LMFDB - Embedded newform 2925.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(1,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, 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("2925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 975)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.08613$ of defining polynomial
Character	$\chi$	$=$	2925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.08613 q^{2} +2.35194 q^{4} -4.08613 q^{7} -0.734191 q^{8} +O(q^{10})$	$q-2.08613 q^{2} +2.35194 q^{4} -4.08613 q^{7} -0.734191 q^{8} +3.43807 q^{11} -1.00000 q^{13} +8.52420 q^{14} -3.17226 q^{16} +7.17226 q^{17} +7.52420 q^{19} -7.17226 q^{22} -2.82032 q^{23} +2.08613 q^{26} -9.61033 q^{28} +5.70388 q^{29} -5.61033 q^{31} +8.08613 q^{32} -14.9623 q^{34} -4.82032 q^{37} -15.6965 q^{38} -5.52420 q^{41} +2.05582 q^{43} +8.08613 q^{44} +5.88356 q^{46} -7.49389 q^{47} +9.69646 q^{49} -2.35194 q^{52} -2.52420 q^{53} +3.00000 q^{56} -11.8990 q^{58} -4.79001 q^{59} -7.87614 q^{61} +11.7039 q^{62} -10.5242 q^{64} +6.20257 q^{67} +16.8687 q^{68} +0.475800 q^{71} -9.35194 q^{73} +10.0558 q^{74} +17.6965 q^{76} -14.0484 q^{77} +11.6965 q^{79} +11.5242 q^{82} +16.4307 q^{83} -4.28870 q^{86} -2.52420 q^{88} +10.1723 q^{89} +4.08613 q^{91} -6.63322 q^{92} +15.6332 q^{94} -14.8761 q^{97} -20.2281 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} - 5 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + 5 * q^4 - 5 * q^7 + 3 * q^8	$ 3 q + q^{2} + 5 q^{4} - 5 q^{7} + 3 q^{8} + q^{11} - 3 q^{13} + 9 q^{14} + 5 q^{16} + 7 q^{17} + 6 q^{19} - 7 q^{22} + 4 q^{23} - q^{26} - 5 q^{28} + 13 q^{29} + 7 q^{31} + 17 q^{32} - 19 q^{34} - 2 q^{37} - 16 q^{38} + 17 q^{44} + 26 q^{46} - 7 q^{47} - 2 q^{49} - 5 q^{52} + 9 q^{53} + 9 q^{56} + 11 q^{58} - 3 q^{59} - 5 q^{61} + 31 q^{62} - 15 q^{64} + 3 q^{67} + 5 q^{68} + 18 q^{71} - 26 q^{73} + 24 q^{74} + 22 q^{76} - 9 q^{77} + 4 q^{79} + 18 q^{82} + 13 q^{83} + 10 q^{86} + 9 q^{88} + 16 q^{89} + 5 q^{91} + 32 q^{92} - 5 q^{94} - 26 q^{97} - 40 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 - 5 * q^7 + 3 * q^8 + q^11 - 3 * q^13 + 9 * q^14 + 5 * q^16 + 7 * q^17 + 6 * q^19 - 7 * q^22 + 4 * q^23 - q^26 - 5 * q^28 + 13 * q^29 + 7 * q^31 + 17 * q^32 - 19 * q^34 - 2 * q^37 - 16 * q^38 + 17 * q^44 + 26 * q^46 - 7 * q^47 - 2 * q^49 - 5 * q^52 + 9 * q^53 + 9 * q^56 + 11 * q^58 - 3 * q^59 - 5 * q^61 + 31 * q^62 - 15 * q^64 + 3 * q^67 + 5 * q^68 + 18 * q^71 - 26 * q^73 + 24 * q^74 + 22 * q^76 - 9 * q^77 + 4 * q^79 + 18 * q^82 + 13 * q^83 + 10 * q^86 + 9 * q^88 + 16 * q^89 + 5 * q^91 + 32 * q^92 - 5 * q^94 - 26 * q^97 - 40 * 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 $

$2$	−2.08613	−1.47512	−0.737558	−	0.675283i	$-0.764021\pi$
$2$	−2.08613	−1.47512	−0.737558	+	0.675283i	$0.764021\pi$
$3$	0	0
$3$	0	0
$4$	2.35194	1.17597
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.08613	−1.54441	−0.772206	−	0.635372i	$-0.780847\pi$
$7$	−4.08613	−1.54441	−0.772206	+	0.635372i	$0.780847\pi$
$8$	−0.734191	−0.259576
$9$	0	0
$10$	0	0
$11$	3.43807	1.03662	0.518308	−	0.855194i	$-0.326562\pi$
$11$	3.43807	1.03662	0.518308	+	0.855194i	$0.326562\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	8.52420	2.27819
$15$	0	0
$16$	−3.17226	−0.793065
$17$	7.17226	1.73953	0.869764	−	0.493467i	$-0.164271\pi$
$17$	7.17226	1.73953	0.869764	+	0.493467i	$0.164271\pi$
$18$	0	0
$19$	7.52420	1.72617	0.863085	−	0.505059i	$-0.168529\pi$
$19$	7.52420	1.72617	0.863085	+	0.505059i	$0.168529\pi$
$20$	0	0
$21$	0	0
$22$	−7.17226	−1.52913
$23$	−2.82032	−0.588078	−0.294039	−	0.955793i	$-0.594999\pi$
$23$	−2.82032	−0.588078	−0.294039	+	0.955793i	$0.594999\pi$
$24$	0	0
$25$	0	0
$26$	2.08613	0.409124
$27$	0	0
$28$	−9.61033	−1.81618
$29$	5.70388	1.05918	0.529592	−	0.848253i	$-0.322345\pi$
$29$	5.70388	1.05918	0.529592	+	0.848253i	$0.322345\pi$
$30$	0	0
$31$	−5.61033	−1.00764	−0.503822	−	0.863807i	$-0.668073\pi$
$31$	−5.61033	−1.00764	−0.503822	+	0.863807i	$0.668073\pi$
$32$	8.08613	1.42944
$33$	0	0
$34$	−14.9623	−2.56601
$35$	0	0
$36$	0	0
$37$	−4.82032	−0.792456	−0.396228	−	0.918152i	$-0.629681\pi$
$37$	−4.82032	−0.792456	−0.396228	+	0.918152i	$0.629681\pi$
$38$	−15.6965	−2.54630
$39$	0	0
$40$	0	0
$41$	−5.52420	−0.862735	−0.431368	−	0.902176i	$-0.641969\pi$
$41$	−5.52420	−0.862735	−0.431368	+	0.902176i	$0.641969\pi$
$42$	0	0
$43$	2.05582	0.313509	0.156755	−	0.987638i	$-0.449897\pi$
$43$	2.05582	0.313509	0.156755	+	0.987638i	$0.449897\pi$
$44$	8.08613	1.21903
$45$	0	0
$46$	5.88356	0.867483
$47$	−7.49389	−1.09310	−0.546548	−	0.837428i	$-0.684058\pi$
$47$	−7.49389	−1.09310	−0.546548	+	0.837428i	$0.684058\pi$
$48$	0	0
$49$	9.69646	1.38521
$50$	0	0
$51$	0	0
$52$	−2.35194	−0.326155
$53$	−2.52420	−0.346725	−0.173363	−	0.984858i	$-0.555463\pi$
$53$	−2.52420	−0.346725	−0.173363	+	0.984858i	$0.555463\pi$
$54$	0	0
