LMFDB - Embedded newform 4275.2.a.z.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4275,2,Mod(1,4275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-4,0,4,0,0,0,-9,0,0,-1,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.44504 q^{2} +0.0881460 q^{4} +1.35690 q^{7} +2.76271 q^{8} -4.85086 q^{11} +0.198062 q^{13} -1.96077 q^{14} -4.16852 q^{16} +1.13706 q^{17} -1.00000 q^{19} +7.00969 q^{22} -2.55496 q^{23} -0.286208 q^{26} +0.119605 q^{28} +10.2349 q^{29} +2.51573 q^{31} +0.498271 q^{32} -1.64310 q^{34} -0.137063 q^{37} +1.44504 q^{38} +11.7506 q^{41} -7.59179 q^{43} -0.427583 q^{44} +3.69202 q^{46} -2.69202 q^{47} -5.15883 q^{49} +0.0174584 q^{52} -12.8780 q^{53} +3.74871 q^{56} -14.7899 q^{58} -5.82371 q^{59} -7.58211 q^{61} -3.63533 q^{62} +7.61702 q^{64} +8.01507 q^{67} +0.100228 q^{68} +8.82371 q^{71} -11.9705 q^{73} +0.198062 q^{74} -0.0881460 q^{76} -6.58211 q^{77} +10.7409 q^{79} -16.9801 q^{82} -3.77479 q^{83} +10.9705 q^{86} -13.4015 q^{88} -9.36658 q^{89} +0.268750 q^{91} -0.225209 q^{92} +3.89008 q^{94} +0.198062 q^{97} +7.45473 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{2} + 4 q^{4} - 9 q^{8} - q^{11} + 5 q^{13} + 7 q^{14} + 18 q^{16} - 2 q^{17} - 3 q^{19} - q^{22} - 8 q^{23} - 9 q^{26} - 21 q^{28} + 7 q^{29} - 5 q^{31} - 27 q^{32} - 9 q^{34} + 5 q^{37} + 4 q^{38}+ \cdots + 5 q^{97}+O(q^{100}) $ 3 * q - 4 * q^2 + 4 * q^4 - 9 * q^8 - q^11 + 5 * q^13 + 7 * q^14 + 18 * q^16 - 2 * q^17 - 3 * q^19 - q^22 - 8 * q^23 - 9 * q^26 - 21 * q^28 + 7 * q^29 - 5 * q^31 - 27 * q^32 - 9 * q^34 + 5 * q^37 + 4 * q^38 - q^41 + 5 * q^43 + 15 * q^44 + 6 * q^46 - 3 * q^47 - 7 * q^49 + 16 * q^52 - 19 * q^53 + 35 * q^56 - 21 * q^58 - 10 * q^59 - 17 * q^61 + 23 * q^62 + 49 * q^64 - q^67 + 23 * q^68 + 19 * q^71 - q^73 + 5 * q^74 - 4 * q^76 - 14 * q^77 + 18 * q^79 + 6 * q^82 - 13 * q^83 - 2 * q^86 - 46 * q^88 - 2 * q^89 - 7 * q^91 + q^92 + 11 * q^94 + 5 * q^97

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 $

$2$	−1.44504	−1.02180	−0.510899	−	0.859640i	$-0.670688\pi$
$2$	−1.44504	−1.02180	−0.510899	+	0.859640i	$0.670688\pi$
$3$	0	0
$3$	0	0
$4$	0.0881460	0.0440730
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.35690	0.512858	0.256429	−	0.966563i	$-0.417454\pi$
$7$	1.35690	0.512858	0.256429	+	0.966563i	$0.417454\pi$
$8$	2.76271	0.976765
$9$	0	0
$10$	0	0
$11$	−4.85086	−1.46259	−0.731294	−	0.682062i	$-0.761083\pi$
$11$	−4.85086	−1.46259	−0.731294	+	0.682062i	$0.761083\pi$
$12$	0	0
$13$	0.198062	0.0549326	0.0274663	−	0.999623i	$-0.491256\pi$
$13$	0.198062	0.0549326	0.0274663	+	0.999623i	$0.491256\pi$
$14$	−1.96077	−0.524038
$15$	0	0
$16$	−4.16852	−1.04213
$17$	1.13706	0.275778	0.137889	−	0.990448i	$-0.455968\pi$
$17$	1.13706	0.275778	0.137889	+	0.990448i	$0.455968\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	7.00969	1.49447
$23$	−2.55496	−0.532746	−0.266373	−	0.963870i	$-0.585825\pi$
$23$	−2.55496	−0.532746	−0.266373	+	0.963870i	$0.585825\pi$
$24$	0	0
$25$	0	0
$26$	−0.286208	−0.0561301
$27$	0	0
$28$	0.119605	0.0226032
$29$	10.2349	1.90057	0.950286	−	0.311377i	$-0.100790\pi$
$29$	10.2349	1.90057	0.950286	+	0.311377i	$0.100790\pi$
$30$	0	0
$31$	2.51573	0.451838	0.225919	−	0.974146i	$-0.427461\pi$
$31$	2.51573	0.451838	0.225919	+	0.974146i	$0.427461\pi$
$32$	0.498271	0.0880827
$33$	0	0
$34$	−1.64310	−0.281790
$35$	0	0
$36$	0	0
$37$	−0.137063	−0.0225331	−0.0112665	−	0.999937i	$-0.503586\pi$
$37$	−0.137063	−0.0225331	−0.0112665	+	0.999937i	$0.503586\pi$
$38$	1.44504	0.234417
$39$	0	0
$40$	0	0
$41$	11.7506	1.83514	0.917570	−	0.397575i	$-0.130148\pi$
$41$	11.7506	1.83514	0.917570	+	0.397575i	$0.130148\pi$
$42$	0	0
$43$	−7.59179	−1.15774	−0.578869	−	0.815421i	$-0.696506\pi$
$43$	−7.59179	−1.15774	−0.578869	+	0.815421i	$0.696506\pi$
$44$	−0.427583	−0.0644606
$45$	0	0
$46$	3.69202	0.544359
$47$	−2.69202	−0.392672	−0.196336	−	0.980537i	$-0.562904\pi$
$47$	−2.69202	−0.392672	−0.196336	+	0.980537i	$0.562904\pi$
$48$	0	0
$49$	−5.15883	−0.736976
$50$	0	0
$51$	0	0
$52$	0.0174584	0.00242104
$53$	−12.8780	−1.76893	−0.884465	−	0.466607i	$-0.845476\pi$
$53$	−12.8780	−1.76893	−0.884465	+	0.466607i	$0.845476\pi$
$54$	0	0
$55$	0	0
$56$	3.74871	0.500942
$57$	0	0
$58$	−14.7899	−1.94200
$59$	−5.82371	−0.758182	−0.379091	−	0.925359i	$-0.623763\pi$
$59$	−5.82371	−0.758182	−0.379091	+	0.925359i	$0.623763\pi$
$60$	0	0
$61$	−7.58211	−0.970789	−0.485395	−	0.874295i	$-0.661324\pi$
$61$	−7.58211	−0.970789	−0.485395	+	0.874295i	$0.661324\pi$
$62$	−3.63533	−0.461688
$63$	0	0
$64$	7.61702	0.952128
$65$	0	0
$66$	0	0
$67$	8.01507	0.979196	0.489598	−	0.871948i	$-0.337144\pi$
$67$	8.01507	0.979196	0.489598	+	0.871948i	$0.337144\pi$
$68$	0.100228	0.0121544
$69$	0	0
$70$	0	0
$71$	8.82371	1.04718	0.523591	−	0.851970i	$-0.324592\pi$
