LMFDB - Embedded newform 4275.2.a.bv.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$34.1360468641$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 8x^{5} + 26x^{4} + 11x^{3} - 51x^{2} + 12x + 4 $ x^7 - 3x^6 - 8x^5 + 26x^4 + 11x^3 - 51x^2 + 12x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.39336$ 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)

$f(q)$	$=$	$q+2.39336 q^{2} +3.72816 q^{4} +4.15221 q^{7} +4.13612 q^{8} +5.71602 q^{11} +3.79886 q^{13} +9.93772 q^{14} +2.44288 q^{16} +2.66579 q^{17} +1.00000 q^{19} +13.6805 q^{22} -8.13886 q^{23} +9.09204 q^{26} +15.4801 q^{28} -4.73061 q^{29} -2.31244 q^{31} -2.42554 q^{32} +6.38019 q^{34} -4.68551 q^{37} +2.39336 q^{38} -10.1869 q^{41} -1.76160 q^{43} +21.3103 q^{44} -19.4792 q^{46} -7.14809 q^{47} +10.2408 q^{49} +14.1628 q^{52} +3.04396 q^{53} +17.1740 q^{56} -11.3221 q^{58} +0.582229 q^{59} -9.54956 q^{61} -5.53450 q^{62} -10.6910 q^{64} -1.20711 q^{67} +9.93850 q^{68} -1.20711 q^{71} +11.5362 q^{73} -11.2141 q^{74} +3.72816 q^{76} +23.7341 q^{77} +5.06076 q^{79} -24.3810 q^{82} -1.83685 q^{83} -4.21613 q^{86} +23.6421 q^{88} -3.36716 q^{89} +15.7737 q^{91} -30.3430 q^{92} -17.1079 q^{94} +0.313344 q^{97} +24.5100 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{2} + 11 q^{4} + 8 q^{7} - 9 q^{8} + 4 q^{11} + 8 q^{13} + 4 q^{14} + 19 q^{16} - 4 q^{17} + 7 q^{19} + 12 q^{22} - 10 q^{23} + 20 q^{26} + 14 q^{28} + 6 q^{29} + 4 q^{31} - 31 q^{32} + 2 q^{34}+ \cdots - 5 q^{98}+O(q^{100}) $ 7 * q - 3 * q^2 + 11 * q^4 + 8 * q^7 - 9 * q^8 + 4 * q^11 + 8 * q^13 + 4 * q^14 + 19 * q^16 - 4 * q^17 + 7 * q^19 + 12 * q^22 - 10 * q^23 + 20 * q^26 + 14 * q^28 + 6 * q^29 + 4 * q^31 - 31 * q^32 + 2 * q^34 + 14 * q^37 - 3 * q^38 - 2 * q^41 - 2 * q^43 + 32 * q^44 - 4 * q^46 - 30 * q^47 + 17 * q^49 + 18 * q^52 + 22 * q^56 - 40 * q^58 + 18 * q^59 + 12 * q^61 - 18 * q^62 + 11 * q^64 + 18 * q^67 + 12 * q^68 + 18 * q^71 + 10 * q^73 - 6 * q^74 + 11 * q^76 + 18 * q^77 - 4 * q^79 - 16 * q^82 - 18 * q^83 + 46 * q^86 - 18 * q^88 + 8 * q^89 + 12 * q^91 - 34 * q^92 - 20 * q^94 + 20 * q^97 - 5 * q^98

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 $

$2$	2.39336	1.69236	0.846180	−	0.532897i	$-0.178897\pi$
$2$	2.39336	1.69236	0.846180	+	0.532897i	$0.178897\pi$
$3$	0	0
$3$	0	0
$4$	3.72816	1.86408
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.15221	1.56939	0.784694	−	0.619884i	$-0.212820\pi$
$7$	4.15221	1.56939	0.784694	+	0.619884i	$0.212820\pi$
$8$	4.13612	1.46234
$9$	0	0
$10$	0	0
$11$	5.71602	1.72344	0.861722	−	0.507380i	$-0.169386\pi$
$11$	5.71602	1.72344	0.861722	+	0.507380i	$0.169386\pi$
$12$	0	0
$13$	3.79886	1.05361	0.526807	−	0.849985i	$-0.323389\pi$
$13$	3.79886	1.05361	0.526807	+	0.849985i	$0.323389\pi$
$14$	9.93772	2.65597
$15$	0	0
$16$	2.44288	0.610721
$17$	2.66579	0.646549	0.323274	−	0.946305i	$-0.395216\pi$
$17$	2.66579	0.646549	0.323274	+	0.946305i	$0.395216\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	13.6805	2.91669
$23$	−8.13886	−1.69707	−0.848535	−	0.529139i	$-0.822515\pi$
$23$	−8.13886	−1.69707	−0.848535	+	0.529139i	$0.822515\pi$
$24$	0	0
$25$	0	0
$26$	9.09204	1.78310
$27$	0	0
$28$	15.4801	2.92547
$29$	−4.73061	−0.878453	−0.439226	−	0.898376i	$-0.644747\pi$
$29$	−4.73061	−0.878453	−0.439226	+	0.898376i	$0.644747\pi$
$30$	0	0
$31$	−2.31244	−0.415327	−0.207663	−	0.978200i	$-0.566586\pi$
$31$	−2.31244	−0.415327	−0.207663	+	0.978200i	$0.566586\pi$
$32$	−2.42554	−0.428779
$33$	0	0
$34$	6.38019	1.09419
$35$	0	0
$36$	0	0
$37$	−4.68551	−0.770293	−0.385146	−	0.922855i	$-0.625849\pi$
$37$	−4.68551	−0.770293	−0.385146	+	0.922855i	$0.625849\pi$
$38$	2.39336	0.388254
$39$	0	0
$40$	0	0
$41$	−10.1869	−1.59093	−0.795467	−	0.605997i	$-0.792774\pi$
$41$	−10.1869	−1.59093	−0.795467	+	0.605997i	$0.792774\pi$
$42$	0	0
$43$	−1.76160	−0.268641	−0.134320	−	0.990938i	$-0.542885\pi$
$43$	−1.76160	−0.268641	−0.134320	+	0.990938i	$0.542885\pi$
$44$	21.3103	3.21264
$45$	0	0
$46$	−19.4792	−2.87205
$47$	−7.14809	−1.04266	−0.521328	−	0.853356i	$-0.674563\pi$
$47$	−7.14809	−1.04266	−0.521328	+	0.853356i	$0.674563\pi$
$48$	0	0
$49$	10.2408	1.46298
$50$	0	0
$51$	0	0
$52$	14.1628	1.96402
$53$	3.04396	0.418120	0.209060	−	0.977903i	$-0.432960\pi$
$53$	3.04396	0.418120	0.209060	+	0.977903i	$0.432960\pi$
$54$	0	0
$55$	0	0
$56$	17.1740	2.29498
$57$	0	0
$58$	−11.3221	−1.48666
$59$	0.582229	0.0757997	0.0378998	−	0.999282i	$-0.487933\pi$
$59$	0.582229	0.0757997	0.0378998	+	0.999282i	$0.487933\pi$
$60$	0	0
$61$	−9.54956	−1.22270	−0.611348	−	0.791362i	$-0.709372\pi$
$61$	−9.54956	−1.22270	−0.611348	+	0.791362i	$0.709372\pi$
$62$	−5.53450	−0.702882
$63$	0	0
$64$	−10.6910	−1.33637
$65$	0	0
$66$	0	0
$67$	−1.20711	−0.147472	−0.0737360	−	0.997278i	$-0.523492\pi$