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	−11.8990	−1.56242
$59$	−4.79001	−0.623606	−0.311803	−	0.950147i	$-0.600933\pi$
$59$	−4.79001	−0.623606	−0.311803	+	0.950147i	$0.600933\pi$
$60$	0	0
$61$	−7.87614	−1.00844	−0.504218	−	0.863576i	$-0.668219\pi$
$61$	−7.87614	−1.00844	−0.504218	+	0.863576i	$0.668219\pi$
$62$	11.7039	1.48639
$63$	0	0
$64$	−10.5242	−1.31552
$65$	0	0
$66$	0	0
$67$	6.20257	0.757765	0.378882	−	0.925445i	$-0.376308\pi$
$67$	6.20257	0.757765	0.378882	+	0.925445i	$0.376308\pi$
$68$	16.8687	2.04563
$69$	0	0
$70$	0	0
$71$	0.475800	0.0564671	0.0282336	−	0.999601i	$-0.491012\pi$
$71$	0.475800	0.0564671	0.0282336	+	0.999601i	$0.491012\pi$
$72$	0	0
$73$	−9.35194	−1.09456	−0.547281	−	0.836949i	$-0.684337\pi$
$73$	−9.35194	−1.09456	−0.547281	+	0.836949i	$0.684337\pi$
$74$	10.0558	1.16897
$75$	0	0
$76$	17.6965	2.02992
$77$	−14.0484	−1.60096
$78$	0	0
$79$	11.6965	1.31595	0.657977	−	0.753038i	$-0.271412\pi$
$79$	11.6965	1.31595	0.657977	+	0.753038i	$0.271412\pi$
$80$	0	0
$81$	0	0
$82$	11.5242	1.27263
$83$	16.4307	1.80350	0.901749	−	0.432260i	$-0.142284\pi$
$83$	16.4307	1.80350	0.901749	+	0.432260i	$0.142284\pi$
$84$	0	0
$85$	0	0
$86$	−4.28870	−0.462463
$87$	0	0
$88$	−2.52420	−0.269081
$89$	10.1723	1.07826	0.539129	−	0.842223i	$-0.318754\pi$
$89$	10.1723	1.07826	0.539129	+	0.842223i	$0.318754\pi$
$90$	0	0
$91$	4.08613	0.428343
$92$	−6.63322	−0.691561
$93$	0	0
$94$	15.6332	1.61244
$95$	0	0
$96$	0	0
$97$	−14.8761	−1.51044	−0.755222	−	0.655470i	$-0.772471\pi$
$97$	−14.8761	−1.51044	−0.755222	+	0.655470i	$0.772471\pi$
$98$	−20.2281	−2.04334
$99$	0	0
$100$	0	0
$101$	4.94418	0.491965	0.245982	−	0.969274i	$-0.420889\pi$
$101$	4.94418	0.491965	0.245982	+	0.969274i	$0.420889\pi$
$102$	0	0
$103$	8.28870	0.816710	0.408355	−	0.912823i	$-0.366103\pi$
$103$	8.28870	0.816710	0.408355	+	0.912823i	$0.366103\pi$
$104$	0.734191	0.0719934
$105$	0	0
$106$	5.26581	0.511461
$107$	12.5168	1.21004	0.605021	−	0.796209i	$-0.293165\pi$
$107$	12.5168	1.21004	0.605021	+	0.796209i	$0.293165\pi$
$108$	0	0
$109$	11.6965	1.12032	0.560159	−	0.828385i	$-0.310740\pi$
$109$	11.6965	1.12032	0.560159	+	0.828385i	$0.310740\pi$
$110$	0	0
$111$	0	0
$112$	12.9623	1.22482
$113$	1.23550	0.116226	0.0581129	−	0.998310i	$-0.481492\pi$
$113$	1.23550	0.116226	0.0581129	+	0.998310i	$0.481492\pi$
$114$	0	0
$115$	0	0
$116$	13.4152	1.24557
$117$	0	0
$118$	9.99258	0.919892
$119$	−29.3068	−2.68655
$120$	0	0
$121$	0.820321	0.0745747
$122$	16.4307	1.48756
$123$	0	0
$124$	−13.1952	−1.18496
$125$	0	0
$126$	0	0
$127$	9.52420	0.845136	0.422568	−	0.906331i	$-0.361129\pi$
$127$	9.52420	0.845136	0.422568	+	0.906331i	$0.361129\pi$
$128$	5.78259	0.511114
$129$	0	0
$130$	0	0
$131$	−10.5726	−0.923732	−0.461866	−	0.886950i	$-0.652820\pi$
$131$	−10.5726	−0.923732	−0.461866	+	0.886950i	$0.652820\pi$
$132$	0	0
$133$	−30.7449	−2.66592
$134$	−12.9394	−1.11779
$135$	0	0
$136$	−5.26581	−0.451539
$137$	4.53162	0.387162	0.193581	−	0.981084i	$-0.437990\pi$
$137$	4.53162	0.387162	0.193581	+	0.981084i	$0.437990\pi$
$138$	0	0
$139$	−14.4562	−1.22616	−0.613078	−	0.790023i	$-0.710069\pi$
$139$	−14.4562	−1.22616	−0.613078	+	0.790023i	$0.710069\pi$
$140$	0	0
$141$	0	0
$142$	−0.992582	−0.0832956
$143$	−3.43807	−0.287506
$144$	0	0
$145$	0	0
$146$	19.5094	1.61461
$147$	0	0
$148$	−11.3371	−0.931904
$149$	−12.2839	−1.00634	−0.503168	−	0.864189i	$-0.667832\pi$
$149$	−12.2839	−1.00634	−0.503168	+	0.864189i	$0.667832\pi$
$150$	0	0
$151$	22.6029	1.83940	0.919699	−	0.392623i	$-0.128432\pi$
$151$	22.6029	1.83940	0.919699	+	0.392623i	$0.128432\pi$
$152$	−5.52420	−0.448072
$153$	0	0
$154$	29.3068	2.36161
$155$	0	0
$156$	0	0
$157$	−3.64806	−0.291147	−0.145573	−	0.989347i	$-0.546503\pi$
$157$	−3.64806	−0.291147	−0.145573	+	0.989347i	$0.546503\pi$
$158$	−24.4003	−1.94119
$159$	0	0
$160$	0	0
$161$	11.5242	0.908234
$162$	0	0
$163$	0.820321	0.0642525	0.0321263	−	0.999484i	$-0.489772\pi$
$163$	0.820321	0.0642525	0.0321263	+	0.999484i	$0.489772\pi$
$164$	−12.9926	−1.01455
$165$	0	0
$166$	−34.2765	−2.66037
$167$	9.46357	0.732313	0.366157	−	0.930553i	$-0.380673\pi$
$167$	9.46357	0.732313	0.366157	+	0.930553i	$0.380673\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	4.83516	0.368677
$173$	−6.69646	−0.509122	−0.254561	−	0.967057i	$-0.581931\pi$
$173$	−6.69646	−0.509122	−0.254561	+	0.967057i	$0.581931\pi$
$174$	0	0
$175$	0	0
$176$	−10.9065	−0.822105
$177$	0	0
$178$	−21.2207	−1.59056
$179$	15.1042	1.12894	0.564471	−	0.825453i	$-0.309080\pi$
$179$	15.1042	1.12894	0.564471	+	0.825453i	$0.309080\pi$
$180$	0	0
$181$	−1.70869	−0.127006	−0.0635028	−	0.997982i	$-0.520227\pi$
$181$	−1.70869	−0.127006	−0.0635028	+	0.997982i	$0.520227\pi$
$182$	−8.52420	−0.631856
$183$	0	0
$184$	2.07065	0.152651
$185$	0	0
$186$	0	0
$187$	24.6587	1.80322
$188$	−17.6252	−1.28545
$189$	0	0
$190$	0	0
$191$	23.6406	1.71058	0.855288	−	0.518152i	$-0.173380\pi$