$71$	8.82371	1.04718	0.523591	+	0.851970i	$0.324592\pi$
$72$	0	0
$73$	−11.9705	−1.40104	−0.700518	−	0.713635i	$-0.747048\pi$
$73$	−11.9705	−1.40104	−0.700518	+	0.713635i	$0.747048\pi$
$74$	0.198062	0.0230243
$75$	0	0
$76$	−0.0881460	−0.0101110
$77$	−6.58211	−0.750101
$78$	0	0
$79$	10.7409	1.20845	0.604225	−	0.796814i	$-0.293483\pi$
$79$	10.7409	1.20845	0.604225	+	0.796814i	$0.293483\pi$
$80$	0	0
$81$	0	0
$82$	−16.9801	−1.87514
$83$	−3.77479	−0.414337	−0.207169	−	0.978305i	$-0.566425\pi$
$83$	−3.77479	−0.414337	−0.207169	+	0.978305i	$0.566425\pi$
$84$	0	0
$85$	0	0
$86$	10.9705	1.18298
$87$	0	0
$88$	−13.4015	−1.42860
$89$	−9.36658	−0.992856	−0.496428	−	0.868078i	$-0.665355\pi$
$89$	−9.36658	−0.992856	−0.496428	+	0.868078i	$0.665355\pi$
$90$	0	0
$91$	0.268750	0.0281726
$92$	−0.225209	−0.0234797
$93$	0	0
$94$	3.89008	0.401232
$95$	0	0
$96$	0	0
$97$	0.198062	0.0201102	0.0100551	−	0.999949i	$-0.496799\pi$
$97$	0.198062	0.0201102	0.0100551	+	0.999949i	$0.496799\pi$
$98$	7.45473	0.753042
$99$	0	0
$100$	0	0
$101$	−11.5090	−1.14519	−0.572595	−	0.819838i	$-0.694063\pi$
$101$	−11.5090	−1.14519	−0.572595	+	0.819838i	$0.694063\pi$
$102$	0	0
$103$	15.1564	1.49341	0.746704	−	0.665156i	$-0.231635\pi$
$103$	15.1564	1.49341	0.746704	+	0.665156i	$0.231635\pi$
$104$	0.547188	0.0536562
$105$	0	0
$106$	18.6093	1.80749
$107$	−2.65279	−0.256455	−0.128228	−	0.991745i	$-0.540929\pi$
$107$	−2.65279	−0.256455	−0.128228	+	0.991745i	$0.540929\pi$
$108$	0	0
$109$	−2.49934	−0.239393	−0.119696	−	0.992811i	$-0.538192\pi$
$109$	−2.49934	−0.239393	−0.119696	+	0.992811i	$0.538192\pi$
$110$	0	0
$111$	0	0
$112$	−5.65625	−0.534465
$113$	−8.52781	−0.802229	−0.401114	−	0.916028i	$-0.631377\pi$
$113$	−8.52781	−0.802229	−0.401114	+	0.916028i	$0.631377\pi$
$114$	0	0
$115$	0	0
$116$	0.902165	0.0837639
$117$	0	0
$118$	8.41550	0.774710
$119$	1.54288	0.141435
$120$	0	0
$121$	12.5308	1.13916
$122$	10.9565	0.991951
$123$	0	0
$124$	0.221751	0.0199139
$125$	0	0
$126$	0	0
$127$	20.4088	1.81099	0.905494	−	0.424359i	$-0.139501\pi$
$127$	20.4088	1.81099	0.905494	+	0.424359i	$0.139501\pi$
$128$	−12.0035	−1.06097
$129$	0	0
$130$	0	0
$131$	13.2131	1.15444	0.577218	−	0.816590i	$-0.304138\pi$
$131$	13.2131	1.15444	0.577218	+	0.816590i	$0.304138\pi$
$132$	0	0
$133$	−1.35690	−0.117658
$134$	−11.5821	−1.00054
$135$	0	0
$136$	3.14138	0.269371
$137$	6.86054	0.586136	0.293068	−	0.956092i	$-0.405324\pi$
$137$	6.86054	0.586136	0.293068	+	0.956092i	$0.405324\pi$
$138$	0	0
$139$	4.28621	0.363551	0.181776	−	0.983340i	$-0.441816\pi$
$139$	4.28621	0.363551	0.181776	+	0.983340i	$0.441816\pi$
$140$	0	0
$141$	0	0
$142$	−12.7506	−1.07001
$143$	−0.960771	−0.0803437
$144$	0	0
$145$	0	0
$146$	17.2978	1.43158
$147$	0	0
$148$	−0.0120816	−0.000993100 0
$149$	15.3545	1.25789	0.628945	−	0.777450i	$-0.283487\pi$
$149$	15.3545	1.25789	0.628945	+	0.777450i	$0.283487\pi$
$150$	0	0
$151$	−10.2295	−0.832467	−0.416233	−	0.909258i	$-0.636650\pi$
$151$	−10.2295	−0.832467	−0.416233	+	0.909258i	$0.636650\pi$
$152$	−2.76271	−0.224085
$153$	0	0
$154$	9.51142	0.766452
$155$	0	0
$156$	0	0
$157$	3.17092	0.253067	0.126533	−	0.991962i	$-0.459615\pi$
$157$	3.17092	0.253067	0.126533	+	0.991962i	$0.459615\pi$
$158$	−15.5211	−1.23479
$159$	0	0
$160$	0	0
$161$	−3.46681	−0.273223
$162$	0	0
$163$	−4.63773	−0.363255	−0.181627	−	0.983367i	$-0.558136\pi$
$163$	−4.63773	−0.363255	−0.181627	+	0.983367i	$0.558136\pi$
$164$	1.03577	0.0808801
$165$	0	0
$166$	5.45473	0.423369
$167$	19.6286	1.51891	0.759454	−	0.650560i	$-0.225466\pi$
$167$	19.6286	1.51891	0.759454	+	0.650560i	$0.225466\pi$
$168$	0	0
$169$	−12.9608	−0.996982
$170$	0	0
$171$	0	0
$172$	−0.669186	−0.0510250
$173$	−20.5646	−1.56350	−0.781751	−	0.623591i	$-0.785673\pi$
$173$	−20.5646	−1.56350	−0.781751	+	0.623591i	$0.785673\pi$
$174$	0	0
$175$	0	0
$176$	20.2209	1.52421
$177$	0	0
$178$	13.5351	1.01450
$179$	−6.92154	−0.517340	−0.258670	−	0.965966i	$-0.583284\pi$
$179$	−6.92154	−0.517340	−0.258670	+	0.965966i	$0.583284\pi$
$180$	0	0
$181$	−17.6461	−1.31162	−0.655812	−	0.754925i	$-0.727673\pi$
$181$	−17.6461	−1.31162	−0.655812	+	0.754925i	$0.727673\pi$
$182$	−0.388355	−0.0287868
$183$	0	0
$184$	−7.05861	−0.520367
$185$	0	0
$186$	0	0
$187$	−5.51573	−0.403350
$188$	−0.237291	−0.0173062
$189$	0	0
$190$	0	0
$191$	−6.92931	−0.501387	−0.250694	−	0.968066i	$-0.580659\pi$
$191$	−6.92931	−0.501387	−0.250694	+	0.968066i	$0.580659\pi$
$192$	0	0
$193$	−21.0368	−1.51426	−0.757132	−	0.653262i	$-0.773400\pi$
$193$	−21.0368	−1.51426	−0.757132	+	0.653262i	$0.773400\pi$
$194$	−0.286208	−0.0205486
$195$	0	0
$196$	−0.454731	−0.0324808
$197$	−21.5646	−1.53642	−0.768209	−	0.640199i	$-0.778852\pi$
$197$	−21.5646	−1.53642	−0.768209	+	0.640199i	$0.778852\pi$
$198$	0	0
$199$	21.9909	1.55889	0.779447	−	0.626468i	$-0.215500\pi$