$67$	−1.20711	−0.147472	−0.0737360	+	0.997278i	$0.523492\pi$
$68$	9.93850	1.20522
$69$	0	0
$70$	0	0
$71$	−1.20711	−0.143258	−0.0716289	−	0.997431i	$-0.522820\pi$
$71$	−1.20711	−0.143258	−0.0716289	+	0.997431i	$0.522820\pi$
$72$	0	0
$73$	11.5362	1.35021	0.675103	−	0.737723i	$-0.264099\pi$
$73$	11.5362	1.35021	0.675103	+	0.737723i	$0.264099\pi$
$74$	−11.2141	−1.30361
$75$	0	0
$76$	3.72816	0.427650
$77$	23.7341	2.70475
$78$	0	0
$79$	5.06076	0.569380	0.284690	−	0.958620i	$-0.408109\pi$
$79$	5.06076	0.569380	0.284690	+	0.958620i	$0.408109\pi$
$80$	0	0
$81$	0	0
$82$	−24.3810	−2.69243
$83$	−1.83685	−0.201620	−0.100810	−	0.994906i	$-0.532143\pi$
$83$	−1.83685	−0.201620	−0.100810	+	0.994906i	$0.532143\pi$
$84$	0	0
$85$	0	0
$86$	−4.21613	−0.454637
$87$	0	0
$88$	23.6421	2.52026
$89$	−3.36716	−0.356918	−0.178459	−	0.983947i	$-0.557111\pi$
$89$	−3.36716	−0.356918	−0.178459	+	0.983947i	$0.557111\pi$
$90$	0	0
$91$	15.7737	1.65353
$92$	−30.3430	−3.16348
$93$	0	0
$94$	−17.1079	−1.76455
$95$	0	0
$96$	0	0
$97$	0.313344	0.0318153	0.0159076	−	0.999873i	$-0.494936\pi$
$97$	0.313344	0.0318153	0.0159076	+	0.999873i	$0.494936\pi$
$98$	24.5100	2.47588
$99$	0	0
$100$	0	0
$101$	14.6926	1.46197	0.730986	−	0.682393i	$-0.239061\pi$
$101$	14.6926	1.46197	0.730986	+	0.682393i	$0.239061\pi$
$102$	0	0
$103$	0.722740	0.0712137	0.0356069	−	0.999366i	$-0.488664\pi$
$103$	0.722740	0.0712137	0.0356069	+	0.999366i	$0.488664\pi$
$104$	15.7125	1.54074
$105$	0	0
$106$	7.28528	0.707609
$107$	4.30442	0.416124	0.208062	−	0.978116i	$-0.433284\pi$
$107$	4.30442	0.416124	0.208062	+	0.978116i	$0.433284\pi$
$108$	0	0
$109$	−3.45633	−0.331056	−0.165528	−	0.986205i	$-0.552933\pi$
$109$	−3.45633	−0.331056	−0.165528	+	0.986205i	$0.552933\pi$
$110$	0	0
$111$	0	0
$112$	10.1434	0.958457
$113$	−5.05220	−0.475271	−0.237636	−	0.971354i	$-0.576372\pi$
$113$	−5.05220	−0.475271	−0.237636	+	0.971354i	$0.576372\pi$
$114$	0	0
$115$	0	0
$116$	−17.6365	−1.63751
$117$	0	0
$118$	1.39348	0.128280
$119$	11.0689	1.01469
$120$	0	0
$121$	21.6729	1.97026
$122$	−22.8555	−2.06924
$123$	0	0
$124$	−8.62116	−0.774203
$125$	0	0
$126$	0	0
$127$	4.37397	0.388127	0.194064	−	0.980989i	$-0.437833\pi$
$127$	4.37397	0.388127	0.194064	+	0.980989i	$0.437833\pi$
$128$	−20.7362	−1.83284
$129$	0	0
$130$	0	0
$131$	2.01872	0.176376	0.0881880	−	0.996104i	$-0.471892\pi$
$131$	2.01872	0.176376	0.0881880	+	0.996104i	$0.471892\pi$
$132$	0	0
$133$	4.15221	0.360042
$134$	−2.88905	−0.249576
$135$	0	0
$136$	11.0260	0.945473
$137$	−2.07984	−0.177693	−0.0888463	−	0.996045i	$-0.528318\pi$
$137$	−2.07984	−0.177693	−0.0888463	+	0.996045i	$0.528318\pi$
$138$	0	0
$139$	−4.43821	−0.376444	−0.188222	−	0.982127i	$-0.560272\pi$
$139$	−4.43821	−0.376444	−0.188222	+	0.982127i	$0.560272\pi$
$140$	0	0
$141$	0	0
$142$	−2.88905	−0.242444
$143$	21.7144	1.81585
$144$	0	0
$145$	0	0
$146$	27.6102	2.28504
$147$	0	0
$148$	−17.4683	−1.43589
$149$	7.62685	0.624816	0.312408	−	0.949948i	$-0.398864\pi$
$149$	7.62685	0.624816	0.312408	+	0.949948i	$0.398864\pi$
$150$	0	0
$151$	19.0193	1.54777	0.773886	−	0.633325i	$-0.218310\pi$
$151$	19.0193	1.54777	0.773886	+	0.633325i	$0.218310\pi$
$152$	4.13612	0.335483
$153$	0	0
$154$	56.8042	4.57742
$155$	0	0
$156$	0	0
$157$	−0.368128	−0.0293798	−0.0146899	−	0.999892i	$-0.504676\pi$
$157$	−0.368128	−0.0293798	−0.0146899	+	0.999892i	$0.504676\pi$
$158$	12.1122	0.963595
$159$	0	0
$160$	0	0
$161$	−33.7943	−2.66336
$162$	0	0
$163$	−3.28438	−0.257252	−0.128626	−	0.991693i	$-0.541057\pi$
$163$	−3.28438	−0.257252	−0.128626	+	0.991693i	$0.541057\pi$
$164$	−37.9786	−2.96563
$165$	0	0
$166$	−4.39623	−0.341214
$167$	−14.1652	−1.09614	−0.548069	−	0.836433i	$-0.684637\pi$
$167$	−14.1652	−1.09614	−0.548069	+	0.836433i	$0.684637\pi$
$168$	0	0
$169$	1.43135	0.110104
$170$	0	0
$171$	0	0
$172$	−6.56752	−0.500769
$173$	−25.0399	−1.90375	−0.951873	−	0.306494i	$-0.900844\pi$
$173$	−25.0399	−1.90375	−0.951873	+	0.306494i	$0.900844\pi$
$174$	0	0
$175$	0	0
$176$	13.9636	1.05254
$177$	0	0
$178$	−8.05882	−0.604034
$179$	−25.8103	−1.92915	−0.964576	−	0.263803i	$-0.915023\pi$
$179$	−25.8103	−1.92915	−0.964576	+	0.263803i	$0.915023\pi$
$180$	0	0
$181$	−4.91266	−0.365155	−0.182578	−	0.983191i	$-0.558444\pi$
$181$	−4.91266	−0.365155	−0.182578	+	0.983191i	$0.558444\pi$
$182$	37.7520	2.79837
$183$	0	0
$184$	−33.6633	−2.48169
$185$	0	0
$186$	0	0
$187$	15.2377	1.11429
$188$	−26.6492	−1.94360
$189$	0	0
$190$	0	0
$191$	13.1806	0.953714	0.476857	−	0.878981i	$-0.341776\pi$
$191$	13.1806	0.953714	0.476857	+	0.878981i	$0.341776\pi$
$192$	0	0
$193$	17.4683	1.25740	0.628700	−	0.777648i	$-0.283588\pi$
$193$	17.4683	1.25740	0.628700	+	0.777648i	$0.283588\pi$
$194$	0.749945	0.0538429
$195$	0	0
$196$	38.1795	2.72711