$191$	23.6406	1.71058	0.855288	+	0.518152i	$0.173380\pi$
$192$	0	0
$193$	21.4487	1.54391	0.771957	−	0.635675i	$-0.219278\pi$
$193$	21.4487	1.54391	0.771957	+	0.635675i	$0.219278\pi$
$194$	31.0336	2.22808
$195$	0	0
$196$	22.8055	1.62896
$197$	4.57260	0.325784	0.162892	−	0.986644i	$-0.447918\pi$
$197$	4.57260	0.325784	0.162892	+	0.986644i	$0.447918\pi$
$198$	0	0
$199$	5.65548	0.400906	0.200453	−	0.979703i	$-0.435759\pi$
$199$	5.65548	0.400906	0.200453	+	0.979703i	$0.435759\pi$
$200$	0	0
$201$	0	0
$202$	−10.3142	−0.725705
$203$	−23.3068	−1.63582
$204$	0	0
$205$	0	0
$206$	−17.2913	−1.20474
$207$	0	0
$208$	3.17226	0.219957
$209$	25.8687	1.78938
$210$	0	0
$211$	−5.63583	−0.387987	−0.193993	−	0.981003i	$-0.562144\pi$
$211$	−5.63583	−0.387987	−0.193993	+	0.981003i	$0.562144\pi$
$212$	−5.93676	−0.407739
$213$	0	0
$214$	−26.1116	−1.78495
$215$	0	0
$216$	0	0
$217$	22.9245	1.55622
$218$	−24.4003	−1.65260
$219$	0	0
$220$	0	0
$221$	−7.17226	−0.482458
$222$	0	0
$223$	6.22808	0.417063	0.208531	−	0.978016i	$-0.433132\pi$
$223$	6.22808	0.417063	0.208531	+	0.978016i	$0.433132\pi$
$224$	−33.0410	−2.20764
$225$	0	0
$226$	−2.57741	−0.171447
$227$	−14.8458	−0.985352	−0.492676	−	0.870213i	$-0.663981\pi$
$227$	−14.8458	−0.985352	−0.492676	+	0.870213i	$0.663981\pi$
$228$	0	0
$229$	−25.5046	−1.68539	−0.842694	−	0.538392i	$-0.819032\pi$
$229$	−25.5046	−1.68539	−0.842694	+	0.538392i	$0.819032\pi$
$230$	0	0
$231$	0	0
$232$	−4.18774	−0.274938
$233$	4.17226	0.273334	0.136667	−	0.990617i	$-0.456361\pi$
$233$	4.17226	0.273334	0.136667	+	0.990617i	$0.456361\pi$
$234$	0	0
$235$	0	0
$236$	−11.2658	−0.733342
$237$	0	0
$238$	61.1378	3.96297
$239$	−23.3068	−1.50759	−0.753795	−	0.657109i	$-0.771779\pi$
$239$	−23.3068	−1.50759	−0.753795	+	0.657109i	$0.771779\pi$
$240$	0	0
$241$	−25.3371	−1.63211	−0.816053	−	0.577977i	$-0.803842\pi$
$241$	−25.3371	−1.63211	−0.816053	+	0.577977i	$0.803842\pi$
$242$	−1.71130	−0.110006
$243$	0	0
$244$	−18.5242	−1.18589
$245$	0	0
$246$	0	0
$247$	−7.52420	−0.478753
$248$	4.11905	0.261560
$249$	0	0
$250$	0	0
$251$	5.24030	0.330765	0.165383	−	0.986229i	$-0.447114\pi$
$251$	5.24030	0.330765	0.165383	+	0.986229i	$0.447114\pi$
$252$	0	0
$253$	−9.69646	−0.609611
$254$	−19.8687	−1.24667
$255$	0	0
$256$	8.98516	0.561573
$257$	23.4610	1.46345	0.731727	−	0.681597i	$-0.238714\pi$
$257$	23.4610	1.46345	0.731727	+	0.681597i	$0.238714\pi$
$258$	0	0
$259$	19.6965	1.22388
$260$	0	0
$261$	0	0
$262$	22.0558	1.36261
$263$	18.0410	1.11245	0.556227	−	0.831030i	$-0.312248\pi$
$263$	18.0410	1.11245	0.556227	+	0.831030i	$0.312248\pi$
$264$	0	0
$265$	0	0
$266$	64.1378	3.93254
$267$	0	0
$268$	14.5881	0.891108
$269$	27.8761	1.69964	0.849819	−	0.527074i	$-0.176711\pi$
$269$	27.8761	1.69964	0.849819	+	0.527074i	$0.176711\pi$
$270$	0	0
$271$	−20.0713	−1.21924	−0.609622	−	0.792692i	$-0.708679\pi$
$271$	−20.0713	−1.21924	−0.609622	+	0.792692i	$0.708679\pi$
$272$	−22.7523	−1.37956
$273$	0	0
$274$	−9.45355	−0.571110
$275$	0	0
$276$	0	0
$277$	8.40515	0.505016	0.252508	−	0.967595i	$-0.418745\pi$
$277$	8.40515	0.505016	0.252508	+	0.967595i	$0.418745\pi$
$278$	30.1574	1.80872
$279$	0	0
$280$	0	0
$281$	10.4562	0.623762	0.311881	−	0.950121i	$-0.399041\pi$
$281$	10.4562	0.623762	0.311881	+	0.950121i	$0.399041\pi$
$282$	0	0
$283$	18.3445	1.09047	0.545234	−	0.838284i	$-0.316441\pi$
$283$	18.3445	1.09047	0.545234	+	0.838284i	$0.316441\pi$
$284$	1.11905	0.0664036
$285$	0	0
$286$	7.17226	0.424105
$287$	22.5726	1.33242
$288$	0	0
$289$	34.4413	2.02596
$290$	0	0
$291$	0	0
$292$	−21.9952	−1.28717
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	3.53904	0.205702
$297$	0	0
$298$	25.6258	1.48446
$299$	2.82032	0.163103
$300$	0	0
$301$	−8.40034	−0.484187
$302$	−47.1526	−2.71333
$303$	0	0
$304$	−23.8687	−1.36897
$305$	0	0
$306$	0	0
$307$	13.9293	0.794990	0.397495	−	0.917604i	$-0.369880\pi$
$307$	13.9293	0.794990	0.397495	+	0.917604i	$0.369880\pi$
$308$	−33.0410	−1.88268
$309$	0	0
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	30.0894	1.70075	0.850376	−	0.526175i	$-0.176374\pi$
$313$	30.0894	1.70075	0.850376	+	0.526175i	$0.176374\pi$
$314$	7.61033	0.429476
$315$	0	0
$316$	27.5094	1.54752
$317$	−4.05582	−0.227797	−0.113899	−	0.993492i	$-0.536334\pi$
$317$	−4.05582	−0.227797	−0.113899	+	0.993492i	$0.536334\pi$
$318$	0	0
$319$	19.6103	1.09797
$320$	0	0
$321$	0	0
$322$	−24.0410	−1.33975
$323$	53.9655	3.00272
$324$	0	0
$325$	0	0
$326$	−1.71130	−0.0947800
$327$	0	0
$328$	4.05582	0.223945
$329$	30.6210	1.68819
$330$	0	0
$331$	−19.4126	−1.06701	−0.533506	−	0.845797i	$-0.679126\pi$
$331$	−19.4126	−1.06701	−0.533506	+	0.845797i	$0.679126\pi$
$332$	38.6439	2.12086
$333$	0	0
$334$	−19.7422	−1.08025
$335$	0	0
$336$	0	0
$337$	−20.1797	−1.09926	−0.549629	−	0.835409i	$-0.685231\pi$
$337$	−20.1797	−1.09926	−0.549629	+	0.835409i	$0.685231\pi$