$199$	21.9909	1.55889	0.779447	+	0.626468i	$0.215500\pi$
$200$	0	0
$201$	0	0
$202$	16.6310	1.17015
$203$	13.8877	0.974725
$204$	0	0
$205$	0	0
$206$	−21.9017	−1.52596
$207$	0	0
$208$	−0.825627	−0.0572469
$209$	4.85086	0.335541
$210$	0	0
$211$	−20.6233	−1.41976	−0.709882	−	0.704321i	$-0.751252\pi$
$211$	−20.6233	−1.41976	−0.709882	+	0.704321i	$0.751252\pi$
$212$	−1.13514	−0.0779620
$213$	0	0
$214$	3.83340	0.262046
$215$	0	0
$216$	0	0
$217$	3.41358	0.231729
$218$	3.61165	0.244611
$219$	0	0
$220$	0	0
$221$	0.225209	0.0151492
$222$	0	0
$223$	7.97716	0.534190	0.267095	−	0.963670i	$-0.413936\pi$
$223$	7.97716	0.534190	0.267095	+	0.963670i	$0.413936\pi$
$224$	0.676102	0.0451740
$225$	0	0
$226$	12.3230	0.819717
$227$	−19.7942	−1.31379	−0.656893	−	0.753984i	$-0.728129\pi$
$227$	−19.7942	−1.31379	−0.656893	+	0.753984i	$0.728129\pi$
$228$	0	0
$229$	−4.03385	−0.266564	−0.133282	−	0.991078i	$-0.542552\pi$
$229$	−4.03385	−0.266564	−0.133282	+	0.991078i	$0.542552\pi$
$230$	0	0
$231$	0	0
$232$	28.2760	1.85641
$233$	−26.8159	−1.75677	−0.878385	−	0.477953i	$-0.841379\pi$
$233$	−26.8159	−1.75677	−0.878385	+	0.477953i	$0.841379\pi$
$234$	0	0
$235$	0	0
$236$	−0.513337	−0.0334154
$237$	0	0
$238$	−2.22952	−0.144518
$239$	−3.36227	−0.217487	−0.108744	−	0.994070i	$-0.534683\pi$
$239$	−3.36227	−0.217487	−0.108744	+	0.994070i	$0.534683\pi$
$240$	0	0
$241$	27.7506	1.78758	0.893788	−	0.448491i	$-0.148038\pi$
$241$	27.7506	1.78758	0.893788	+	0.448491i	$0.148038\pi$
$242$	−18.1075	−1.16400
$243$	0	0
$244$	−0.668332	−0.0427856
$245$	0	0
$246$	0	0
$247$	−0.198062	−0.0126024
$248$	6.95023	0.441340
$249$	0	0
$250$	0	0
$251$	5.59419	0.353102	0.176551	−	0.984291i	$-0.443506\pi$
$251$	5.59419	0.353102	0.176551	+	0.984291i	$0.443506\pi$
$252$	0	0
$253$	12.3937	0.779187
$254$	−29.4916	−1.85047
$255$	0	0
$256$	2.11146	0.131966
$257$	−10.4668	−0.652902	−0.326451	−	0.945214i	$-0.605853\pi$
$257$	−10.4668	−0.652902	−0.326451	+	0.945214i	$0.605853\pi$
$258$	0	0
$259$	−0.185981	−0.0115563
$260$	0	0
$261$	0	0
$262$	−19.0935	−1.17960
$263$	−15.4795	−0.954506	−0.477253	−	0.878766i	$-0.658367\pi$
$263$	−15.4795	−0.954506	−0.477253	+	0.878766i	$0.658367\pi$
$264$	0	0
$265$	0	0
$266$	1.96077	0.120223
$267$	0	0
$268$	0.706496	0.0431561
$269$	−9.13036	−0.556688	−0.278344	−	0.960481i	$-0.589785\pi$
$269$	−9.13036	−0.556688	−0.278344	+	0.960481i	$0.589785\pi$
$270$	0	0
$271$	7.44265	0.452109	0.226054	−	0.974115i	$-0.427417\pi$
$271$	7.44265	0.452109	0.226054	+	0.974115i	$0.427417\pi$
$272$	−4.73987	−0.287397
$273$	0	0
$274$	−9.91377	−0.598913
$275$	0	0
$276$	0	0
$277$	−11.4155	−0.685891	−0.342946	−	0.939355i	$-0.611425\pi$
$277$	−11.4155	−0.685891	−0.342946	+	0.939355i	$0.611425\pi$
$278$	−6.19375	−0.371476
$279$	0	0
$280$	0	0
$281$	−21.5060	−1.28294	−0.641471	−	0.767147i	$-0.721676\pi$
$281$	−21.5060	−1.28294	−0.641471	+	0.767147i	$0.721676\pi$
$282$	0	0
$283$	−5.45712	−0.324392	−0.162196	−	0.986759i	$-0.551858\pi$
$283$	−5.45712	−0.324392	−0.162196	+	0.986759i	$0.551858\pi$
$284$	0.777775	0.0461524
$285$	0	0
$286$	1.38835	0.0820951
$287$	15.9444	0.941167
$288$	0	0
$289$	−15.7071	−0.923946
$290$	0	0
$291$	0	0
$292$	−1.05515	−0.0617479
$293$	−7.39075	−0.431772	−0.215886	−	0.976419i	$-0.569264\pi$
$293$	−7.39075	−0.431772	−0.215886	+	0.976419i	$0.569264\pi$
$294$	0	0
$295$	0	0
$296$	−0.378666	−0.0220095
$297$	0	0
$298$	−22.1879	−1.28531
$299$	−0.506041	−0.0292651
$300$	0	0
$301$	−10.3013	−0.593756
$302$	14.7821	0.850613
$303$	0	0
$304$	4.16852	0.239081
$305$	0	0
$306$	0	0
$307$	−32.2131	−1.83850	−0.919250	−	0.393674i	$-0.871204\pi$
$307$	−32.2131	−1.83850	−0.919250	+	0.393674i	$0.871204\pi$
$308$	−0.580186	−0.0330592
$309$	0	0
$310$	0	0
$311$	−14.8442	−0.841735	−0.420867	−	0.907122i	$-0.638274\pi$
$311$	−14.8442	−0.841735	−0.420867	+	0.907122i	$0.638274\pi$
$312$	0	0
$313$	−13.1491	−0.743234	−0.371617	−	0.928386i	$-0.621196\pi$
$313$	−13.1491	−0.743234	−0.371617	+	0.928386i	$0.621196\pi$
$314$	−4.58211	−0.258583
$315$	0	0
$316$	0.946771	0.0532600
$317$	5.13467	0.288392	0.144196	−	0.989549i	$-0.453940\pi$
$317$	5.13467	0.288392	0.144196	+	0.989549i	$0.453940\pi$
$318$	0	0
$319$	−49.6480	−2.77975
$320$	0	0
$321$	0	0
$322$	5.00969	0.279179
$323$	−1.13706	−0.0632679
$324$	0	0
$325$	0	0
$326$	6.70171	0.371173
$327$	0	0
$328$	32.4636	1.79250
$329$	−3.65279	−0.201385
$330$	0	0
$331$	2.00969	0.110462	0.0552312	−	0.998474i	$-0.482410\pi$
$331$	2.00969	0.110462	0.0552312	+	0.998474i	$0.482410\pi$
$332$	−0.332733	−0.0182611
$333$	0	0
$334$	−28.3642	−1.55202
$335$	0	0
$336$	0	0
$337$	−1.31873	−0.0718359	−0.0359180	−	0.999355i	$-0.511436\pi$
$337$	−1.31873	−0.0718359	−0.0359180	+	0.999355i	$0.511436\pi$
$338$	18.7289	1.01872
$339$	0	0
$340$	0	0
$341$	−12.2034	−0.660853
$342$	0	0
$343$	−16.4983	−0.890823
$344$	−20.9739	−1.13084