$197$	0.743906	0.0530011	0.0265006	−	0.999649i	$-0.491564\pi$
$197$	0.743906	0.0530011	0.0265006	+	0.999649i	$0.491564\pi$
$198$	0	0
$199$	3.78079	0.268013	0.134007	−	0.990980i	$-0.457216\pi$
$199$	3.78079	0.268013	0.134007	+	0.990980i	$0.457216\pi$
$200$	0	0
$201$	0	0
$202$	35.1647	2.47418
$203$	−19.6425	−1.37863
$204$	0	0
$205$	0	0
$206$	1.72978	0.120519
$207$	0	0
$208$	9.28017	0.643464
$209$	5.71602	0.395385
$210$	0	0
$211$	13.1712	0.906740	0.453370	−	0.891322i	$-0.350222\pi$
$211$	13.1712	0.906740	0.453370	+	0.891322i	$0.350222\pi$
$212$	11.3484	0.779410
$213$	0	0
$214$	10.3020	0.704231
$215$	0	0
$216$	0	0
$217$	−9.60174	−0.651809
$218$	−8.27223	−0.560266
$219$	0	0
$220$	0	0
$221$	10.1270	0.681213
$222$	0	0
$223$	6.87668	0.460497	0.230248	−	0.973132i	$-0.426046\pi$
$223$	6.87668	0.460497	0.230248	+	0.973132i	$0.426046\pi$
$224$	−10.0714	−0.672921
$225$	0	0
$226$	−12.0917	−0.804330
$227$	−5.43069	−0.360448	−0.180224	−	0.983626i	$-0.557682\pi$
$227$	−5.43069	−0.360448	−0.180224	+	0.983626i	$0.557682\pi$
$228$	0	0
$229$	21.7789	1.43919	0.719594	−	0.694395i	$-0.244328\pi$
$229$	21.7789	1.43919	0.719594	+	0.694395i	$0.244328\pi$
$230$	0	0
$231$	0	0
$232$	−19.5664	−1.28460
$233$	−2.11751	−0.138723	−0.0693615	−	0.997592i	$-0.522096\pi$
$233$	−2.11751	−0.138723	−0.0693615	+	0.997592i	$0.522096\pi$
$234$	0	0
$235$	0	0
$236$	2.17064	0.141297
$237$	0	0
$238$	26.4919	1.71721
$239$	8.63535	0.558575	0.279287	−	0.960208i	$-0.409902\pi$
$239$	8.63535	0.558575	0.279287	+	0.960208i	$0.409902\pi$
$240$	0	0
$241$	−20.9476	−1.34935	−0.674676	−	0.738114i	$-0.735717\pi$
$241$	−20.9476	−1.34935	−0.674676	+	0.738114i	$0.735717\pi$
$242$	51.8710	3.33439
$243$	0	0
$244$	−35.6023	−2.27921
$245$	0	0
$246$	0	0
$247$	3.79886	0.241716
$248$	−9.56453	−0.607348
$249$	0	0
$250$	0	0
$251$	1.30618	0.0824453	0.0412227	−	0.999150i	$-0.486875\pi$
$251$	1.30618	0.0824453	0.0412227	+	0.999150i	$0.486875\pi$
$252$	0	0
$253$	−46.5219	−2.92481
$254$	10.4685	0.656851
$255$	0	0
$256$	−28.2473	−1.76545
$257$	13.8228	0.862241	0.431121	−	0.902294i	$-0.358118\pi$
$257$	13.8228	0.862241	0.431121	+	0.902294i	$0.358118\pi$
$258$	0	0
$259$	−19.4552	−1.20889
$260$	0	0
$261$	0	0
$262$	4.83151	0.298492
$263$	31.1035	1.91793	0.958963	−	0.283530i	$-0.0915056\pi$
$263$	31.1035	1.91793	0.958963	+	0.283530i	$0.0915056\pi$
$264$	0	0
$265$	0	0
$266$	9.93772	0.609321
$267$	0	0
$268$	−4.50031	−0.274900
$269$	1.03331	0.0630021	0.0315010	−	0.999504i	$-0.489971\pi$
$269$	1.03331	0.0630021	0.0315010	+	0.999504i	$0.489971\pi$
$270$	0	0
$271$	−15.6939	−0.953336	−0.476668	−	0.879084i	$-0.658156\pi$
$271$	−15.6939	−0.953336	−0.476668	+	0.879084i	$0.658156\pi$
$272$	6.51221	0.394861
$273$	0	0
$274$	−4.97780	−0.300720
$275$	0	0
$276$	0	0
$277$	−25.4343	−1.52820	−0.764099	−	0.645099i	$-0.776816\pi$
$277$	−25.4343	−1.52820	−0.764099	+	0.645099i	$0.776816\pi$
$278$	−10.6222	−0.637079
$279$	0	0
$280$	0	0
$281$	−8.36249	−0.498864	−0.249432	−	0.968392i	$-0.580244\pi$
$281$	−8.36249	−0.498864	−0.249432	+	0.968392i	$0.580244\pi$
$282$	0	0
$283$	−4.50627	−0.267870	−0.133935	−	0.990990i	$-0.542761\pi$
$283$	−4.50627	−0.267870	−0.133935	+	0.990990i	$0.542761\pi$
$284$	−4.50031	−0.267044
$285$	0	0
$286$	51.9703	3.07307
$287$	−42.2983	−2.49679
$288$	0	0
$289$	−9.89357	−0.581975
$290$	0	0
$291$	0	0
$292$	43.0087	2.51690
$293$	17.9925	1.05113	0.525566	−	0.850753i	$-0.323854\pi$
$293$	17.9925	1.05113	0.525566	+	0.850753i	$0.323854\pi$
$294$	0	0
$295$	0	0
$296$	−19.3798	−1.12643
$297$	0	0
$298$	18.2538	1.05741
$299$	−30.9184	−1.78806
$300$	0	0
$301$	−7.31452	−0.421602
$302$	45.5201	2.61939
$303$	0	0
$304$	2.44288	0.140109
$305$	0	0
$306$	0	0
$307$	−17.8530	−1.01892	−0.509461	−	0.860494i	$-0.670155\pi$
$307$	−17.8530	−1.01892	−0.509461	+	0.860494i	$0.670155\pi$
$308$	88.4847	5.04188
$309$	0	0
$310$	0	0
$311$	23.8024	1.34971	0.674856	−	0.737950i	$-0.264206\pi$
$311$	23.8024	1.34971	0.674856	+	0.737950i	$0.264206\pi$
$312$	0	0
$313$	7.21536	0.407836	0.203918	−	0.978988i	$-0.434632\pi$
$313$	7.21536	0.407836	0.203918	+	0.978988i	$0.434632\pi$
$314$	−0.881063	−0.0497213
$315$	0	0
$316$	18.8673	1.06137
$317$	−28.3490	−1.59224	−0.796118	−	0.605142i	$-0.793116\pi$
$317$	−28.3490	−1.59224	−0.796118	+	0.605142i	$0.793116\pi$
$318$	0	0
$319$	−27.0403	−1.51397
$320$	0	0
$321$	0	0
$322$	−80.8818	−4.50737
$323$	2.66579	0.148328
$324$	0	0
$325$	0	0
$326$	−7.86070	−0.435364
$327$	0	0
$328$	−42.1344	−2.32648
$329$	−29.6804	−1.63633
$330$	0	0
$331$	19.3169	1.06175	0.530875	−	0.847450i	$-0.321863\pi$
$331$	19.3169	1.06175	0.530875	+	0.847450i	$0.321863\pi$
$332$	−6.84807	−0.375837
$333$	0	0
$334$	−33.9025	−1.85506
$335$	0	0
$336$	0	0
$337$	−19.4944	−1.06193	−0.530963	−	0.847395i	$-0.678170\pi$
$337$	−19.4944	−1.06193	−0.530963	+	0.847395i	$0.678170\pi$
