
$969$	0	0
$970$	−31.8677	−1.02321
$971$	−13.3090	−0.427106	−0.213553	−	0.976931i	$-0.568504\pi$
$971$	−13.3090	−0.427106	−0.213553	+	0.976931i	$0.568504\pi$
$972$	0	0
$973$	19.6013	0.628389
$974$	−58.5536	−1.87618
$975$	0	0
$976$	−20.5683	−0.658376
$977$	−1.37314	−0.0439307	−0.0219654	−	0.999759i	$-0.506992\pi$
$977$	−1.37314	−0.0439307	−0.0219654	+	0.999759i	$0.506992\pi$
$978$	0	0
$979$	0	0
$980$	2.37134	0.0757497
$981$	0	0
$982$	−63.7194	−2.03337
$983$	10.7743	0.343646	0.171823	−	0.985128i	$-0.445034\pi$
$983$	10.7743	0.343646	0.171823	+	0.985128i	$0.445034\pi$
$984$	0	0
$985$	−6.49357	−0.206902
$986$	−47.7139	−1.51952
$987$	0	0
$988$	−1.53694	−0.0488966
$989$	0.700175	0.0222643
$990$	0	0
$991$	5.59499	0.177731	0.0888654	−	0.996044i	$-0.471676\pi$
$991$	5.59499	0.177731	0.0888654	+	0.996044i	$0.471676\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	25.7350	0.816265
$995$	21.7040	0.688062
$996$	0	0
$997$	45.7503	1.44893	0.724463	−	0.689314i	$-0.242088\pi$
$997$	45.7503	1.44893	0.724463	+	0.689314i	$0.242088\pi$
$998$	70.5711	2.23389
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cm.1.1		4
3.2	odd	2		2541.2.a.bk.1.4	✓	4
11.10	odd	2		7623.2.a.cf.1.4		4
33.32	even	2		2541.2.a.bo.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.bk.1.4	✓	4	3.2	odd	2
2541.2.a.bo.1.1	yes	4	33.32	even	2
7623.2.a.cf.1.4		4	11.10	odd	2
7623.2.a.cm.1.1		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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-2.05896 q^{2} +2.23931 q^{4} +1.05896 q^{5} -1.00000 q^{7} -0.492737 q^{8} +O(q^{10})\)	\(q-2.05896 q^{2} +2.23931 q^{4} +1.05896 q^{5} -1.00000 q^{7} -0.492737 q^{8} -2.18035 q^{10} +3.80554 q^{13} +2.05896 q^{14} -3.46410 q^{16} +2.56622 q^{17} -0.180354 q^{19} +2.37134 q^{20} -0.433778 q^{23} -3.87861 q^{25} -7.83544 q^{26} -2.23931 q^{28} +9.03032 q^{29} -0.492737 q^{31} +8.11792 q^{32} -5.28375 q^{34} -1.05896 q^{35} -0.775212 q^{37} +0.371342 q^{38} -0.521789 q^{40} +3.09276 q^{41} -1.61413 q^{43} +0.893131 q^{46} -0.502098 q^{47} +1.00000 q^{49} +7.98589 q^{50} +8.52179 q^{52} +9.94273 q^{53} +0.492737 q^{56} -18.5931 q^{58} -13.4513 q^{59} +5.93756 q^{61} +1.01453 q^{62} -9.78626 q^{64} +4.02991 q^{65} -14.4068 q^{67} +5.74658 q^{68} +2.18035 q^{70} +12.4990 q^{71} -4.27383 q^{73} +1.59613 q^{74} -0.403870 q^{76} +1.37176 q^{79} -3.66834 q^{80} -6.36787 q^{82} +7.47346 q^{83} +2.71753 q^{85} +3.32343 q^{86} +16.1341 q^{89} -3.80554 q^{91} -0.971364 q^{92} +1.03380 q^{94} -0.190988 q^{95} +14.6158 q^{97} -2.05896 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 4 * q^7	\( 4 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 4 q^{7} - 14 q^{10} + 2 q^{13} - 2 q^{14} + 2 q^{17} - 6 q^{19} - 8 q^{20} - 10 q^{23} - 4 q^{28} + 14 q^{29} + 12 q^{32} - 2 q^{34} + 6 q^{35} - 12 q^{37} - 16 q^{38} - 8 q^{40} + 16 q^{41} - 20 q^{43} - 8 q^{46} - 2 q^{47} + 4 q^{49} + 24 q^{50} + 40 q^{52} + 16 q^{53} - 8 q^{58} - 2 q^{59} - 2 q^{61} + 8 q^{62} - 16 q^{64} - 2 q^{65} - 20 q^{67} + 20 q^{68} + 14 q^{70} + 10 q^{71} + 10 q^{73} - 20 q^{74} - 28 q^{76} - 16 q^{79} - 12 q^{80} + 28 q^{82} + 18 q^{83} - 26 q^{86} + 14 q^{89} - 2 q^{91} + 8 q^{92} + 18 q^{94} + 22 q^{95} + 38 q^{97} + 2 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 4 * q^7 - 14 * q^10 + 2 * q^13 - 2 * q^14 + 2 * q^17 - 6 * q^19 - 8 * q^20 - 10 * q^23 - 4 * q^28 + 14 * q^29 + 12 * q^32 - 2 * q^34 + 6 * q^35 - 12 * q^37 - 16 * q^38 - 8 * q^40 + 16 * q^41 - 20 * q^43 - 8 * q^46 - 2 * q^47 + 4 * q^49 + 24 * q^50 + 40 * q^52 + 16 * q^53 - 8 * q^58 - 2 * q^59 - 2 * q^61 + 8 * q^62 - 16 * q^64 - 2 * q^65 - 20 * q^67 + 20 * q^68 + 14 * q^70 + 10 * q^71 + 10 * q^73 - 20 * q^74 - 28 * q^76 - 16 * q^79 - 12 * q^80 + 28 * q^82 + 18 * q^83 - 26 * q^86 + 14 * q^89 - 2 * q^91 + 8 * q^92 + 18 * q^94 + 22 * q^95 + 38 * q^97 + 2 * q^98

Introduction

L-functions

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