$345$	0	0
$346$	29.7168	1.59758
$347$	−12.0151	−0.645003	−0.322501	−	0.946569i	$-0.604524\pi$
$347$	−12.0151	−0.645003	−0.322501	+	0.946569i	$0.604524\pi$
$348$	0	0
$349$	−10.5579	−0.565154	−0.282577	−	0.959245i	$-0.591189\pi$
$349$	−10.5579	−0.565154	−0.282577	+	0.959245i	$0.591189\pi$
$350$	0	0
$351$	0	0
$352$	−2.41704	−0.128829
$353$	1.01102	0.0538110	0.0269055	−	0.999638i	$-0.491435\pi$
$353$	1.01102	0.0538110	0.0269055	+	0.999638i	$0.491435\pi$
$354$	0	0
$355$	0	0
$356$	−0.825627	−0.0437581
$357$	0	0
$358$	10.0019	0.528618
$359$	5.14244	0.271408	0.135704	−	0.990749i	$-0.456670\pi$
$359$	5.14244	0.271408	0.135704	+	0.990749i	$0.456670\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	25.4993	1.34022
$363$	0	0
$364$	0.0236892	0.00124165
$365$	0	0
$366$	0	0
$367$	−29.5308	−1.54149	−0.770747	−	0.637141i	$-0.780117\pi$
$367$	−29.5308	−1.54149	−0.770747	+	0.637141i	$0.780117\pi$
$368$	10.6504	0.555190
$369$	0	0
$370$	0	0
$371$	−17.4741	−0.907210
$372$	0	0
$373$	8.78986	0.455121	0.227561	−	0.973764i	$-0.426925\pi$
$373$	8.78986	0.455121	0.227561	+	0.973764i	$0.426925\pi$
$374$	7.97046	0.412143
$375$	0	0
$376$	−7.43727	−0.383548
$377$	2.02715	0.104403
$378$	0	0
$379$	19.1511	0.983724	0.491862	−	0.870673i	$-0.336316\pi$
$379$	19.1511	0.983724	0.491862	+	0.870673i	$0.336316\pi$
$380$	0	0
$381$	0	0
$382$	10.0131	0.512317
$383$	6.99894	0.357629	0.178814	−	0.983883i	$-0.442774\pi$
$383$	6.99894	0.357629	0.178814	+	0.983883i	$0.442774\pi$
$384$	0	0
$385$	0	0
$386$	30.3991	1.54727
$387$	0	0
$388$	0.0174584	0.000886316 0
$389$	−8.08575	−0.409964	−0.204982	−	0.978766i	$-0.565714\pi$
$389$	−8.08575	−0.409964	−0.204982	+	0.978766i	$0.565714\pi$
$390$	0	0
$391$	−2.90515	−0.146920
$392$	−14.2524	−0.719853
$393$	0	0
$394$	31.1618	1.56991
$395$	0	0
$396$	0	0
$397$	−17.7006	−0.888370	−0.444185	−	0.895935i	$-0.646507\pi$
$397$	−17.7006	−0.888370	−0.444185	+	0.895935i	$0.646507\pi$
$398$	−31.7778	−1.59288
$399$	0	0
$400$	0	0
$401$	15.5418	0.776121	0.388061	−	0.921634i	$-0.373145\pi$
$401$	15.5418	0.776121	0.388061	+	0.921634i	$0.373145\pi$
$402$	0	0
$403$	0.498271	0.0248207
$404$	−1.01447	−0.0504720
$405$	0	0
$406$	−20.0683	−0.995973
$407$	0.664874	0.0329566
$408$	0	0
$409$	13.1661	0.651023	0.325512	−	0.945538i	$-0.394463\pi$
$409$	13.1661	0.651023	0.325512	+	0.945538i	$0.394463\pi$
$410$	0	0
$411$	0	0
$412$	1.33598	0.0658190
$413$	−7.90217	−0.388840
$414$	0	0
$415$	0	0
$416$	0.0986887	0.00483861
$417$	0	0
$418$	−7.00969	−0.342855
$419$	25.1739	1.22983	0.614913	−	0.788595i	$-0.289191\pi$
$419$	25.1739	1.22983	0.614913	+	0.788595i	$0.289191\pi$
$420$	0	0
$421$	26.9420	1.31307	0.656536	−	0.754295i	$-0.272021\pi$
$421$	26.9420	1.31307	0.656536	+	0.754295i	$0.272021\pi$
$422$	29.8015	1.45071
$423$	0	0
$424$	−35.5782	−1.72783
$425$	0	0
$426$	0	0
$427$	−10.2881	−0.497877
$428$	−0.233833	−0.0113027
$429$	0	0
$430$	0	0
$431$	−18.9487	−0.912726	−0.456363	−	0.889794i	$-0.650848\pi$
$431$	−18.9487	−0.912726	−0.456363	+	0.889794i	$0.650848\pi$
$432$	0	0
$433$	−3.68904	−0.177284	−0.0886419	−	0.996064i	$-0.528253\pi$
$433$	−3.68904	−0.177284	−0.0886419	+	0.996064i	$0.528253\pi$
$434$	−4.93277	−0.236781
$435$	0	0
$436$	−0.220306	−0.0105508
$437$	2.55496	0.122220
$438$	0	0
$439$	−25.6926	−1.22624	−0.613121	−	0.789989i	$-0.710086\pi$
$439$	−25.6926	−1.22624	−0.613121	+	0.789989i	$0.710086\pi$
$440$	0	0
$441$	0	0
$442$	−0.325437	−0.0154795
$443$	27.3653	1.30016	0.650081	−	0.759865i	$-0.274735\pi$
$443$	27.3653	1.30016	0.650081	+	0.759865i	$0.274735\pi$
$444$	0	0
$445$	0	0
$446$	−11.5273	−0.545835
$447$	0	0
$448$	10.3355	0.488307
$449$	−7.55794	−0.356681	−0.178341	−	0.983969i	$-0.557073\pi$
$449$	−7.55794	−0.356681	−0.178341	+	0.983969i	$0.557073\pi$
$450$	0	0
$451$	−57.0006	−2.68405
$452$	−0.751692	−0.0353566
$453$	0	0
$454$	28.6034	1.34242
$455$	0	0
$456$	0	0
$457$	−7.85623	−0.367499	−0.183750	−	0.982973i	$-0.558824\pi$
$457$	−7.85623	−0.367499	−0.183750	+	0.982973i	$0.558824\pi$
$458$	5.82908	0.272375
$459$	0	0
$460$	0	0
$461$	−3.87907	−0.180666	−0.0903331	−	0.995912i	$-0.528793\pi$
$461$	−3.87907	−0.180666	−0.0903331	+	0.995912i	$0.528793\pi$
$462$	0	0
$463$	−13.0954	−0.608597	−0.304298	−	0.952577i	$-0.598422\pi$
$463$	−13.0954	−0.608597	−0.304298	+	0.952577i	$0.598422\pi$
$464$	−42.6644	−1.98065
$465$	0	0
$466$	38.7502	1.79507
$467$	−6.44026	−0.298020	−0.149010	−	0.988836i	$-0.547609\pi$
$467$	−6.44026	−0.298020	−0.149010	+	0.988836i	$0.547609\pi$
$468$	0	0
$469$	10.8756	0.502189
$470$	0	0
$471$	0	0
$472$	−16.0892	−0.740566
$473$	36.8267	1.69329
$474$	0	0
$475$	0	0
$476$	0.135998	0.00623348
$477$	0	0
$478$	4.85862	0.222228
$479$	−5.69096	−0.260026	−0.130013	−	0.991512i	$-0.541502\pi$
$479$	−5.69096	−0.260026	−0.130013	+	0.991512i	$0.541502\pi$
$480$	0	0