$338$	−2.08613	−0.113471
$339$	0	0
$340$	0	0
$341$	−19.2887	−1.04454
$342$	0	0
$343$	−11.0181	−0.594921
$344$	−1.50936	−0.0813794
$345$	0	0
$346$	13.9697	0.751015
$347$	18.8007	1.00927	0.504637	−	0.863332i	$-0.331626\pi$
$347$	18.8007	1.00927	0.504637	+	0.863332i	$0.331626\pi$
$348$	0	0
$349$	−4.47580	−0.239584	−0.119792	−	0.992799i	$-0.538223\pi$
$349$	−4.47580	−0.239584	−0.119792	+	0.992799i	$0.538223\pi$
$350$	0	0
$351$	0	0
$352$	27.8007	1.48178
$353$	26.8203	1.42750	0.713751	−	0.700400i	$-0.246995\pi$
$353$	26.8203	1.42750	0.713751	+	0.700400i	$0.246995\pi$
$354$	0	0
$355$	0	0
$356$	23.9245	1.26800
$357$	0	0
$358$	−31.5094	−1.66532
$359$	5.89903	0.311339	0.155670	−	0.987809i	$-0.450247\pi$
$359$	5.89903	0.311339	0.155670	+	0.987809i	$0.450247\pi$
$360$	0	0
$361$	37.6136	1.97966
$362$	3.56454	0.187348
$363$	0	0
$364$	9.61033	0.503718
$365$	0	0
$366$	0	0
$367$	−18.0968	−0.944645	−0.472323	−	0.881426i	$-0.656584\pi$
$367$	−18.0968	−0.944645	−0.472323	+	0.881426i	$0.656584\pi$
$368$	8.94679	0.466384
$369$	0	0
$370$	0	0
$371$	10.3142	0.535487
$372$	0	0
$373$	12.8836	0.667085	0.333543	−	0.942735i	$-0.391756\pi$
$373$	12.8836	0.667085	0.333543	+	0.942735i	$0.391756\pi$
$374$	−51.4413	−2.65997
$375$	0	0
$376$	5.50194	0.283741
$377$	−5.70388	−0.293765
$378$	0	0
$379$	−19.0032	−0.976131	−0.488066	−	0.872807i	$-0.662297\pi$
$379$	−19.0032	−0.976131	−0.488066	+	0.872807i	$0.662297\pi$
$380$	0	0
$381$	0	0
$382$	−49.3175	−2.52330
$383$	9.62100	0.491610	0.245805	−	0.969319i	$-0.420948\pi$
$383$	9.62100	0.491610	0.245805	+	0.969319i	$0.420948\pi$
$384$	0	0
$385$	0	0
$386$	−44.7449	−2.27745
$387$	0	0
$388$	−34.9878	−1.77624
$389$	2.41998	0.122698	0.0613490	−	0.998116i	$-0.480460\pi$
$389$	2.41998	0.122698	0.0613490	+	0.998116i	$0.480460\pi$
$390$	0	0
$391$	−20.2281	−1.02298
$392$	−7.11905	−0.359567
$393$	0	0
$394$	−9.53904	−0.480570
$395$	0	0
$396$	0	0
$397$	26.6136	1.33570	0.667849	−	0.744297i	$-0.267215\pi$
$397$	26.6136	1.33570	0.667849	+	0.744297i	$0.267215\pi$
$398$	−11.7981	−0.591384
$399$	0	0
$400$	0	0
$401$	−12.4758	−0.623012	−0.311506	−	0.950244i	$-0.600833\pi$
$401$	−12.4758	−0.623012	−0.311506	+	0.950244i	$0.600833\pi$
$402$	0	0
$403$	5.61033	0.279470
$404$	11.6284	0.578535
$405$	0	0
$406$	48.6210	2.41302
$407$	−16.5726	−0.821473
$408$	0	0
$409$	16.2281	0.802427	0.401213	−	0.915985i	$-0.368589\pi$
$409$	16.2281	0.802427	0.401213	+	0.915985i	$0.368589\pi$
$410$	0	0
$411$	0	0
$412$	19.4945	0.960426
$413$	19.5726	0.963105
$414$	0	0
$415$	0	0
$416$	−8.08613	−0.396455
$417$	0	0
$418$	−53.9655	−2.63954
$419$	38.3855	1.87525	0.937627	−	0.347642i	$-0.113018\pi$
$419$	38.3855	1.87525	0.937627	+	0.347642i	$0.113018\pi$
$420$	0	0
$421$	8.95160	0.436274	0.218137	−	0.975918i	$-0.430002\pi$
$421$	8.95160	0.436274	0.218137	+	0.975918i	$0.430002\pi$
$422$	11.7571	0.572326
$423$	0	0
$424$	1.85324	0.0900015
$425$	0	0
$426$	0	0
$427$	32.1829	1.55744
$428$	29.4387	1.42297
$429$	0	0
$430$	0	0
$431$	1.99519	0.0961050	0.0480525	−	0.998845i	$-0.484699\pi$
$431$	1.99519	0.0961050	0.0480525	+	0.998845i	$0.484699\pi$
$432$	0	0
$433$	−36.9729	−1.77681	−0.888403	−	0.459064i	$-0.848185\pi$
$433$	−36.9729	−1.77681	−0.888403	+	0.459064i	$0.848185\pi$
$434$	−47.8236	−2.29560
$435$	0	0
$436$	27.5094	1.31746
$437$	−21.2207	−1.01512
$438$	0	0
$439$	−5.06804	−0.241885	−0.120942	−	0.992660i	$-0.538592\pi$
$439$	−5.06804	−0.241885	−0.120942	+	0.992660i	$0.538592\pi$
$440$	0	0
$441$	0	0
$442$	14.9623	0.711683
$443$	5.59966	0.266048	0.133024	−	0.991113i	$-0.457531\pi$
$443$	5.59966	0.266048	0.133024	+	0.991113i	$0.457531\pi$
$444$	0	0
$445$	0	0
$446$	−12.9926	−0.615217
$447$	0	0
$448$	43.0032	2.03171
$449$	−13.9852	−0.660001	−0.330000	−	0.943981i	$-0.607049\pi$
$449$	−13.9852	−0.660001	−0.330000	+	0.943981i	$0.607049\pi$
$450$	0	0
$451$	−18.9926	−0.894326
$452$	2.90581	0.136678
$453$	0	0
$454$	30.9703	1.45351
$455$	0	0
$456$	0	0
$457$	31.6210	1.47917	0.739584	−	0.673064i	$-0.235022\pi$
$457$	31.6210	1.47917	0.739584	+	0.673064i	$0.235022\pi$
$458$	53.2058	2.48614
$459$	0	0
$460$	0	0
$461$	−29.0484	−1.35292	−0.676459	−	0.736480i	$-0.736487\pi$
$461$	−29.0484	−1.35292	−0.676459	+	0.736480i	$0.736487\pi$
$462$	0	0
$463$	30.8458	1.43353	0.716764	−	0.697316i	$-0.245623\pi$
$463$	30.8458	1.43353	0.716764	+	0.697316i	$0.245623\pi$
$464$	−18.0942	−0.840002
$465$	0	0
$466$	−8.70388	−0.403199
$467$	−17.3323	−0.802043	−0.401021	−	0.916069i	$-0.631345\pi$
$467$	−17.3323	−0.802043	−0.401021	+	0.916069i	$0.631345\pi$
$468$	0	0
$469$	−25.3445	−1.17030
$470$	0	0
$471$	0	0
$472$	3.51678	0.161873
$473$	7.06804	0.324989
$474$	0	0
$475$	0	0
$476$	−68.9278	−3.15930
$477$	0	0
$478$	48.6210	2.22387
$479$	9.20518	0.420596	0.210298	−	0.977637i	$-0.432557\pi$
$479$	9.20518	0.420596	0.210298	+	0.977637i	$0.432557\pi$
$480$	0	0
$481$	4.82032	0.219788
$482$	52.8565	2.40755
$483$	0	0