$338$	3.42574	0.186336
$339$	0	0
$340$	0	0
$341$	−13.2180	−0.715793
$342$	0	0
$343$	13.4567	0.726591
$344$	−7.28617	−0.392844
$345$	0	0
$346$	−59.9294	−3.22182
$347$	16.1592	0.867471	0.433736	−	0.901040i	$-0.357195\pi$
$347$	16.1592	0.867471	0.433736	+	0.901040i	$0.357195\pi$
$348$	0	0
$349$	−6.37305	−0.341141	−0.170571	−	0.985345i	$-0.554561\pi$
$349$	−6.37305	−0.341141	−0.170571	+	0.985345i	$0.554561\pi$
$350$	0	0
$351$	0	0
$352$	−13.8644	−0.738977
$353$	−31.7273	−1.68867	−0.844337	−	0.535813i	$-0.820005\pi$
$353$	−31.7273	−1.68867	−0.844337	+	0.535813i	$0.820005\pi$
$354$	0	0
$355$	0	0
$356$	−12.5533	−0.665325
$357$	0	0
$358$	−61.7733	−3.26482
$359$	14.7365	0.777761	0.388881	−	0.921288i	$-0.372862\pi$
$359$	14.7365	0.777761	0.388881	+	0.921288i	$0.372862\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−11.7578	−0.617974
$363$	0	0
$364$	58.8068	3.08232
$365$	0	0
$366$	0	0
$367$	−9.75466	−0.509189	−0.254595	−	0.967048i	$-0.581942\pi$
$367$	−9.75466	−0.509189	−0.254595	+	0.967048i	$0.581942\pi$
$368$	−19.8823	−1.03644
$369$	0	0
$370$	0	0
$371$	12.6392	0.656192
$372$	0	0
$373$	2.62601	0.135969	0.0679847	−	0.997686i	$-0.478343\pi$
$373$	2.62601	0.135969	0.0679847	+	0.997686i	$0.478343\pi$
$374$	36.4693	1.88578
$375$	0	0
$376$	−29.5653	−1.52472
$377$	−17.9709	−0.925551
$378$	0	0
$379$	−0.835168	−0.0428997	−0.0214498	−	0.999770i	$-0.506828\pi$
$379$	−0.835168	−0.0428997	−0.0214498	+	0.999770i	$0.506828\pi$
$380$	0	0
$381$	0	0
$382$	31.5459	1.61403
$383$	9.14340	0.467206	0.233603	−	0.972332i	$-0.424948\pi$
$383$	9.14340	0.467206	0.233603	+	0.972332i	$0.424948\pi$
$384$	0	0
$385$	0	0
$386$	41.8080	2.12797
$387$	0	0
$388$	1.16820	0.0593063
$389$	15.8729	0.804790	0.402395	−	0.915466i	$-0.368178\pi$
$389$	15.8729	0.804790	0.402395	+	0.915466i	$0.368178\pi$
$390$	0	0
$391$	−21.6965	−1.09724
$392$	42.3573	2.13937
$393$	0	0
$394$	1.78043	0.0896970
$395$	0	0
$396$	0	0
$397$	25.0535	1.25740	0.628701	−	0.777647i	$-0.283587\pi$
$397$	25.0535	1.25740	0.628701	+	0.777647i	$0.283587\pi$
$398$	9.04879	0.453575
$399$	0	0
$400$	0	0
$401$	33.7989	1.68784	0.843918	−	0.536472i	$-0.180243\pi$
$401$	33.7989	1.68784	0.843918	+	0.536472i	$0.180243\pi$
$402$	0	0
$403$	−8.78464	−0.437594
$404$	54.7765	2.72523
$405$	0	0
$406$	−47.0115	−2.33314
$407$	−26.7825	−1.32756
$408$	0	0
$409$	16.6942	0.825475	0.412737	−	0.910850i	$-0.364573\pi$
$409$	16.6942	0.825475	0.412737	+	0.910850i	$0.364573\pi$
$410$	0	0
$411$	0	0
$412$	2.69450	0.132748
$413$	2.41753	0.118959
$414$	0	0
$415$	0	0
$416$	−9.21430	−0.451768
$417$	0	0
$418$	13.6805	0.669134
$419$	13.3371	0.651562	0.325781	−	0.945445i	$-0.394373\pi$
$419$	13.3371	0.651562	0.325781	+	0.945445i	$0.394373\pi$
$420$	0	0
$421$	34.8398	1.69799	0.848994	−	0.528403i	$-0.177209\pi$
$421$	34.8398	1.69799	0.848994	+	0.528403i	$0.177209\pi$
$422$	31.5233	1.53453
$423$	0	0
$424$	12.5902	0.611433
$425$	0	0
$426$	0	0
$427$	−39.6518	−1.91888
$428$	16.0476	0.775689
$429$	0	0
$430$	0	0
$431$	16.8542	0.811838	0.405919	−	0.913909i	$-0.366952\pi$
$431$	16.8542	0.811838	0.405919	+	0.913909i	$0.366952\pi$
$432$	0	0
$433$	−29.2583	−1.40606	−0.703032	−	0.711159i	$-0.748171\pi$
$433$	−29.2583	−1.40606	−0.703032	+	0.711159i	$0.748171\pi$
$434$	−22.9804	−1.10309
$435$	0	0
$436$	−12.8858	−0.617116
$437$	−8.13886	−0.389335
$438$	0	0
$439$	32.9453	1.57239	0.786196	−	0.617977i	$-0.212047\pi$
$439$	32.9453	1.57239	0.786196	+	0.617977i	$0.212047\pi$
$440$	0	0
$441$	0	0
$442$	24.2375	1.15286
$443$	−24.1805	−1.14885	−0.574424	−	0.818558i	$-0.694774\pi$
$443$	−24.1805	−1.14885	−0.574424	+	0.818558i	$0.694774\pi$
$444$	0	0
$445$	0	0
$446$	16.4584	0.779326
$447$	0	0
$448$	−44.3911	−2.09728
$449$	2.76399	0.130441	0.0652203	−	0.997871i	$-0.479225\pi$
$449$	2.76399	0.130441	0.0652203	+	0.997871i	$0.479225\pi$
$450$	0	0
$451$	−58.2288	−2.74189
$452$	−18.8354	−0.885945
$453$	0	0
$454$	−12.9976	−0.610007
$455$	0	0
$456$	0	0
$457$	11.5998	0.542615	0.271308	−	0.962493i	$-0.412544\pi$
$457$	11.5998	0.542615	0.271308	+	0.962493i	$0.412544\pi$
$458$	52.1246	2.43562
$459$	0	0
$460$	0	0
$461$	11.5257	0.536803	0.268402	−	0.963307i	$-0.413505\pi$
$461$	11.5257	0.536803	0.268402	+	0.963307i	$0.413505\pi$
$462$	0	0
$463$	4.62404	0.214897	0.107449	−	0.994211i	$-0.465732\pi$
$463$	4.62404	0.214897	0.107449	+	0.994211i	$0.465732\pi$
$464$	−11.5563	−0.536489
$465$	0	0
$466$	−5.06797	−0.234769
$467$	−16.4171	−0.759693	−0.379846	−	0.925050i	$-0.624023\pi$
$467$	−16.4171	−0.759693	−0.379846	+	0.925050i	$0.624023\pi$
$468$	0	0
$469$	−5.01218	−0.231441
$470$	0	0
$471$	0	0
$472$	2.40817	0.110845
$473$	−10.0693	−0.462988
$474$	0	0
$475$	0	0
$476$	41.2667	1.89146
$477$	0	0
$478$	20.6675	0.945309
$479$	37.8775	1.73067	0.865333	−	0.501197i	$-0.167107\pi$