$481$	−0.0271471	−0.00123780
$482$	−40.1008	−1.82654
$483$	0	0
$484$	1.10454	0.0502063
$485$	0	0
$486$	0	0
$487$	−12.9632	−0.587417	−0.293709	−	0.955895i	$-0.594890\pi$
$487$	−12.9632	−0.587417	−0.293709	+	0.955895i	$0.594890\pi$
$488$	−20.9472	−0.948233
$489$	0	0
$490$	0	0
$491$	19.6045	0.884737	0.442369	−	0.896833i	$-0.354138\pi$
$491$	19.6045	0.884737	0.442369	+	0.896833i	$0.354138\pi$
$492$	0	0
$493$	11.6377	0.524137
$494$	0.286208	0.0128771
$495$	0	0
$496$	−10.4869	−0.470875
$497$	11.9729	0.537056
$498$	0	0
$499$	5.49827	0.246136	0.123068	−	0.992398i	$-0.460727\pi$
$499$	5.49827	0.246136	0.123068	+	0.992398i	$0.460727\pi$
$500$	0	0
$501$	0	0
$502$	−8.08383	−0.360799
$503$	35.6939	1.59151	0.795757	−	0.605616i	$-0.207073\pi$
$503$	35.6939	1.59151	0.795757	+	0.605616i	$0.207073\pi$
$504$	0	0
$505$	0	0
$506$	−17.9095	−0.796173
$507$	0	0
$508$	1.79895	0.0798157
$509$	16.4450	0.728914	0.364457	−	0.931220i	$-0.381255\pi$
$509$	16.4450	0.728914	0.364457	+	0.931220i	$0.381255\pi$
$510$	0	0
$511$	−16.2427	−0.718533
$512$	20.9558	0.926123
$513$	0	0
$514$	15.1250	0.667134
$515$	0	0
$516$	0	0
$517$	13.0586	0.574317
$518$	0.268750	0.0118082
$519$	0	0
$520$	0	0
$521$	26.5435	1.16289	0.581445	−	0.813586i	$-0.302487\pi$
$521$	26.5435	1.16289	0.581445	+	0.813586i	$0.302487\pi$
$522$	0	0
$523$	−24.1685	−1.05682	−0.528408	−	0.848991i	$-0.677211\pi$
$523$	−24.1685	−1.05682	−0.528408	+	0.848991i	$0.677211\pi$
$524$	1.16468	0.0508795
$525$	0	0
$526$	22.3685	0.975313
$527$	2.86054	0.124607
$528$	0	0
$529$	−16.4722	−0.716182
$530$	0	0
$531$	0	0
$532$	−0.119605	−0.00518553
$533$	2.32736	0.100809
$534$	0	0
$535$	0	0
$536$	22.1433	0.956445
$537$	0	0
$538$	13.1938	0.568823
$539$	25.0248	1.07789
$540$	0	0
$541$	9.80386	0.421501	0.210750	−	0.977540i	$-0.432409\pi$
$541$	9.80386	0.421501	0.210750	+	0.977540i	$0.432409\pi$
$542$	−10.7549	−0.461964
$543$	0	0
$544$	0.566566	0.0242913
$545$	0	0
$546$	0	0
$547$	21.1739	0.905331	0.452665	−	0.891681i	$-0.350473\pi$
$547$	21.1739	0.905331	0.452665	+	0.891681i	$0.350473\pi$
$548$	0.604729	0.0258328
$549$	0	0
$550$	0	0
$551$	−10.2349	−0.436021
$552$	0	0
$553$	14.5743	0.619764
$554$	16.4959	0.700843
$555$	0	0
$556$	0.377812	0.0160228
$557$	−24.4077	−1.03419	−0.517094	−	0.855928i	$-0.672986\pi$
$557$	−24.4077	−1.03419	−0.517094	+	0.855928i	$0.672986\pi$
$558$	0	0
$559$	−1.50365	−0.0635975
$560$	0	0
$561$	0	0
$562$	31.0771	1.31091
$563$	−14.4849	−0.610464	−0.305232	−	0.952278i	$-0.598734\pi$
$563$	−14.4849	−0.610464	−0.305232	+	0.952278i	$0.598734\pi$
$564$	0	0
$565$	0	0
$566$	7.88577	0.331464
$567$	0	0
$568$	24.3773	1.02285
$569$	−0.811626	−0.0340251	−0.0170126	−	0.999855i	$-0.505416\pi$
$569$	−0.811626	−0.0340251	−0.0170126	+	0.999855i	$0.505416\pi$
$570$	0	0
$571$	−37.6588	−1.57597	−0.787985	−	0.615694i	$-0.788876\pi$
$571$	−37.6588	−1.57597	−0.787985	+	0.615694i	$0.788876\pi$
$572$	−0.0846882	−0.00354099
$573$	0	0
$574$	−23.0403	−0.961683
$575$	0	0
$576$	0	0
$577$	8.89307	0.370223	0.185112	−	0.982718i	$-0.440735\pi$
$577$	8.89307	0.370223	0.185112	+	0.982718i	$0.440735\pi$
$578$	22.6974	0.944087
$579$	0	0
$580$	0	0
$581$	−5.12200	−0.212496
$582$	0	0
$583$	62.4693	2.58721
$584$	−33.0709	−1.36848
$585$	0	0
$586$	10.6799	0.441184
$587$	−20.3327	−0.839222	−0.419611	−	0.907704i	$-0.637833\pi$
$587$	−20.3327	−0.839222	−0.419611	+	0.907704i	$0.637833\pi$
$588$	0	0
$589$	−2.51573	−0.103659
$590$	0	0
$591$	0	0
$592$	0.571352	0.0234824
$593$	−17.0049	−0.698308	−0.349154	−	0.937065i	$-0.613531\pi$
$593$	−17.0049	−0.698308	−0.349154	+	0.937065i	$0.613531\pi$
$594$	0	0
$595$	0	0
$596$	1.35344	0.0554390
$597$	0	0
$598$	0.731250	0.0299030
$599$	12.4004	0.506668	0.253334	−	0.967379i	$-0.418473\pi$
$599$	12.4004	0.506668	0.253334	+	0.967379i	$0.418473\pi$
$600$	0	0
$601$	−9.73125	−0.396946	−0.198473	−	0.980106i	$-0.563598\pi$
$601$	−9.73125	−0.396946	−0.198473	+	0.980106i	$0.563598\pi$
$602$	14.8858	0.606699
$603$	0	0
$604$	−0.901691	−0.0366893
$605$	0	0
$606$	0	0
$607$	−23.4282	−0.950920	−0.475460	−	0.879737i	$-0.657718\pi$
$607$	−23.4282	−0.950920	−0.475460	+	0.879737i	$0.657718\pi$
$608$	−0.498271	−0.0202076
$609$	0	0
$610$	0	0
$611$	−0.533188	−0.0215705
$612$	0	0
$613$	28.1424	1.13666	0.568331	−	0.822800i	$-0.307589\pi$
$613$	28.1424	1.13666	0.568331	+	0.822800i	$0.307589\pi$
$614$	46.5493	1.87858
$615$	0	0
$616$	−18.1844	−0.732672
$617$	−36.4805	−1.46865	−0.734326	−	0.678797i	$-0.762502\pi$
$617$	−36.4805	−1.46865	−0.734326	+	0.678797i	$0.762502\pi$
$618$	0	0
$619$	−38.2097	−1.53578	−0.767888	−	0.640584i	$-0.778692\pi$
$619$	−38.2097	−1.53578	−0.767888	+	0.640584i	$0.778692\pi$
$620$	0	0
$621$	0	0
$622$	21.4504	0.860083
$623$	−12.7095	−0.509195
$624$	0	0
$625$	0	0
$626$	19.0011	0.759435
$627$	0	0
$628$	0.279503	0.0111534
$629$	−0.155850	−0.00621413