$484$	1.92935	0.0876975
$485$	0	0
$486$	0	0
$487$	−27.8432	−1.26170	−0.630848	−	0.775906i	$-0.717293\pi$
$487$	−27.8432	−1.26170	−0.630848	+	0.775906i	$0.717293\pi$
$488$	5.78259	0.261766
$489$	0	0
$490$	0	0
$491$	23.6406	1.06689	0.533444	−	0.845836i	$-0.320898\pi$
$491$	23.6406	1.06689	0.533444	+	0.845836i	$0.320898\pi$
$492$	0	0
$493$	40.9097	1.84248
$494$	15.6965	0.706217
$495$	0	0
$496$	17.7974	0.799128
$497$	−1.94418	−0.0872085
$498$	0	0
$499$	−16.1829	−0.724447	−0.362224	−	0.932091i	$-0.617982\pi$
$499$	−16.1829	−0.724447	−0.362224	+	0.932091i	$0.617982\pi$
$500$	0	0
$501$	0	0
$502$	−10.9320	−0.487917
$503$	−12.4758	−0.556268	−0.278134	−	0.960542i	$-0.589716\pi$
$503$	−12.4758	−0.556268	−0.278134	+	0.960542i	$0.589716\pi$
$504$	0	0
$505$	0	0
$506$	20.2281	0.899248
$507$	0	0
$508$	22.4003	0.993854
$509$	10.0558	0.445716	0.222858	−	0.974851i	$-0.428461\pi$
$509$	10.0558	0.445716	0.222858	+	0.974851i	$0.428461\pi$
$510$	0	0
$511$	38.2132	1.69045
$512$	−30.3094	−1.33950
$513$	0	0
$514$	−48.9426	−2.15877
$515$	0	0
$516$	0	0
$517$	−25.7645	−1.13312
$518$	−41.0894	−1.80536
$519$	0	0
$520$	0	0
$521$	−31.7523	−1.39109	−0.695546	−	0.718481i	$-0.744838\pi$
$521$	−31.7523	−1.39109	−0.695546	+	0.718481i	$0.744838\pi$
$522$	0	0
$523$	18.9878	0.830277	0.415139	−	0.909758i	$-0.363733\pi$
$523$	18.9878	0.830277	0.415139	+	0.909758i	$0.363733\pi$
$524$	−24.8661	−1.08628
$525$	0	0
$526$	−37.6358	−1.64100
$527$	−40.2387	−1.75283
$528$	0	0
$529$	−15.0458	−0.654165
$530$	0	0
$531$	0	0
$532$	−72.3100	−3.13504
$533$	5.52420	0.239280
$534$	0	0
$535$	0	0
$536$	−4.55387	−0.196697
$537$	0	0
$538$	−58.1533	−2.50716
$539$	33.3371	1.43593
$540$	0	0
$541$	1.00742	0.0433123	0.0216562	−	0.999765i	$-0.493106\pi$
$541$	1.00742	0.0433123	0.0216562	+	0.999765i	$0.493106\pi$
$542$	41.8713	1.79853
$543$	0	0
$544$	57.9958	2.48655
$545$	0	0
$546$	0	0
$547$	0.662898	0.0283435	0.0141717	−	0.999900i	$-0.495489\pi$
$547$	0.662898	0.0283435	0.0141717	+	0.999900i	$0.495489\pi$
$548$	10.6581	0.455291
$549$	0	0
$550$	0	0
$551$	42.9171	1.82833
$552$	0	0
$553$	−47.7933	−2.03238
$554$	−17.5342	−0.744958
$555$	0	0
$556$	−34.0000	−1.44192
$557$	−16.8809	−0.715269	−0.357634	−	0.933862i	$-0.616417\pi$
$557$	−16.8809	−0.715269	−0.357634	+	0.933862i	$0.616417\pi$
$558$	0	0
$559$	−2.05582	−0.0869518
$560$	0	0
$561$	0	0
$562$	−21.8129	−0.920122
$563$	18.7087	0.788477	0.394239	−	0.919008i	$-0.371008\pi$
$563$	18.7087	0.788477	0.394239	+	0.919008i	$0.371008\pi$
$564$	0	0
$565$	0	0
$566$	−38.2691	−1.60857
$567$	0	0
$568$	−0.349328	−0.0146575
$569$	16.5848	0.695272	0.347636	−	0.937630i	$-0.386984\pi$
$569$	16.5848	0.695272	0.347636	+	0.937630i	$0.386984\pi$
$570$	0	0
$571$	−0.460964	−0.0192907	−0.00964536	−	0.999953i	$-0.503070\pi$
$571$	−0.460964	−0.0192907	−0.00964536	+	0.999953i	$0.503070\pi$
$572$	−8.08613	−0.338098
$573$	0	0
$574$	−47.0894	−1.96547
$575$	0	0
$576$	0	0
$577$	−23.9394	−0.996609	−0.498305	−	0.867002i	$-0.666044\pi$
$577$	−23.9394	−0.996609	−0.498305	+	0.867002i	$0.666044\pi$
$578$	−71.8491	−2.98853
$579$	0	0
$580$	0	0
$581$	−67.1378	−2.78534
$582$	0	0
$583$	−8.67837	−0.359422
$584$	6.86611	0.284122
$585$	0	0
$586$	−12.5168	−0.517063
$587$	−32.7194	−1.35047	−0.675236	−	0.737602i	$-0.735958\pi$
$587$	−32.7194	−1.35047	−0.675236	+	0.737602i	$0.735958\pi$
$588$	0	0
$589$	−42.2132	−1.73937
$590$	0	0
$591$	0	0
$592$	15.2913	0.628469
$593$	0.825129	0.0338840	0.0169420	−	0.999856i	$-0.494607\pi$
$593$	0.825129	0.0338840	0.0169420	+	0.999856i	$0.494607\pi$
$594$	0	0
$595$	0	0
$596$	−28.8910	−1.18342
$597$	0	0
$598$	−5.88356	−0.240597
$599$	11.2913	0.461351	0.230675	−	0.973031i	$-0.425906\pi$
$599$	11.2913	0.461351	0.230675	+	0.973031i	$0.425906\pi$
$600$	0	0
$601$	9.64806	0.393553	0.196776	−	0.980448i	$-0.436953\pi$
$601$	9.64806	0.393553	0.196776	+	0.980448i	$0.436953\pi$
$602$	17.5242	0.714233
$603$	0	0
$604$	53.1607	2.16308
$605$	0	0
$606$	0	0
$607$	24.6284	0.999637	0.499818	−	0.866130i	$-0.333400\pi$
$607$	24.6284	0.999637	0.499818	+	0.866130i	$0.333400\pi$
$608$	60.8417	2.46746
$609$	0	0
$610$	0	0
$611$	7.49389	0.303170
$612$	0	0
$613$	−10.3445	−0.417811	−0.208906	−	0.977936i	$-0.566990\pi$
$613$	−10.3445	−0.417811	−0.208906	+	0.977936i	$0.566990\pi$
$614$	−29.0584	−1.17270
$615$	0	0
$616$	10.3142	0.415571
$617$	32.8613	1.32295	0.661473	−	0.749969i	$-0.269932\pi$
$617$	32.8613	1.32295	0.661473	+	0.749969i	$0.269932\pi$
$618$	0	0
$619$	36.8565	1.48139	0.740694	−	0.671843i	$-0.234497\pi$
$619$	36.8565	1.48139	0.740694	+	0.671843i	$0.234497\pi$
$620$	0	0
$621$	0	0
$622$	−37.5503	−1.50563
$623$	−41.5652	−1.66527
$624$	0	0
$625$	0	0
$626$	−62.7704	−2.50881
$627$	0	0
$628$	−8.58002	−0.342380
$629$	−34.5726	−1.37850
$630$	0	0
$631$	−25.6965	−1.02296	−0.511480	−	0.859295i	$-0.670903\pi$
$631$	−25.6965	−1.02296	−0.511480	+	0.859295i	$0.670903\pi$