$479$	37.8775	1.73067	0.865333	+	0.501197i	$0.167107\pi$
$480$	0	0
$481$	−17.7996	−0.811592
$482$	−50.1351	−2.28359
$483$	0	0
$484$	80.8001	3.67273
$485$	0	0
$486$	0	0
$487$	−26.5135	−1.20144	−0.600721	−	0.799459i	$-0.705120\pi$
$487$	−26.5135	−1.20144	−0.600721	+	0.799459i	$0.705120\pi$
$488$	−39.4981	−1.78800
$489$	0	0
$490$	0	0
$491$	−22.7926	−1.02862	−0.514308	−	0.857605i	$-0.671951\pi$
$491$	−22.7926	−1.02862	−0.514308	+	0.857605i	$0.671951\pi$
$492$	0	0
$493$	−12.6108	−0.567963
$494$	9.09204	0.409070
$495$	0	0
$496$	−5.64902	−0.253649
$497$	−5.01218	−0.224827
$498$	0	0
$499$	−31.7340	−1.42061	−0.710305	−	0.703894i	$-0.751443\pi$
$499$	−31.7340	−1.42061	−0.710305	+	0.703894i	$0.751443\pi$
$500$	0	0
$501$	0	0
$502$	3.12616	0.139527
$503$	−34.8198	−1.55254	−0.776270	−	0.630401i	$-0.782891\pi$
$503$	−34.8198	−1.55254	−0.776270	+	0.630401i	$0.782891\pi$
$504$	0	0
$505$	0	0
$506$	−111.344	−4.94983
$507$	0	0
$508$	16.3069	0.723501
$509$	30.6872	1.36019	0.680093	−	0.733126i	$-0.261940\pi$
$509$	30.6872	1.36019	0.680093	+	0.733126i	$0.261940\pi$
$510$	0	0
$511$	47.9006	2.11900
$512$	−26.1334	−1.15494
$513$	0	0
$514$	33.0829	1.45922
$515$	0	0
$516$	0	0
$517$	−40.8586	−1.79696
$518$	−46.5633	−2.04587
$519$	0	0
$520$	0	0
$521$	6.02661	0.264030	0.132015	−	0.991248i	$-0.457855\pi$
$521$	6.02661	0.264030	0.132015	+	0.991248i	$0.457855\pi$
$522$	0	0
$523$	−13.1962	−0.577030	−0.288515	−	0.957475i	$-0.593161\pi$
$523$	−13.1962	−0.577030	−0.288515	+	0.957475i	$0.593161\pi$
$524$	7.52611	0.328780
$525$	0	0
$526$	74.4419	3.24582
$527$	−6.16448	−0.268529
$528$	0	0
$529$	43.2411	1.88005
$530$	0	0
$531$	0	0
$532$	15.4801	0.671148
$533$	−38.6988	−1.67623
$534$	0	0
$535$	0	0
$536$	−4.99275	−0.215654
$537$	0	0
$538$	2.47308	0.106622
$539$	58.5368	2.52136
$540$	0	0
$541$	−23.7490	−1.02105	−0.510526	−	0.859863i	$-0.670549\pi$
$541$	−23.7490	−1.02105	−0.510526	+	0.859863i	$0.670549\pi$
$542$	−37.5611	−1.61339
$543$	0	0
$544$	−6.46598	−0.277227
$545$	0	0
$546$	0	0
$547$	11.2331	0.480294	0.240147	−	0.970737i	$-0.422804\pi$
$547$	11.2331	0.480294	0.240147	+	0.970737i	$0.422804\pi$
$548$	−7.75398	−0.331234
$549$	0	0
$550$	0	0
$551$	−4.73061	−0.201531
$552$	0	0
$553$	21.0133	0.893577
$554$	−60.8734	−2.58626
$555$	0	0
$556$	−16.5464	−0.701722
$557$	11.8681	0.502868	0.251434	−	0.967874i	$-0.419098\pi$
$557$	11.8681	0.502868	0.251434	+	0.967874i	$0.419098\pi$
$558$	0	0
$559$	−6.69206	−0.283044
$560$	0	0
$561$	0	0
$562$	−20.0144	−0.844258
$563$	18.9867	0.800193	0.400097	−	0.916473i	$-0.368977\pi$
$563$	18.9867	0.800193	0.400097	+	0.916473i	$0.368977\pi$
$564$	0	0
$565$	0	0
$566$	−10.7851	−0.453333
$567$	0	0
$568$	−4.99275	−0.209491
$569$	38.8613	1.62915	0.814575	−	0.580058i	$-0.196970\pi$
$569$	38.8613	1.62915	0.814575	+	0.580058i	$0.196970\pi$
$570$	0	0
$571$	28.5840	1.19620	0.598102	−	0.801420i	$-0.295922\pi$
$571$	28.5840	1.19620	0.598102	+	0.801420i	$0.295922\pi$
$572$	80.9547	3.38489
$573$	0	0
$574$	−101.235	−4.22547
$575$	0	0
$576$	0	0
$577$	−42.4311	−1.76643	−0.883215	−	0.468969i	$-0.844626\pi$
$577$	−42.4311	−1.76643	−0.883215	+	0.468969i	$0.844626\pi$
$578$	−23.6789	−0.984911
$579$	0	0
$580$	0	0
$581$	−7.62697	−0.316420
$582$	0	0
$583$	17.3993	0.720606
$584$	47.7149	1.97446
$585$	0	0
$586$	43.0625	1.77889
$587$	−26.0549	−1.07540	−0.537700	−	0.843136i	$-0.680707\pi$
$587$	−26.0549	−1.07540	−0.537700	+	0.843136i	$0.680707\pi$
$588$	0	0
$589$	−2.31244	−0.0952825
$590$	0	0
$591$	0	0
$592$	−11.4461	−0.470434
$593$	−14.1580	−0.581401	−0.290701	−	0.956814i	$-0.593888\pi$
$593$	−14.1580	−0.581401	−0.290701	+	0.956814i	$0.593888\pi$
$594$	0	0
$595$	0	0
$596$	28.4342	1.16471
$597$	0	0
$598$	−73.9989	−3.02604
$599$	23.6371	0.965785	0.482892	−	0.875680i	$-0.339586\pi$
$599$	23.6371	0.965785	0.482892	+	0.875680i	$0.339586\pi$
$600$	0	0
$601$	−42.0887	−1.71683	−0.858417	−	0.512952i	$-0.828552\pi$
$601$	−42.0887	−1.71683	−0.858417	+	0.512952i	$0.828552\pi$
$602$	−17.5063	−0.713502
$603$	0	0
$604$	70.9072	2.88517
$605$	0	0
$606$	0	0
$607$	31.9023	1.29487	0.647436	−	0.762119i	$-0.275841\pi$
$607$	31.9023	1.29487	0.647436	+	0.762119i	$0.275841\pi$
$608$	−2.42554	−0.0983687
$609$	0	0
$610$	0	0
$611$	−27.1546	−1.09856
$612$	0	0
$613$	26.5468	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$613$	26.5468	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$614$	−42.7285	−1.72438
$615$	0	0
$616$	98.1671	3.95526
$617$	−23.4262	−0.943102	−0.471551	−	0.881839i	$-0.656306\pi$
$617$	−23.4262	−0.943102	−0.471551	+	0.881839i	$0.656306\pi$
$618$	0	0
$619$	−19.7893	−0.795401	−0.397700	−	0.917515i	$-0.630192\pi$
$619$	−19.7893	−0.795401	−0.397700	+	0.917515i	$0.630192\pi$
$620$	0	0
$621$	0	0
$622$	56.9677	2.28420
$623$	−13.9812	−0.560143