$630$	0	0
$631$	−12.5235	−0.498553	−0.249276	−	0.968432i	$-0.580193\pi$
$631$	−12.5235	−0.498553	−0.249276	+	0.968432i	$0.580193\pi$
$632$	29.6741	1.18037
$633$	0	0
$634$	−7.41981	−0.294678
$635$	0	0
$636$	0	0
$637$	−1.02177	−0.0404840
$638$	71.7434	2.84035
$639$	0	0
$640$	0	0
$641$	−37.3564	−1.47549	−0.737745	−	0.675080i	$-0.764109\pi$
$641$	−37.3564	−1.47549	−0.737745	+	0.675080i	$0.764109\pi$
$642$	0	0
$643$	−14.5483	−0.573727	−0.286864	−	0.957971i	$-0.592613\pi$
$643$	−14.5483	−0.573727	−0.286864	+	0.957971i	$0.592613\pi$
$644$	−0.305586	−0.0120418
$645$	0	0
$646$	1.64310	0.0646471
$647$	23.4403	0.921532	0.460766	−	0.887522i	$-0.347575\pi$
$647$	23.4403	0.921532	0.460766	+	0.887522i	$0.347575\pi$
$648$	0	0
$649$	28.2500	1.10891
$650$	0	0
$651$	0	0
$652$	−0.408797	−0.0160097
$653$	−27.1903	−1.06404	−0.532019	−	0.846732i	$-0.678567\pi$
$653$	−27.1903	−1.06404	−0.532019	+	0.846732i	$0.678567\pi$
$654$	0	0
$655$	0	0
$656$	−48.9828	−1.91246
$657$	0	0
$658$	5.27844	0.205775
$659$	−2.71486	−0.105756	−0.0528779	−	0.998601i	$-0.516839\pi$
$659$	−2.71486	−0.105756	−0.0528779	+	0.998601i	$0.516839\pi$
$660$	0	0
$661$	−9.41311	−0.366128	−0.183064	−	0.983101i	$-0.558601\pi$
$661$	−9.41311	−0.366128	−0.183064	+	0.983101i	$0.558601\pi$
$662$	−2.90408	−0.112870
$663$	0	0
$664$	−10.4286	−0.404710
$665$	0	0
$666$	0	0
$667$	−26.1497	−1.01252
$668$	1.73019	0.0669429
$669$	0	0
$670$	0	0
$671$	36.7797	1.41986
$672$	0	0
$673$	−44.8001	−1.72692	−0.863459	−	0.504419i	$-0.831707\pi$
$673$	−44.8001	−1.72692	−0.863459	+	0.504419i	$0.831707\pi$
$674$	1.90562	0.0734019
$675$	0	0
$676$	−1.14244	−0.0439400
$677$	32.9197	1.26521	0.632604	−	0.774475i	$-0.281986\pi$
$677$	32.9197	1.26521	0.632604	+	0.774475i	$0.281986\pi$
$678$	0	0
$679$	0.268750	0.0103137
$680$	0	0
$681$	0	0
$682$	17.6345	0.675259
$683$	−20.3448	−0.778473	−0.389236	−	0.921138i	$-0.627261\pi$
$683$	−20.3448	−0.778473	−0.389236	+	0.921138i	$0.627261\pi$
$684$	0	0
$685$	0	0
$686$	23.8407	0.910242
$687$	0	0
$688$	31.6466	1.20651
$689$	−2.55065	−0.0971719
$690$	0	0
$691$	−46.1473	−1.75553	−0.877764	−	0.479094i	$-0.840965\pi$
$691$	−46.1473	−1.75553	−0.877764	+	0.479094i	$0.840965\pi$
$692$	−1.81269	−0.0689082
$693$	0	0
$694$	17.3623	0.659063
$695$	0	0
$696$	0	0
$697$	13.3612	0.506092
$698$	15.2567	0.577473
$699$	0	0
$700$	0	0
$701$	−13.1933	−0.498303	−0.249152	−	0.968464i	$-0.580152\pi$
$701$	−13.1933	−0.498303	−0.249152	+	0.968464i	$0.580152\pi$
$702$	0	0
$703$	0.137063	0.00516944
$704$	−36.9491	−1.39257
$705$	0	0
$706$	−1.46096	−0.0549840
$707$	−15.6165	−0.587321
$708$	0	0
$709$	41.1860	1.54677	0.773386	−	0.633935i	$-0.218562\pi$
$709$	41.1860	1.54677	0.773386	+	0.633935i	$0.218562\pi$
$710$	0	0
$711$	0	0
$712$	−25.8771	−0.969787
$713$	−6.42758	−0.240715
$714$	0	0
$715$	0	0
$716$	−0.610106	−0.0228007
$717$	0	0
$718$	−7.43104	−0.277324
$719$	35.6256	1.32861	0.664306	−	0.747461i	$-0.268727\pi$
$719$	35.6256	1.32861	0.664306	+	0.747461i	$0.268727\pi$
$720$	0	0
$721$	20.5657	0.765907
$722$	−1.44504	−0.0537789
$723$	0	0
$724$	−1.55543	−0.0578072
$725$	0	0
$726$	0	0
$727$	20.0116	0.742189	0.371095	−	0.928595i	$-0.378983\pi$
$727$	20.0116	0.742189	0.371095	+	0.928595i	$0.378983\pi$
$728$	0.742478	0.0275181
$729$	0	0
$730$	0	0
$731$	−8.63235	−0.319279
$732$	0	0
$733$	−18.9952	−0.701604	−0.350802	−	0.936450i	$-0.614091\pi$
$733$	−18.9952	−0.701604	−0.350802	+	0.936450i	$0.614091\pi$
$734$	42.6732	1.57510
$735$	0	0
$736$	−1.27306	−0.0469257
$737$	−38.8799	−1.43216
$738$	0	0
$739$	48.1704	1.77198	0.885989	−	0.463706i	$-0.153481\pi$
$739$	48.1704	1.77198	0.885989	+	0.463706i	$0.153481\pi$
$740$	0	0
$741$	0	0
$742$	25.2508	0.926987
$743$	13.2446	0.485897	0.242948	−	0.970039i	$-0.421885\pi$
$743$	13.2446	0.485897	0.242948	+	0.970039i	$0.421885\pi$
$744$	0	0
$745$	0	0
$746$	−12.7017	−0.465043
$747$	0	0
$748$	−0.486189	−0.0177768
$749$	−3.59956	−0.131525
$750$	0	0
$751$	12.0562	0.439937	0.219969	−	0.975507i	$-0.429404\pi$
$751$	12.0562	0.439937	0.219969	+	0.975507i	$0.429404\pi$
$752$	11.2218	0.409215
$753$	0	0
$754$	−2.92931	−0.106679
$755$	0	0
$756$	0	0
$757$	15.0054	0.545380	0.272690	−	0.962102i	$-0.412087\pi$
$757$	15.0054	0.545380	0.272690	+	0.962102i	$0.412087\pi$
$758$	−27.6741	−1.00517
$759$	0	0
$760$	0	0
$761$	−44.3967	−1.60938	−0.804690	−	0.593695i	$-0.797668\pi$
$761$	−44.3967	−1.60938	−0.804690	+	0.593695i	$0.797668\pi$
$762$	0	0
$763$	−3.39134	−0.122775
$764$	−0.610791	−0.0220976
$765$	0	0
$766$	−10.1138	−0.365425
$767$	−1.15346	−0.0416489
$768$	0	0
$769$	−39.7211	−1.43238	−0.716190	−	0.697906i	$-0.754115\pi$
$769$	−39.7211	−1.43238	−0.716190	+	0.697906i	$0.754115\pi$
$770$	0	0
$771$	0	0
$772$	−1.85431	−0.0667382
$773$	1.72779	0.0621444	0.0310722	−	0.999517i	$-0.490108\pi$
$773$	1.72779	0.0621444	0.0310722	+	0.999517i	$0.490108\pi$