$632$	−8.58744	−0.341590
$633$	0	0
$634$	8.46096	0.336028
$635$	0	0
$636$	0	0
$637$	−9.69646	−0.384188
$638$	−40.9097	−1.61963
$639$	0	0
$640$	0	0
$641$	−18.4535	−0.728871	−0.364436	−	0.931229i	$-0.618738\pi$
$641$	−18.4535	−0.728871	−0.364436	+	0.931229i	$0.618738\pi$
$642$	0	0
$643$	16.7087	0.658926	0.329463	−	0.944168i	$-0.393132\pi$
$643$	16.7087	0.658926	0.329463	+	0.944168i	$0.393132\pi$
$644$	27.1042	1.06806
$645$	0	0
$646$	−112.579	−4.42937
$647$	28.6136	1.12492	0.562458	−	0.826826i	$-0.309856\pi$
$647$	28.6136	1.12492	0.562458	+	0.826826i	$0.309856\pi$
$648$	0	0
$649$	−16.4684	−0.646441
$650$	0	0
$651$	0	0
$652$	1.92935	0.0755590
$653$	−39.7252	−1.55457	−0.777284	−	0.629150i	$-0.783403\pi$
$653$	−39.7252	−1.55457	−0.777284	+	0.629150i	$0.783403\pi$
$654$	0	0
$655$	0	0
$656$	17.5242	0.684205
$657$	0	0
$658$	−63.8794	−2.49028
$659$	−5.21585	−0.203181	−0.101590	−	0.994826i	$-0.532393\pi$
$659$	−5.21585	−0.203181	−0.101590	+	0.994826i	$0.532393\pi$
$660$	0	0
$661$	20.3493	0.791497	0.395749	−	0.918359i	$-0.370485\pi$
$661$	20.3493	0.791497	0.395749	+	0.918359i	$0.370485\pi$
$662$	40.4971	1.57397
$663$	0	0
$664$	−12.0632	−0.468144
$665$	0	0
$666$	0	0
$667$	−16.0868	−0.622882
$668$	22.2578	0.861178
$669$	0	0
$670$	0	0
$671$	−27.0787	−1.04536
$672$	0	0
$673$	−18.2765	−0.704506	−0.352253	−	0.935905i	$-0.614584\pi$
$673$	−18.2765	−0.704506	−0.352253	+	0.935905i	$0.614584\pi$
$674$	42.0974	1.62153
$675$	0	0
$676$	2.35194	0.0904592
$677$	−43.2664	−1.66286	−0.831432	−	0.555626i	$-0.812479\pi$
$677$	−43.2664	−1.66286	−0.831432	+	0.555626i	$0.812479\pi$
$678$	0	0
$679$	60.7858	2.33275
$680$	0	0
$681$	0	0
$682$	40.2387	1.54082
$683$	−25.8432	−0.988863	−0.494432	−	0.869217i	$-0.664624\pi$
$683$	−25.8432	−0.988863	−0.494432	+	0.869217i	$0.664624\pi$
$684$	0	0
$685$	0	0
$686$	22.9852	0.877578
$687$	0	0
$688$	−6.52159	−0.248633
$689$	2.52420	0.0961643
$690$	0	0
$691$	−8.94743	−0.340376	−0.170188	−	0.985412i	$-0.554438\pi$
$691$	−8.94743	−0.340376	−0.170188	+	0.985412i	$0.554438\pi$
$692$	−15.7497	−0.598712
$693$	0	0
$694$	−39.2207	−1.48880
$695$	0	0
$696$	0	0
$697$	−39.6210	−1.50075
$698$	9.33710	0.353415
$699$	0	0
$700$	0	0
$701$	26.4897	1.00050	0.500251	−	0.865880i	$-0.333241\pi$
$701$	26.4897	1.00050	0.500251	+	0.865880i	$0.333241\pi$
$702$	0	0
$703$	−36.2691	−1.36791
$704$	−36.1829	−1.36370
$705$	0	0
$706$	−55.9507	−2.10573
$707$	−20.2026	−0.759796
$708$	0	0
$709$	19.3977	0.728497	0.364248	−	0.931302i	$-0.381326\pi$
$709$	19.3977	0.728497	0.364248	+	0.931302i	$0.381326\pi$
$710$	0	0
$711$	0	0
$712$	−7.46838	−0.279889
$713$	15.8229	0.592573
$714$	0	0
$715$	0	0
$716$	35.5242	1.32760
$717$	0	0
$718$	−12.3062	−0.459261
$719$	32.0261	1.19437	0.597187	−	0.802102i	$-0.296285\pi$
$719$	32.0261	1.19437	0.597187	+	0.802102i	$0.296285\pi$
$720$	0	0
$721$	−33.8687	−1.26134
$722$	−78.4668	−2.92023
$723$	0	0
$724$	−4.01873	−0.149355
$725$	0	0
$726$	0	0
$727$	−5.46357	−0.202633	−0.101316	−	0.994854i	$-0.532305\pi$
$727$	−5.46357	−0.202633	−0.101316	+	0.994854i	$0.532305\pi$
$728$	−3.00000	−0.111187
$729$	0	0
$730$	0	0
$731$	14.7449	0.545358
$732$	0	0
$733$	45.6406	1.68578	0.842888	−	0.538089i	$-0.180854\pi$
$733$	45.6406	1.68578	0.842888	+	0.538089i	$0.180854\pi$
$734$	37.7523	1.39346
$735$	0	0
$736$	−22.8055	−0.840621
$737$	21.3249	0.785512
$738$	0	0
$739$	34.9278	1.28484	0.642420	−	0.766353i	$-0.277931\pi$
$739$	34.9278	1.28484	0.642420	+	0.766353i	$0.277931\pi$
$740$	0	0
$741$	0	0
$742$	−21.5168	−0.789906
$743$	−52.3962	−1.92223	−0.961115	−	0.276150i	$-0.910941\pi$
$743$	−52.3962	−1.92223	−0.961115	+	0.276150i	$0.910941\pi$
$744$	0	0
$745$	0	0
$746$	−26.8768	−0.984029
$747$	0	0
$748$	57.9958	2.12054
$749$	−51.1452	−1.86880
$750$	0	0
$751$	31.9655	1.16644	0.583219	−	0.812315i	$-0.301793\pi$
$751$	31.9655	1.16644	0.583219	+	0.812315i	$0.301793\pi$
$752$	23.7726	0.866896
$753$	0	0
$754$	11.8990	0.433337
$755$	0	0
$756$	0	0
$757$	−34.4897	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$757$	−34.4897	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$758$	39.6433	1.43991
$759$	0	0
$760$	0	0
$761$	19.7933	0.717505	0.358753	−	0.933433i	$-0.383202\pi$
$761$	19.7933	0.717505	0.358753	+	0.933433i	$0.383202\pi$
$762$	0	0
$763$	−47.7933	−1.73023
$764$	55.6014	2.01159
$765$	0	0
$766$	−20.0707	−0.725182
$767$	4.79001	0.172957
$768$	0	0
$769$	4.65287	0.167787	0.0838934	−	0.996475i	$-0.473264\pi$
$769$	4.65287	0.167787	0.0838934	+	0.996475i	$0.473264\pi$
$770$	0	0
$771$	0	0
$772$	50.4461	1.81560
$773$	20.6284	0.741953	0.370976	−	0.928642i	$-0.379023\pi$
$773$	20.6284	0.741953	0.370976	+	0.928642i	$0.379023\pi$
$774$	0	0
$775$	0	0
$776$	10.9219	0.392074
$777$	0	0
$778$	−5.04840	−0.180994
$779$	−41.5652	−1.48923
$780$	0	0
$781$	1.63583	0.0585348
$782$	42.1984	1.50901
$783$	0	0
$784$	−30.7597	−1.09856
$785$	0	0
$786$	0	0