$624$	0	0
$625$	0	0
$626$	17.2689	0.690205
$627$	0	0
$628$	−1.37244	−0.0547664
$629$	−12.4906	−0.498032
$630$	0	0
$631$	45.2620	1.80185	0.900925	−	0.433975i	$-0.142889\pi$
$631$	45.2620	1.80185	0.900925	+	0.433975i	$0.142889\pi$
$632$	20.9319	0.832626
$633$	0	0
$634$	−67.8492	−2.69464
$635$	0	0
$636$	0	0
$637$	38.9035	1.54141
$638$	−64.7171	−2.56217
$639$	0	0
$640$	0	0
$641$	−3.60938	−0.142562	−0.0712810	−	0.997456i	$-0.522709\pi$
$641$	−3.60938	−0.142562	−0.0712810	+	0.997456i	$0.522709\pi$
$642$	0	0
$643$	33.2955	1.31305	0.656524	−	0.754305i	$-0.272026\pi$
$643$	33.2955	1.31305	0.656524	+	0.754305i	$0.272026\pi$
$644$	−125.991	−4.96472
$645$	0	0
$646$	6.38019	0.251025
$647$	27.9119	1.09733	0.548665	−	0.836042i	$-0.315136\pi$
$647$	27.9119	1.09733	0.548665	+	0.836042i	$0.315136\pi$
$648$	0	0
$649$	3.32803	0.130637
$650$	0	0
$651$	0	0
$652$	−12.2447	−0.479540
$653$	19.8672	0.777463	0.388731	−	0.921351i	$-0.372913\pi$
$653$	19.8672	0.777463	0.388731	+	0.921351i	$0.372913\pi$
$654$	0	0
$655$	0	0
$656$	−24.8855	−0.971616
$657$	0	0
$658$	−71.0357	−2.76926
$659$	2.96382	0.115454	0.0577271	−	0.998332i	$-0.481615\pi$
$659$	2.96382	0.115454	0.0577271	+	0.998332i	$0.481615\pi$
$660$	0	0
$661$	−15.4066	−0.599249	−0.299625	−	0.954057i	$-0.596861\pi$
$661$	−15.4066	−0.599249	−0.299625	+	0.954057i	$0.596861\pi$
$662$	46.2321	1.79686
$663$	0	0
$664$	−7.59741	−0.294837
$665$	0	0
$666$	0	0
$667$	38.5018	1.49080
$668$	−52.8103	−2.04329
$669$	0	0
$670$	0	0
$671$	−54.5855	−2.10725
$672$	0	0
$673$	−23.0136	−0.887108	−0.443554	−	0.896248i	$-0.646283\pi$
$673$	−23.0136	−0.887108	−0.443554	+	0.896248i	$0.646283\pi$
$674$	−46.6570	−1.79716
$675$	0	0
$676$	5.33631	0.205243
$677$	29.6077	1.13791	0.568957	−	0.822367i	$-0.307347\pi$
$677$	29.6077	1.13791	0.568957	+	0.822367i	$0.307347\pi$
$678$	0	0
$679$	1.30107	0.0499305
$680$	0	0
$681$	0	0
$682$	−31.6353	−1.21138
$683$	−5.86030	−0.224238	−0.112119	−	0.993695i	$-0.535764\pi$
$683$	−5.86030	−0.224238	−0.112119	+	0.993695i	$0.535764\pi$
$684$	0	0
$685$	0	0
$686$	32.2066	1.22965
$687$	0	0
$688$	−4.30337	−0.164065
$689$	11.5636	0.440537
$690$	0	0
$691$	0.0661786	0.00251755	0.00125878	−	0.999999i	$-0.499599\pi$
$691$	0.0661786	0.00251755	0.00125878	+	0.999999i	$0.499599\pi$
$692$	−93.3527	−3.54874
$693$	0	0
$694$	38.6748	1.46807
$695$	0	0
$696$	0	0
$697$	−27.1562	−1.02862
$698$	−15.2530	−0.577334
$699$	0	0
$700$	0	0
$701$	26.4938	1.00066	0.500329	−	0.865835i	$-0.333212\pi$
$701$	26.4938	1.00066	0.500329	+	0.865835i	$0.333212\pi$
$702$	0	0
$703$	−4.68551	−0.176717
$704$	−61.1097	−2.30316
$705$	0	0
$706$	−75.9348	−2.85784
$707$	61.0069	2.29440
$708$	0	0
$709$	36.5674	1.37332	0.686659	−	0.726980i	$-0.259077\pi$
$709$	36.5674	1.37332	0.686659	+	0.726980i	$0.259077\pi$
$710$	0	0
$711$	0	0
$712$	−13.9270	−0.521935
$713$	18.8206	0.704839
$714$	0	0
$715$	0	0
$716$	−96.2251	−3.59610
$717$	0	0
$718$	35.2697	1.31625
$719$	−29.2859	−1.09218	−0.546090	−	0.837727i	$-0.683884\pi$
$719$	−29.2859	−1.09218	−0.546090	+	0.837727i	$0.683884\pi$
$720$	0	0
$721$	3.00097	0.111762
$722$	2.39336	0.0890716
$723$	0	0
$724$	−18.3152	−0.680679
$725$	0	0
$726$	0	0
$727$	2.74936	0.101968	0.0509840	−	0.998699i	$-0.483764\pi$
$727$	2.74936	0.101968	0.0509840	+	0.998699i	$0.483764\pi$
$728$	65.2417	2.41802
$729$	0	0
$730$	0	0
$731$	−4.69604	−0.173689
$732$	0	0
$733$	−21.9833	−0.811971	−0.405985	−	0.913880i	$-0.633072\pi$
$733$	−21.9833	−0.811971	−0.405985	+	0.913880i	$0.633072\pi$
$734$	−23.3464	−0.861732
$735$	0	0
$736$	19.7412	0.727669
$737$	−6.89987	−0.254160
$738$	0	0
$739$	30.1074	1.10752	0.553759	−	0.832677i	$-0.313193\pi$
$739$	30.1074	1.10752	0.553759	+	0.832677i	$0.313193\pi$
$740$	0	0
$741$	0	0
$742$	30.2500	1.11051
$743$	17.7818	0.652351	0.326175	−	0.945309i	$-0.394240\pi$
$743$	17.7818	0.652351	0.326175	+	0.945309i	$0.394240\pi$
$744$	0	0
$745$	0	0
$746$	6.28498	0.230109
$747$	0	0
$748$	56.8087	2.07713
$749$	17.8728	0.653060
$750$	0	0
$751$	−44.3608	−1.61875	−0.809375	−	0.587293i	$-0.800194\pi$
$751$	−44.3608	−1.61875	−0.809375	+	0.587293i	$0.800194\pi$
$752$	−17.4619	−0.636771
$753$	0	0
$754$	−43.0109	−1.56637
$755$	0	0
$756$	0	0
$757$	−28.3533	−1.03052	−0.515260	−	0.857034i	$-0.672305\pi$
$757$	−28.3533	−1.03052	−0.515260	+	0.857034i	$0.672305\pi$
$758$	−1.99886	−0.0726017
$759$	0	0
$760$	0	0
$761$	38.7292	1.40393	0.701967	−	0.712209i	$-0.252305\pi$
$761$	38.7292	1.40393	0.701967	+	0.712209i	$0.252305\pi$
$762$	0	0
$763$	−14.3514	−0.519556
$764$	49.1394	1.77780
$765$	0	0
$766$	21.8834	0.790681
$767$	2.21181	0.0798637
$768$	0	0
$769$	−2.62828	−0.0947783	−0.0473892	−	0.998877i	$-0.515090\pi$
$769$	−2.62828	−0.0947783	−0.0473892	+	0.998877i	$0.515090\pi$
$770$	0	0
$771$	0	0
$772$	65.1249	2.34390