$774$	0	0
$775$	0	0
$776$	0.547188	0.0196429
$777$	0	0
$778$	11.6843	0.418901
$779$	−11.7506	−0.421010
$780$	0	0
$781$	−42.8025	−1.53159
$782$	4.19806	0.150122
$783$	0	0
$784$	21.5047	0.768025
$785$	0	0
$786$	0	0
$787$	42.6329	1.51970	0.759850	−	0.650098i	$-0.225272\pi$
$787$	42.6329	1.51970	0.759850	+	0.650098i	$0.225272\pi$
$788$	−1.90084	−0.0677145
$789$	0	0
$790$	0	0
$791$	−11.5714	−0.411430
$792$	0	0
$793$	−1.50173	−0.0533280
$794$	25.5782	0.907735
$795$	0	0
$796$	1.93841	0.0687051
$797$	−38.5864	−1.36680	−0.683401	−	0.730044i	$-0.739500\pi$
$797$	−38.5864	−1.36680	−0.683401	+	0.730044i	$0.739500\pi$
$798$	0	0
$799$	−3.06100	−0.108290
$800$	0	0
$801$	0	0
$802$	−22.4586	−0.793040
$803$	58.0670	2.04914
$804$	0	0
$805$	0	0
$806$	−0.720023	−0.0253617
$807$	0	0
$808$	−31.7961	−1.11858
$809$	8.38298	0.294730	0.147365	−	0.989082i	$-0.452921\pi$
$809$	8.38298	0.294730	0.147365	+	0.989082i	$0.452921\pi$
$810$	0	0
$811$	−0.340765	−0.0119659	−0.00598295	−	0.999982i	$-0.501904\pi$
$811$	−0.340765	−0.0119659	−0.00598295	+	0.999982i	$0.501904\pi$
$812$	1.22414	0.0429590
$813$	0	0
$814$	−0.960771	−0.0336750
$815$	0	0
$816$	0	0
$817$	7.59179	0.265603
$818$	−19.0256	−0.665215
$819$	0	0
$820$	0	0
$821$	33.7506	1.17791	0.588953	−	0.808168i	$-0.299540\pi$
$821$	33.7506	1.17791	0.588953	+	0.808168i	$0.299540\pi$
$822$	0	0
$823$	37.2669	1.29904	0.649522	−	0.760343i	$-0.274969\pi$
$823$	37.2669	1.29904	0.649522	+	0.760343i	$0.274969\pi$
$824$	41.8728	1.45871
$825$	0	0
$826$	11.4190	0.397316
$827$	21.1691	0.736122	0.368061	−	0.929802i	$-0.380022\pi$
$827$	21.1691	0.736122	0.368061	+	0.929802i	$0.380022\pi$
$828$	0	0
$829$	−31.0374	−1.07797	−0.538987	−	0.842314i	$-0.681193\pi$
$829$	−31.0374	−1.07797	−0.538987	+	0.842314i	$0.681193\pi$
$830$	0	0
$831$	0	0
$832$	1.50864	0.0523028
$833$	−5.86592	−0.203242
$834$	0	0
$835$	0	0
$836$	0.427583	0.0147883
$837$	0	0
$838$	−36.3773	−1.25663
$839$	−33.2403	−1.14758	−0.573791	−	0.819002i	$-0.694528\pi$
$839$	−33.2403	−1.14758	−0.573791	+	0.819002i	$0.694528\pi$
$840$	0	0
$841$	75.7531	2.61218
$842$	−38.9323	−1.34170
$843$	0	0
$844$	−1.81786	−0.0625732
$845$	0	0
$846$	0	0
$847$	17.0030	0.584229
$848$	53.6822	1.84346
$849$	0	0
$850$	0	0
$851$	0.350191	0.0120044
$852$	0	0
$853$	−24.3086	−0.832310	−0.416155	−	0.909294i	$-0.636623\pi$
$853$	−24.3086	−0.832310	−0.416155	+	0.909294i	$0.636623\pi$
$854$	14.8668	0.508731
$855$	0	0
$856$	−7.32889	−0.250496
$857$	57.3889	1.96037	0.980185	−	0.198087i	$-0.0634727\pi$
$857$	57.3889	1.96037	0.980185	+	0.198087i	$0.0634727\pi$
$858$	0	0
$859$	−13.8135	−0.471312	−0.235656	−	0.971837i	$-0.575724\pi$
$859$	−13.8135	−0.471312	−0.235656	+	0.971837i	$0.575724\pi$
$860$	0	0
$861$	0	0
$862$	27.3817	0.932623
$863$	9.01938	0.307023	0.153512	−	0.988147i	$-0.450942\pi$
$863$	9.01938	0.307023	0.153512	+	0.988147i	$0.450942\pi$
$864$	0	0
$865$	0	0
$866$	5.33081	0.181148
$867$	0	0
$868$	0.300894	0.0102130
$869$	−52.1027	−1.76746
$870$	0	0
$871$	1.58748	0.0537898
$872$	−6.90494	−0.233831
$873$	0	0
$874$	−3.69202	−0.124884
$875$	0	0
$876$	0	0
$877$	−2.37675	−0.0802570	−0.0401285	−	0.999195i	$-0.512777\pi$
$877$	−2.37675	−0.0802570	−0.0401285	+	0.999195i	$0.512777\pi$
$878$	37.1269	1.25297
$879$	0	0
$880$	0	0
$881$	7.99330	0.269301	0.134650	−	0.990893i	$-0.457009\pi$
$881$	7.99330	0.269301	0.134650	+	0.990893i	$0.457009\pi$
$882$	0	0
$883$	−1.76377	−0.0593557	−0.0296779	−	0.999560i	$-0.509448\pi$
$883$	−1.76377	−0.0593557	−0.0296779	+	0.999560i	$0.509448\pi$
$884$	0.0198513	0.000667672 0
$885$	0	0
$886$	−39.5439	−1.32850
$887$	−45.9197	−1.54183	−0.770917	−	0.636936i	$-0.780202\pi$
$887$	−45.9197	−1.54183	−0.770917	+	0.636936i	$0.780202\pi$
$888$	0	0
$889$	27.6926	0.928780
$890$	0	0
$891$	0	0
$892$	0.703155	0.0235434
$893$	2.69202	0.0900851
$894$	0	0
$895$	0	0
$896$	−16.2874	−0.544125
$897$	0	0
$898$	10.9215	0.364457
$899$	25.7482	0.858752
$900$	0	0
$901$	−14.6431	−0.487833
$902$	82.3682	2.74256
$903$	0	0
$904$	−23.5599	−0.783589
$905$	0	0
$906$	0	0
$907$	55.0549	1.82807	0.914034	−	0.405638i	$-0.132951\pi$
$907$	55.0549	1.82807	0.914034	+	0.405638i	$0.132951\pi$
$908$	−1.74478	−0.0579024
$909$	0	0
$910$	0	0
$911$	−51.8998	−1.71952	−0.859758	−	0.510702i	$-0.829386\pi$
$911$	−51.8998	−1.71952	−0.859758	+	0.510702i	$0.829386\pi$
$912$	0	0
$913$	18.3110	0.606004
$914$	11.3526	0.375510
$915$	0	0
$916$	−0.355568	−0.0117483
$917$	17.9288	0.592062
$918$	0	0
$919$	11.4614	0.378078	0.189039	−	0.981970i	$-0.439463\pi$
$919$	11.4614	0.378078	0.189039	+	0.981970i	$0.439463\pi$
$920$	0	0
$921$	0	0
$922$	5.60541	0.184604
$923$	1.74764	0.0575244
$924$	0	0
$925$	0	0
$926$	18.9235	0.621864
$927$	0	0
$928$	5.09975	0.167408
$929$	−36.5295	−1.19849	−0.599246	−	0.800565i	$-0.704533\pi$