$787$	26.1877	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$787$	26.1877	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$788$	10.7545	0.383112
$789$	0	0
$790$	0	0
$791$	−5.04840	−0.179500
$792$	0	0
$793$	7.87614	0.279690
$794$	−55.5194	−1.97031
$795$	0	0
$796$	13.3013	0.471454
$797$	17.2180	0.609895	0.304947	−	0.952369i	$-0.401361\pi$
$797$	17.2180	0.609895	0.304947	+	0.952369i	$0.401361\pi$
$798$	0	0
$799$	−53.7481	−1.90147
$800$	0	0
$801$	0	0
$802$	26.0261	0.919015
$803$	−32.1526	−1.13464
$804$	0	0
$805$	0	0
$806$	−11.7039	−0.412252
$807$	0	0
$808$	−3.62997	−0.127702
$809$	−13.4684	−0.473523	−0.236762	−	0.971568i	$-0.576086\pi$
$809$	−13.4684	−0.473523	−0.236762	+	0.971568i	$0.576086\pi$
$810$	0	0
$811$	−7.83841	−0.275244	−0.137622	−	0.990485i	$-0.543946\pi$
$811$	−7.83841	−0.275244	−0.137622	+	0.990485i	$0.543946\pi$
$812$	−54.8162	−1.92367
$813$	0	0
$814$	34.5726	1.21177
$815$	0	0
$816$	0	0
$817$	15.4684	0.541170
$818$	−33.8539	−1.18367
$819$	0	0
$820$	0	0
$821$	−32.8203	−1.14544	−0.572719	−	0.819752i	$-0.694111\pi$
$821$	−32.8203	−1.14544	−0.572719	+	0.819752i	$0.694111\pi$
$822$	0	0
$823$	−24.1478	−0.841740	−0.420870	−	0.907121i	$-0.638275\pi$
$823$	−24.1478	−0.841740	−0.420870	+	0.907121i	$0.638275\pi$
$824$	−6.08549	−0.211998
$825$	0	0
$826$	−40.8310	−1.42069
$827$	−10.1058	−0.351412	−0.175706	−	0.984443i	$-0.556221\pi$
$827$	−10.1058	−0.351412	−0.175706	+	0.984443i	$0.556221\pi$
$828$	0	0
$829$	−3.54645	−0.123173	−0.0615867	−	0.998102i	$-0.519616\pi$
$829$	−3.54645	−0.123173	−0.0615867	+	0.998102i	$0.519616\pi$
$830$	0	0
$831$	0	0
$832$	10.5242	0.364861
$833$	69.5455	2.40961
$834$	0	0
$835$	0	0
$836$	60.8417	2.10425
$837$	0	0
$838$	−80.0772	−2.76622
$839$	27.1797	0.938347	0.469173	−	0.883106i	$-0.344552\pi$
$839$	27.1797	0.938347	0.469173	+	0.883106i	$0.344552\pi$
$840$	0	0
$841$	3.53423	0.121870
$842$	−18.6742	−0.643556
$843$	0	0
$844$	−13.2551	−0.456261
$845$	0	0
$846$	0	0
$847$	−3.35194	−0.115174
$848$	8.00742	0.274976
$849$	0	0
$850$	0	0
$851$	13.5949	0.466026
$852$	0	0
$853$	−14.4413	−0.494461	−0.247231	−	0.968957i	$-0.579521\pi$
$853$	−14.4413	−0.494461	−0.247231	+	0.968957i	$0.579521\pi$
$854$	−67.1378	−2.29741
$855$	0	0
$856$	−9.18971	−0.314098
$857$	−20.7858	−0.710031	−0.355015	−	0.934860i	$-0.615524\pi$
$857$	−20.7858	−0.710031	−0.355015	+	0.934860i	$0.615524\pi$
$858$	0	0
$859$	20.9926	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$859$	20.9926	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$860$	0	0
$861$	0	0
$862$	−4.16223	−0.141766
$863$	−25.3020	−0.861289	−0.430645	−	0.902522i	$-0.641714\pi$
$863$	−25.3020	−0.861289	−0.430645	+	0.902522i	$0.641714\pi$
$864$	0	0
$865$	0	0
$866$	77.1304	2.62100
$867$	0	0
$868$	53.9171	1.83007
$869$	40.2132	1.36414
$870$	0	0
$871$	−6.20257	−0.210166
$872$	−8.58744	−0.290807
$873$	0	0
$874$	44.2691	1.49742
$875$	0	0
$876$	0	0
$877$	−27.9342	−0.943269	−0.471635	−	0.881794i	$-0.656336\pi$
$877$	−27.9342	−0.943269	−0.471635	+	0.881794i	$0.656336\pi$
$878$	10.5726	0.356808
$879$	0	0
$880$	0	0
$881$	38.9900	1.31361	0.656803	−	0.754062i	$-0.271908\pi$
$881$	38.9900	1.31361	0.656803	+	0.754062i	$0.271908\pi$
$882$	0	0
$883$	12.1871	0.410128	0.205064	−	0.978749i	$-0.434260\pi$
$883$	12.1871	0.410128	0.205064	+	0.978749i	$0.434260\pi$
$884$	−16.8687	−0.567356
$885$	0	0
$886$	−11.6816	−0.392452
$887$	32.1526	1.07958	0.539790	−	0.841800i	$-0.318504\pi$
$887$	32.1526	1.07958	0.539790	+	0.841800i	$0.318504\pi$
$888$	0	0
$889$	−38.9171	−1.30524
$890$	0	0
$891$	0	0
$892$	14.6481	0.490453
$893$	−56.3855	−1.88687
$894$	0	0
$895$	0	0
$896$	−23.6284	−0.789370
$897$	0	0
$898$	29.1749	0.973578
$899$	−32.0006	−1.06728
$900$	0	0
$901$	−18.1042	−0.603139
$902$	39.6210	1.31923
$903$	0	0
$904$	−0.907090	−0.0301694
$905$	0	0
$906$	0	0
$907$	42.9368	1.42569	0.712846	−	0.701321i	$-0.247406\pi$
$907$	42.9368	1.42569	0.712846	+	0.701321i	$0.247406\pi$
$908$	−34.9165	−1.15874
$909$	0	0
$910$	0	0
$911$	−8.22808	−0.272608	−0.136304	−	0.990667i	$-0.543522\pi$
$911$	−8.22808	−0.272608	−0.136304	+	0.990667i	$0.543522\pi$
$912$	0	0
$913$	56.4897	1.86954
$914$	−65.9655	−2.18195
$915$	0	0
$916$	−59.9852	−1.98197
$917$	43.2010	1.42662
$918$	0	0
$919$	−27.5652	−0.909291	−0.454646	−	0.890672i	$-0.650234\pi$
$919$	−27.5652	−0.909291	−0.454646	+	0.890672i	$0.650234\pi$
$920$	0	0
$921$	0	0
$922$	60.5987	1.99571
$923$	−0.475800	−0.0156612
$924$	0	0
$925$	0	0
$926$	−64.3484	−2.11462
$927$	0	0
$928$	46.1223	1.51404
$929$	24.2084	0.794253	0.397126	−	0.917764i	$-0.370007\pi$
$929$	24.2084	0.794253	0.397126	+	0.917764i	$0.370007\pi$
$930$	0	0
$931$	72.9581	2.39111
$932$	9.81290	0.321432
$933$	0	0
$934$	36.1574	1.18311
$935$	0	0
$936$	0	0
$937$	−2.58002	−0.0842855	−0.0421427	−	0.999112i	$-0.513418\pi$
$937$	−2.58002	−0.0842855	−0.0421427	+	0.999112i	$0.513418\pi$
$938$	52.8720	1.72633
$939$	0	0