$773$	10.8541	0.390396	0.195198	−	0.980764i	$-0.437465\pi$
$773$	10.8541	0.390396	0.195198	+	0.980764i	$0.437465\pi$
$774$	0	0
$775$	0	0
$776$	1.29603	0.0465247
$777$	0	0
$778$	37.9896	1.36199
$779$	−10.1869	−0.364985
$780$	0	0
$781$	−6.89987	−0.246897
$782$	−51.9275	−1.85692
$783$	0	0
$784$	25.0172	0.893470
$785$	0	0
$786$	0	0
$787$	−44.9026	−1.60061	−0.800303	−	0.599596i	$-0.795328\pi$
$787$	−44.9026	−1.60061	−0.800303	+	0.599596i	$0.795328\pi$
$788$	2.77340	0.0987984
$789$	0	0
$790$	0	0
$791$	−20.9778	−0.745885
$792$	0	0
$793$	−36.2775	−1.28825
$794$	59.9621	2.12798
$795$	0	0
$796$	14.0954	0.499599
$797$	2.80317	0.0992932	0.0496466	−	0.998767i	$-0.484190\pi$
$797$	2.80317	0.0992932	0.0496466	+	0.998767i	$0.484190\pi$
$798$	0	0
$799$	−19.0553	−0.674128
$800$	0	0
$801$	0	0
$802$	80.8929	2.85643
$803$	65.9410	2.32701
$804$	0	0
$805$	0	0
$806$	−21.0248	−0.740567
$807$	0	0
$808$	60.7704	2.13790
$809$	24.2210	0.851564	0.425782	−	0.904826i	$-0.359999\pi$
$809$	24.2210	0.851564	0.425782	+	0.904826i	$0.359999\pi$
$810$	0	0
$811$	−37.2046	−1.30643	−0.653215	−	0.757173i	$-0.726580\pi$
$811$	−37.2046	−1.30643	−0.653215	+	0.757173i	$0.726580\pi$
$812$	−73.2305	−2.56989
$813$	0	0
$814$	−64.1000	−2.24670
$815$	0	0
$816$	0	0
$817$	−1.76160	−0.0616305
$818$	39.9552	1.39700
$819$	0	0
$820$	0	0
$821$	12.8833	0.449632	0.224816	−	0.974401i	$-0.427822\pi$
$821$	12.8833	0.449632	0.224816	+	0.974401i	$0.427822\pi$
$822$	0	0
$823$	2.28995	0.0798227	0.0399114	−	0.999203i	$-0.487292\pi$
$823$	2.28995	0.0798227	0.0399114	+	0.999203i	$0.487292\pi$
$824$	2.98934	0.104139
$825$	0	0
$826$	5.78603	0.201322
$827$	45.1481	1.56995	0.784977	−	0.619525i	$-0.212675\pi$
$827$	45.1481	1.56995	0.784977	+	0.619525i	$0.212675\pi$
$828$	0	0
$829$	−50.8866	−1.76736	−0.883682	−	0.468088i	$-0.844943\pi$
$829$	−50.8866	−1.76736	−0.883682	+	0.468088i	$0.844943\pi$
$830$	0	0
$831$	0	0
$832$	−40.6135	−1.40802
$833$	27.2999	0.945886
$834$	0	0
$835$	0	0
$836$	21.3103	0.737031
$837$	0	0
$838$	31.9205	1.10268
$839$	−45.6605	−1.57638	−0.788188	−	0.615434i	$-0.788981\pi$
$839$	−45.6605	−1.57638	−0.788188	+	0.615434i	$0.788981\pi$
$840$	0	0
$841$	−6.62129	−0.228320
$842$	83.3841	2.87361
$843$	0	0
$844$	49.1042	1.69024
$845$	0	0
$846$	0	0
$847$	89.9903	3.09210
$848$	7.43603	0.255354
$849$	0	0
$850$	0	0
$851$	38.1347	1.30724
$852$	0	0
$853$	−22.9225	−0.784853	−0.392426	−	0.919783i	$-0.628364\pi$
$853$	−22.9225	−0.784853	−0.392426	+	0.919783i	$0.628364\pi$
$854$	−94.9009	−3.24744
$855$	0	0
$856$	17.8036	0.608514
$857$	22.3212	0.762478	0.381239	−	0.924476i	$-0.375497\pi$
$857$	22.3212	0.762478	0.381239	+	0.924476i	$0.375497\pi$
$858$	0	0
$859$	1.02386	0.0349335	0.0174668	−	0.999847i	$-0.494440\pi$
$859$	1.02386	0.0349335	0.0174668	+	0.999847i	$0.494440\pi$
$860$	0	0
$861$	0	0
$862$	40.3381	1.37392
$863$	−4.40228	−0.149855	−0.0749276	−	0.997189i	$-0.523873\pi$
$863$	−4.40228	−0.149855	−0.0749276	+	0.997189i	$0.523873\pi$
$864$	0	0
$865$	0	0
$866$	−70.0256	−2.37957
$867$	0	0
$868$	−35.7969	−1.21502
$869$	28.9274	0.981294
$870$	0	0
$871$	−4.58565	−0.155379
$872$	−14.2958	−0.484116
$873$	0	0
$874$	−19.4792	−0.658894
$875$	0	0
$876$	0	0
$877$	−13.9049	−0.469534	−0.234767	−	0.972052i	$-0.575433\pi$
$877$	−13.9049	−0.469534	−0.234767	+	0.972052i	$0.575433\pi$
$878$	78.8499	2.66105
$879$	0	0
$880$	0	0
$881$	−22.9771	−0.774120	−0.387060	−	0.922055i	$-0.626509\pi$
$881$	−22.9771	−0.774120	−0.387060	+	0.922055i	$0.626509\pi$
$882$	0	0
$883$	1.38507	0.0466113	0.0233057	−	0.999728i	$-0.492581\pi$
$883$	1.38507	0.0466113	0.0233057	+	0.999728i	$0.492581\pi$
$884$	37.7550	1.26984
$885$	0	0
$886$	−57.8725	−1.94426
$887$	−41.9736	−1.40934	−0.704669	−	0.709537i	$-0.748904\pi$
$887$	−41.9736	−1.40934	−0.704669	+	0.709537i	$0.748904\pi$
$888$	0	0
$889$	18.1616	0.609122
$890$	0	0
$891$	0	0
$892$	25.6374	0.858404
$893$	−7.14809	−0.239202
$894$	0	0
$895$	0	0
$896$	−86.1011	−2.87643
$897$	0	0
$898$	6.61521	0.220752
$899$	10.9393	0.364845
$900$	0	0
$901$	8.11455	0.270335
$902$	−139.362	−4.64026
$903$	0	0
$904$	−20.8965	−0.695007
$905$	0	0
$906$	0	0
$907$	35.3835	1.17489	0.587446	−	0.809264i	$-0.300134\pi$
$907$	35.3835	1.17489	0.587446	+	0.809264i	$0.300134\pi$
$908$	−20.2465	−0.671904
$909$	0	0
$910$	0	0
$911$	−20.4675	−0.678118	−0.339059	−	0.940765i	$-0.610109\pi$
$911$	−20.4675	−0.678118	−0.339059	+	0.940765i	$0.610109\pi$
$912$	0	0
$913$	−10.4995	−0.347481
$914$	27.7625	0.918301
$915$	0	0
$916$	81.1952	2.68277
$917$	8.38213	0.276802
$918$	0	0
$919$	52.0875	1.71821	0.859104	−	0.511801i	$-0.171022\pi$
$919$	52.0875	1.71821	0.859104	+	0.511801i	$0.171022\pi$
$920$	0	0
$921$	0	0
$922$	27.5850	0.908465
$923$	−4.58565	−0.150938
$924$	0	0
$925$	0	0
$926$	11.0670	0.363683
$927$	0	0
$928$	11.4743	0.376662