$929$	−36.5295	−1.19849	−0.599246	+	0.800565i	$0.704533\pi$
$930$	0	0
$931$	5.15883	0.169074
$932$	−2.36372	−0.0774261
$933$	0	0
$934$	9.30644	0.304516
$935$	0	0
$936$	0	0
$937$	−48.7845	−1.59372	−0.796860	−	0.604164i	$-0.793507\pi$
$937$	−48.7845	−1.59372	−0.796860	+	0.604164i	$0.793507\pi$
$938$	−15.7157	−0.513136
$939$	0	0
$940$	0	0
$941$	18.1817	0.592705	0.296353	−	0.955079i	$-0.404230\pi$
$941$	18.1817	0.592705	0.296353	+	0.955079i	$0.404230\pi$
$942$	0	0
$943$	−30.0224	−0.977663
$944$	24.2763	0.790125
$945$	0	0
$946$	−53.2161	−1.73021
$947$	7.55150	0.245391	0.122695	−	0.992444i	$-0.460846\pi$
$947$	7.55150	0.245391	0.122695	+	0.992444i	$0.460846\pi$
$948$	0	0
$949$	−2.37090	−0.0769626
$950$	0	0
$951$	0	0
$952$	4.26252	0.138149
$953$	−13.8592	−0.448944	−0.224472	−	0.974481i	$-0.572066\pi$
$953$	−13.8592	−0.448944	−0.224472	+	0.974481i	$0.572066\pi$
$954$	0	0
$955$	0	0
$956$	−0.296371	−0.00958532
$957$	0	0
$958$	8.22367	0.265695
$959$	9.30904	0.300605
$960$	0	0
$961$	−24.6711	−0.795842
$962$	0.0392287	0.00126478
$963$	0	0
$964$	2.44611	0.0787838
$965$	0	0
$966$	0	0
$967$	4.89977	0.157566	0.0787830	−	0.996892i	$-0.474897\pi$
$967$	4.89977	0.157566	0.0787830	+	0.996892i	$0.474897\pi$
$968$	34.6189	1.11269
$969$	0	0
$970$	0	0
$971$	−14.5133	−0.465755	−0.232878	−	0.972506i	$-0.574814\pi$
$971$	−14.5133	−0.465755	−0.232878	+	0.972506i	$0.574814\pi$
$972$	0	0
$973$	5.81594	0.186450
$974$	18.7323	0.600222
$975$	0	0
$976$	31.6062	1.01169
$977$	19.6644	0.629120	0.314560	−	0.949238i	$-0.398143\pi$
$977$	19.6644	0.629120	0.314560	+	0.949238i	$0.398143\pi$
$978$	0	0
$979$	45.4359	1.45214
$980$	0	0
$981$	0	0
$982$	−28.3293	−0.904023
$983$	15.4397	0.492449	0.246224	−	0.969213i	$-0.420810\pi$
$983$	15.4397	0.492449	0.246224	+	0.969213i	$0.420810\pi$
$984$	0	0
$985$	0	0
$986$	−16.8170	−0.535562
$987$	0	0
$988$	−0.0174584	−0.000555426 0
$989$	19.3967	0.616780
$990$	0	0
$991$	11.9377	0.379213	0.189606	−	0.981860i	$-0.439279\pi$
$991$	11.9377	0.379213	0.189606	+	0.981860i	$0.439279\pi$
$992$	1.25352	0.0397991
$993$	0	0
$994$	−17.3013	−0.548763
$995$	0	0
$996$	0	0
$997$	17.9390	0.568134	0.284067	−	0.958804i	$-0.408316\pi$
$997$	17.9390	0.568134	0.284067	+	0.958804i	$0.408316\pi$
$998$	−7.94523	−0.251502
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.z.1.2		3
3.2	odd	2		475.2.a.h.1.2	yes	3
5.4	even	2		4275.2.a.bn.1.2		3
12.11	even	2		7600.2.a.bn.1.1		3
15.2	even	4		475.2.b.c.324.5		6
15.8	even	4		475.2.b.c.324.2		6
15.14	odd	2		475.2.a.d.1.2	✓	3
57.56	even	2		9025.2.a.w.1.2		3
60.59	even	2		7600.2.a.bw.1.3		3
285.284	even	2		9025.2.a.be.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.d.1.2	✓	3	15.14	odd	2
475.2.a.h.1.2	yes	3	3.2	odd	2
475.2.b.c.324.2		6	15.8	even	4
475.2.b.c.324.5		6	15.2	even	4
4275.2.a.z.1.2		3	1.1	even	1	trivial
4275.2.a.bn.1.2		3	5.4	even	2
7600.2.a.bn.1.1		3	12.11	even	2
7600.2.a.bw.1.3		3	60.59	even	2
9025.2.a.w.1.2		3	57.56	even	2
9025.2.a.be.1.2		3	285.284	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q-1.44504 q^{2} +0.0881460 q^{4} +1.35690 q^{7} +2.76271 q^{8} -4.85086 q^{11} +0.198062 q^{13} -1.96077 q^{14} -4.16852 q^{16} +1.13706 q^{17} -1.00000 q^{19} +7.00969 q^{22} -2.55496 q^{23} -0.286208 q^{26} +0.119605 q^{28} +10.2349 q^{29} +2.51573 q^{31} +0.498271 q^{32} -1.64310 q^{34} -0.137063 q^{37} +1.44504 q^{38} +11.7506 q^{41} -7.59179 q^{43} -0.427583 q^{44} +3.69202 q^{46} -2.69202 q^{47} -5.15883 q^{49} +0.0174584 q^{52} -12.8780 q^{53} +3.74871 q^{56} -14.7899 q^{58} -5.82371 q^{59} -7.58211 q^{61} -3.63533 q^{62} +7.61702 q^{64} +8.01507 q^{67} +0.100228 q^{68} +8.82371 q^{71} -11.9705 q^{73} +0.198062 q^{74} -0.0881460 q^{76} -6.58211 q^{77} +10.7409 q^{79} -16.9801 q^{82} -3.77479 q^{83} +10.9705 q^{86} -13.4015 q^{88} -9.36658 q^{89} +0.268750 q^{91} -0.225209 q^{92} +3.89008 q^{94} +0.198062 q^{97} +7.45473 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{2} + 4 q^{4} - 9 q^{8} - q^{11} + 5 q^{13} + 7 q^{14} + 18 q^{16} - 2 q^{17} - 3 q^{19} - q^{22} - 8 q^{23} - 9 q^{26} - 21 q^{28} + 7 q^{29} - 5 q^{31} - 27 q^{32} - 9 q^{34} + 5 q^{37} + 4 q^{38}+ \cdots + 5 q^{97}+O(q^{100}) \) 3 * q - 4 * q^2 + 4 * q^4 - 9 * q^8 - q^11 + 5 * q^13 + 7 * q^14 + 18 * q^16 - 2 * q^17 - 3 * q^19 - q^22 - 8 * q^23 - 9 * q^26 - 21 * q^28 + 7 * q^29 - 5 * q^31 - 27 * q^32 - 9 * q^34 + 5 * q^37 + 4 * q^38 - q^41 + 5 * q^43 + 15 * q^44 + 6 * q^46 - 3 * q^47 - 7 * q^49 + 16 * q^52 - 19 * q^53 + 35 * q^56 - 21 * q^58 - 10 * q^59 - 17 * q^61 + 23 * q^62 + 49 * q^64 - q^67 + 23 * q^68 + 19 * q^71 - q^73 + 5 * q^74 - 4 * q^76 - 14 * q^77 + 18 * q^79 + 6 * q^82 - 13 * q^83 - 2 * q^86 - 46 * q^88 - 2 * q^89 - 7 * q^91 + q^92 + 11 * q^94 + 5 * q^97

Introduction

L-functions

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