$940$	0	0
$941$	−30.6088	−0.997817	−0.498909	−	0.866655i	$-0.666266\pi$
$941$	−30.6088	−0.997817	−0.498909	+	0.866655i	$0.666266\pi$
$942$	0	0
$943$	15.5800	0.507355
$944$	15.1952	0.494560
$945$	0	0
$946$	−14.7449	−0.479397
$947$	−51.3626	−1.66906	−0.834530	−	0.550962i	$-0.814261\pi$
$947$	−51.3626	−1.66906	−0.834530	+	0.550962i	$0.814261\pi$
$948$	0	0
$949$	9.35194	0.303577
$950$	0	0
$951$	0	0
$952$	21.5168	0.697363
$953$	9.06543	0.293658	0.146829	−	0.989162i	$-0.453093\pi$
$953$	9.06543	0.293658	0.146829	+	0.989162i	$0.453093\pi$
$954$	0	0
$955$	0	0
$956$	−54.8162	−1.77288
$957$	0	0
$958$	−19.2032	−0.620428
$959$	−18.5168	−0.597938
$960$	0	0
$961$	0.475800	0.0153484
$962$	−10.0558	−0.324213
$963$	0	0
$964$	−59.5913	−1.91931
$965$	0	0
$966$	0	0
$967$	47.8942	1.54017	0.770087	−	0.637939i	$-0.220213\pi$
$967$	47.8942	1.54017	0.770087	+	0.637939i	$0.220213\pi$
$968$	−0.602272	−0.0193578
$969$	0	0
$970$	0	0
$971$	7.59485	0.243730	0.121865	−	0.992547i	$-0.461112\pi$
$971$	7.59485	0.243730	0.121865	+	0.992547i	$0.461112\pi$
$972$	0	0
$973$	59.0697	1.89369
$974$	58.0846	1.86115
$975$	0	0
$976$	24.9852	0.799756
$977$	−23.0484	−0.737384	−0.368692	−	0.929552i	$-0.620194\pi$
$977$	−23.0484	−0.737384	−0.368692	+	0.929552i	$0.620194\pi$
$978$	0	0
$979$	34.9729	1.11774
$980$	0	0
$981$	0	0
$982$	−49.3175	−1.57378
$983$	−26.2026	−0.835732	−0.417866	−	0.908509i	$-0.637222\pi$
$983$	−26.2026	−0.835732	−0.417866	+	0.908509i	$0.637222\pi$
$984$	0	0
$985$	0	0
$986$	−85.3430	−2.71787
$987$	0	0
$988$	−17.6965	−0.562999
$989$	−5.79807	−0.184368
$990$	0	0
$991$	46.9023	1.48990	0.744950	−	0.667120i	$-0.232473\pi$
$991$	46.9023	1.48990	0.744950	+	0.667120i	$0.232473\pi$
$992$	−45.3659	−1.44037
$993$	0	0
$994$	4.05582	0.128643
$995$	0	0
$996$	0	0
$997$	−36.5503	−1.15756	−0.578780	−	0.815483i	$-0.696471\pi$
$997$	−36.5503	−1.15756	−0.578780	+	0.815483i	$0.696471\pi$
$998$	33.7597	1.06864
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.a.bi.1.1		3
3.2	odd	2		975.2.a.n.1.3	✓	3
5.2	odd	4		2925.2.c.x.2224.2		6
5.3	odd	4		2925.2.c.x.2224.5		6
5.4	even	2		2925.2.a.bg.1.3		3
15.2	even	4		975.2.c.j.274.5		6
15.8	even	4		975.2.c.j.274.2		6
15.14	odd	2		975.2.a.p.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.a.n.1.3	✓	3	3.2	odd	2
975.2.a.p.1.1	yes	3	15.14	odd	2
975.2.c.j.274.2		6	15.8	even	4
975.2.c.j.274.5		6	15.2	even	4
2925.2.a.bg.1.3		3	5.4	even	2
2925.2.a.bi.1.1		3	1.1	even	1	trivial
2925.2.c.x.2224.2		6	5.2	odd	4
2925.2.c.x.2224.5		6	5.3	odd	4

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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.a (trivial)

\(f(q)\)	\(=\)	\(q-2.08613 q^{2} +2.35194 q^{4} -4.08613 q^{7} -0.734191 q^{8} +O(q^{10})\)	\(q-2.08613 q^{2} +2.35194 q^{4} -4.08613 q^{7} -0.734191 q^{8} +3.43807 q^{11} -1.00000 q^{13} +8.52420 q^{14} -3.17226 q^{16} +7.17226 q^{17} +7.52420 q^{19} -7.17226 q^{22} -2.82032 q^{23} +2.08613 q^{26} -9.61033 q^{28} +5.70388 q^{29} -5.61033 q^{31} +8.08613 q^{32} -14.9623 q^{34} -4.82032 q^{37} -15.6965 q^{38} -5.52420 q^{41} +2.05582 q^{43} +8.08613 q^{44} +5.88356 q^{46} -7.49389 q^{47} +9.69646 q^{49} -2.35194 q^{52} -2.52420 q^{53} +3.00000 q^{56} -11.8990 q^{58} -4.79001 q^{59} -7.87614 q^{61} +11.7039 q^{62} -10.5242 q^{64} +6.20257 q^{67} +16.8687 q^{68} +0.475800 q^{71} -9.35194 q^{73} +10.0558 q^{74} +17.6965 q^{76} -14.0484 q^{77} +11.6965 q^{79} +11.5242 q^{82} +16.4307 q^{83} -4.28870 q^{86} -2.52420 q^{88} +10.1723 q^{89} +4.08613 q^{91} -6.63322 q^{92} +15.6332 q^{94} -14.8761 q^{97} -20.2281 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + 5 * q^4 - 5 * q^7 + 3 * q^8	\( 3 q + q^{2} + 5 q^{4} - 5 q^{7} + 3 q^{8} + q^{11} - 3 q^{13} + 9 q^{14} + 5 q^{16} + 7 q^{17} + 6 q^{19} - 7 q^{22} + 4 q^{23} - q^{26} - 5 q^{28} + 13 q^{29} + 7 q^{31} + 17 q^{32} - 19 q^{34} - 2 q^{37} - 16 q^{38} + 17 q^{44} + 26 q^{46} - 7 q^{47} - 2 q^{49} - 5 q^{52} + 9 q^{53} + 9 q^{56} + 11 q^{58} - 3 q^{59} - 5 q^{61} + 31 q^{62} - 15 q^{64} + 3 q^{67} + 5 q^{68} + 18 q^{71} - 26 q^{73} + 24 q^{74} + 22 q^{76} - 9 q^{77} + 4 q^{79} + 18 q^{82} + 13 q^{83} + 10 q^{86} + 9 q^{88} + 16 q^{89} + 5 q^{91} + 32 q^{92} - 5 q^{94} - 26 q^{97} - 40 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 - 5 * q^7 + 3 * q^8 + q^11 - 3 * q^13 + 9 * q^14 + 5 * q^16 + 7 * q^17 + 6 * q^19 - 7 * q^22 + 4 * q^23 - q^26 - 5 * q^28 + 13 * q^29 + 7 * q^31 + 17 * q^32 - 19 * q^34 - 2 * q^37 - 16 * q^38 + 17 * q^44 + 26 * q^46 - 7 * q^47 - 2 * q^49 - 5 * q^52 + 9 * q^53 + 9 * q^56 + 11 * q^58 - 3 * q^59 - 5 * q^61 + 31 * q^62 - 15 * q^64 + 3 * q^67 + 5 * q^68 + 18 * q^71 - 26 * q^73 + 24 * q^74 + 22 * q^76 - 9 * q^77 + 4 * q^79 + 18 * q^82 + 13 * q^83 + 10 * q^86 + 9 * q^88 + 16 * q^89 + 5 * q^91 + 32 * q^92 - 5 * q^94 - 26 * q^97 - 40 * q^98

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