$929$	25.6384	0.841169	0.420585	−	0.907253i	$-0.361825\pi$
$929$	25.6384	0.841169	0.420585	+	0.907253i	$0.361825\pi$
$930$	0	0
$931$	10.2408	0.335630
$932$	−7.89444	−0.258591
$933$	0	0
$934$	−39.2920	−1.28567
$935$	0	0
$936$	0	0
$937$	−23.0135	−0.751819	−0.375910	−	0.926656i	$-0.622670\pi$
$937$	−23.0135	−0.751819	−0.375910	+	0.926656i	$0.622670\pi$
$938$	−11.9959	−0.391681
$939$	0	0
$940$	0	0
$941$	18.9987	0.619340	0.309670	−	0.950844i	$-0.399781\pi$
$941$	18.9987	0.619340	0.309670	+	0.950844i	$0.399781\pi$
$942$	0	0
$943$	82.9101	2.69993
$944$	1.42232	0.0462924
$945$	0	0
$946$	−24.0995	−0.783542
$947$	36.2442	1.17778	0.588890	−	0.808213i	$-0.299565\pi$
$947$	36.2442	1.17778	0.588890	+	0.808213i	$0.299565\pi$
$948$	0	0
$949$	43.8243	1.42260
$950$	0	0
$951$	0	0
$952$	45.7823	1.48381
$953$	−14.9590	−0.484570	−0.242285	−	0.970205i	$-0.577897\pi$
$953$	−14.9590	−0.484570	−0.242285	+	0.970205i	$0.577897\pi$
$954$	0	0
$955$	0	0
$956$	32.1940	1.04123
$957$	0	0
$958$	90.6544	2.92891
$959$	−8.63593	−0.278869
$960$	0	0
$961$	−25.6526	−0.827504
$962$	−42.6008	−1.37351
$963$	0	0
$964$	−78.0960	−2.51530
$965$	0	0
$966$	0	0
$967$	−25.6465	−0.824734	−0.412367	−	0.911018i	$-0.635298\pi$
$967$	−25.6465	−0.824734	−0.412367	+	0.911018i	$0.635298\pi$
$968$	89.6416	2.88119
$969$	0	0
$970$	0	0
$971$	−56.6249	−1.81718	−0.908590	−	0.417689i	$-0.862840\pi$
$971$	−56.6249	−1.81718	−0.908590	+	0.417689i	$0.862840\pi$
$972$	0	0
$973$	−18.4284	−0.590786
$974$	−63.4563	−2.03327
$975$	0	0
$976$	−23.3285	−0.746726
$977$	10.8909	0.348432	0.174216	−	0.984707i	$-0.444261\pi$
$977$	10.8909	0.348432	0.174216	+	0.984707i	$0.444261\pi$
$978$	0	0
$979$	−19.2468	−0.615129
$980$	0	0
$981$	0	0
$982$	−54.5509	−1.74079
$983$	−43.3279	−1.38195	−0.690973	−	0.722881i	$-0.742818\pi$
$983$	−43.3279	−1.38195	−0.690973	+	0.722881i	$0.742818\pi$
$984$	0	0
$985$	0	0
$986$	−30.1822	−0.961197
$987$	0	0
$988$	14.1628	0.450578
$989$	14.3374	0.455903
$990$	0	0
$991$	−12.9921	−0.412707	−0.206353	−	0.978478i	$-0.566160\pi$
$991$	−12.9921	−0.412707	−0.206353	+	0.978478i	$0.566160\pi$
$992$	5.60892	0.178084
$993$	0	0
$994$	−11.9959	−0.380488
$995$	0	0
$996$	0	0
$997$	−54.8983	−1.73865	−0.869323	−	0.494244i	$-0.835445\pi$
$997$	−54.8983	−1.73865	−0.869323	+	0.494244i	$0.835445\pi$
$998$	−75.9509	−2.40418
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bv.1.7		7
3.2	odd	2		1425.2.a.z.1.1		7
5.2	odd	4		855.2.c.g.514.12		14
5.3	odd	4		855.2.c.g.514.3		14
5.4	even	2		4275.2.a.bw.1.1		7
15.2	even	4		285.2.c.b.229.3	✓	14
15.8	even	4		285.2.c.b.229.12	yes	14
15.14	odd	2		1425.2.a.y.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.c.b.229.3	✓	14	15.2	even	4
285.2.c.b.229.12	yes	14	15.8	even	4
855.2.c.g.514.3		14	5.3	odd	4
855.2.c.g.514.12		14	5.2	odd	4
1425.2.a.y.1.7		7	15.14	odd	2
1425.2.a.z.1.1		7	3.2	odd	2
4275.2.a.bv.1.7		7	1.1	even	1	trivial
4275.2.a.bw.1.1		7	5.4	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.39336 q^{2} +3.72816 q^{4} +4.15221 q^{7} +4.13612 q^{8} +5.71602 q^{11} +3.79886 q^{13} +9.93772 q^{14} +2.44288 q^{16} +2.66579 q^{17} +1.00000 q^{19} +13.6805 q^{22} -8.13886 q^{23} +9.09204 q^{26} +15.4801 q^{28} -4.73061 q^{29} -2.31244 q^{31} -2.42554 q^{32} +6.38019 q^{34} -4.68551 q^{37} +2.39336 q^{38} -10.1869 q^{41} -1.76160 q^{43} +21.3103 q^{44} -19.4792 q^{46} -7.14809 q^{47} +10.2408 q^{49} +14.1628 q^{52} +3.04396 q^{53} +17.1740 q^{56} -11.3221 q^{58} +0.582229 q^{59} -9.54956 q^{61} -5.53450 q^{62} -10.6910 q^{64} -1.20711 q^{67} +9.93850 q^{68} -1.20711 q^{71} +11.5362 q^{73} -11.2141 q^{74} +3.72816 q^{76} +23.7341 q^{77} +5.06076 q^{79} -24.3810 q^{82} -1.83685 q^{83} -4.21613 q^{86} +23.6421 q^{88} -3.36716 q^{89} +15.7737 q^{91} -30.3430 q^{92} -17.1079 q^{94} +0.313344 q^{97} +24.5100 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{2} + 11 q^{4} + 8 q^{7} - 9 q^{8} + 4 q^{11} + 8 q^{13} + 4 q^{14} + 19 q^{16} - 4 q^{17} + 7 q^{19} + 12 q^{22} - 10 q^{23} + 20 q^{26} + 14 q^{28} + 6 q^{29} + 4 q^{31} - 31 q^{32} + 2 q^{34}+ \cdots - 5 q^{98}+O(q^{100}) \) 7 * q - 3 * q^2 + 11 * q^4 + 8 * q^7 - 9 * q^8 + 4 * q^11 + 8 * q^13 + 4 * q^14 + 19 * q^16 - 4 * q^17 + 7 * q^19 + 12 * q^22 - 10 * q^23 + 20 * q^26 + 14 * q^28 + 6 * q^29 + 4 * q^31 - 31 * q^32 + 2 * q^34 + 14 * q^37 - 3 * q^38 - 2 * q^41 - 2 * q^43 + 32 * q^44 - 4 * q^46 - 30 * q^47 + 17 * q^49 + 18 * q^52 + 22 * q^56 - 40 * q^58 + 18 * q^59 + 12 * q^61 - 18 * q^62 + 11 * q^64 + 18 * q^67 + 12 * q^68 + 18 * q^71 + 10 * q^73 - 6 * q^74 + 11 * q^76 + 18 * q^77 - 4 * q^79 - 16 * q^82 - 18 * q^83 + 46 * q^86 - 18 * q^88 + 8 * q^89 + 12 * q^91 - 34 * q^92 - 20 * q^94 + 20 * q^97 - 5 * q^98

Introduction

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