LMFDB - Embedded newform 1225.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.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:	$9.78167424761$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.874032$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966 q^{2} -0.874032 q^{3} -1.85410 q^{4} +0.333851 q^{6} +1.47214 q^{8} -2.23607 q^{9} +O(q^{10})$	\(q-0.381966 q^{2} -0.874032 q^{3} -1.85410 q^{4} +0.333851 q^{6} +1.47214 q^{8} -2.23607 q^{9} +2.23607 q^{11} +1.62054 q^{12} +4.03631 q^{13} +3.14590 q^{16} -7.40492 q^{17} +0.854102 q^{18} +4.24264 q^{19} -0.854102 q^{22} -3.76393 q^{23} -1.28669 q^{24} -1.54173 q^{26} +4.57649 q^{27} +2.23607 q^{29} +6.86474 q^{31} -4.14590 q^{32} -1.95440 q^{33} +2.82843 q^{34} +4.14590 q^{36} -10.7082 q^{37} -1.62054 q^{38} -3.52786 q^{39} -4.78282 q^{41} -5.00000 q^{43} -4.14590 q^{44} +1.43769 q^{46} +9.48683 q^{47} -2.74962 q^{48} +6.47214 q^{51} -7.48373 q^{52} -9.70820 q^{53} -1.74806 q^{54} -3.70820 q^{57} -0.854102 q^{58} -13.1893 q^{59} -3.03476 q^{61} -2.62210 q^{62} -4.70820 q^{64} +0.746512 q^{66} -8.70820 q^{67} +13.7295 q^{68} +3.28980 q^{69} -7.47214 q^{71} -3.29180 q^{72} -2.62210 q^{73} +4.09017 q^{74} -7.86629 q^{76} +1.34752 q^{78} -4.70820 q^{79} +2.70820 q^{81} +1.82688 q^{82} -6.86474 q^{83} +1.90983 q^{86} -1.95440 q^{87} +3.29180 q^{88} +7.94510 q^{89} +6.97871 q^{92} -6.00000 q^{93} -3.62365 q^{94} +3.62365 q^{96} +10.1058 q^{97} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} + 6 q^{4} - 12 q^{8}+O(q^{10}) $ 4 * q - 6 * q^2 + 6 * q^4 - 12 * q^8	$ 4 q - 6 q^{2} + 6 q^{4} - 12 q^{8} + 26 q^{16} - 10 q^{18} + 10 q^{22} - 24 q^{23} - 30 q^{32} + 30 q^{36} - 16 q^{37} - 32 q^{39} - 20 q^{43} - 30 q^{44} + 46 q^{46} + 8 q^{51} - 12 q^{53} + 12 q^{57} + 10 q^{58} + 8 q^{64} - 8 q^{67} - 12 q^{71} - 40 q^{72} - 6 q^{74} + 68 q^{78} + 8 q^{79} - 16 q^{81} + 30 q^{86} + 40 q^{88} - 66 q^{92} - 24 q^{93} - 20 q^{99}+O(q^{100}) $ 4 * q - 6 * q^2 + 6 * q^4 - 12 * q^8 + 26 * q^16 - 10 * q^18 + 10 * q^22 - 24 * q^23 - 30 * q^32 + 30 * q^36 - 16 * q^37 - 32 * q^39 - 20 * q^43 - 30 * q^44 + 46 * q^46 + 8 * q^51 - 12 * q^53 + 12 * q^57 + 10 * q^58 + 8 * q^64 - 8 * q^67 - 12 * q^71 - 40 * q^72 - 6 * q^74 + 68 * q^78 + 8 * q^79 - 16 * q^81 + 30 * q^86 + 40 * q^88 - 66 * q^92 - 24 * q^93 - 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.381966	−0.270091	−0.135045	−	0.990839i	$-0.543118\pi$
$2$	−0.381966	−0.270091	−0.135045	+	0.990839i	$0.543118\pi$
$3$	−0.874032	−0.504623	−0.252311	−	0.967646i	$-0.581191\pi$
$3$	−0.874032	−0.504623	−0.252311	+	0.967646i	$0.581191\pi$
$4$	−1.85410	−0.927051
$5$	0	0
$5$	0	0
$6$	0.333851	0.136294
$7$	0	0
$7$	0	0
$8$	1.47214	0.520479
$9$	−2.23607	−0.745356
$10$	0	0
$11$	2.23607	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$11$	2.23607	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$12$	1.62054	0.467811
$13$	4.03631	1.11947	0.559735	−	0.828671i	$-0.310903\pi$
$13$	4.03631	1.11947	0.559735	+	0.828671i	$0.310903\pi$
$14$	0	0
$15$	0	0
$16$	3.14590	0.786475
$17$	−7.40492	−1.79596	−0.897978	−	0.440040i	$-0.854964\pi$
$17$	−7.40492	−1.79596	−0.897978	+	0.440040i	$0.854964\pi$
$18$	0.854102	0.201314
$19$	4.24264	0.973329	0.486664	−	0.873589i	$-0.338214\pi$
$19$	4.24264	0.973329	0.486664	+	0.873589i	$0.338214\pi$
$20$	0	0
$21$	0	0
$22$	−0.854102	−0.182095
$23$	−3.76393	−0.784834	−0.392417	−	0.919787i	$-0.628361\pi$
$23$	−3.76393	−0.784834	−0.392417	+	0.919787i	$0.628361\pi$
$24$	−1.28669	−0.262645
$25$	0	0
$26$	−1.54173	−0.302359
$27$	4.57649	0.880746
$28$	0	0
$29$	2.23607	0.415227	0.207614	−	0.978211i	$-0.433430\pi$
$29$	2.23607	0.415227	0.207614	+	0.978211i	$0.433430\pi$
$30$	0	0
$31$	6.86474	1.23294	0.616472	−	0.787377i	$-0.288562\pi$
$31$	6.86474	1.23294	0.616472	+	0.787377i	$0.288562\pi$
$32$	−4.14590	−0.732898
$33$	−1.95440	−0.340217
$34$	2.82843	0.485071
$35$	0	0
$36$	4.14590	0.690983
$37$	−10.7082	−1.76042	−0.880209	−	0.474586i	$-0.842598\pi$
$37$	−10.7082	−1.76042	−0.880209	+	0.474586i	$0.842598\pi$
$38$	−1.62054	−0.262887
$39$	−3.52786	−0.564910
$40$	0	0
$41$	−4.78282	−0.746951	−0.373476	−	0.927640i	$-0.621834\pi$
$41$	−4.78282	−0.746951	−0.373476	+	0.927640i	$0.621834\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	−4.14590	−0.625018
$45$	0	0
$46$	1.43769	0.211976
$47$	9.48683	1.38380	0.691898	−	0.721995i	$-0.256775\pi$
$47$	9.48683	1.38380	0.691898	+	0.721995i	$0.256775\pi$
$48$	−2.74962	−0.396873
$49$	0	0
$50$	0	0
$51$	6.47214	0.906280
$52$	−7.48373	−1.03781
$53$	−9.70820	−1.33352	−0.666762	−	0.745271i	$-0.732320\pi$
$53$	−9.70820	−1.33352	−0.666762	+	0.745271i	$0.732320\pi$
$54$	−1.74806	−0.237881
$55$	0	0
$56$	0	0
$57$	−3.70820	−0.491164
$58$	−0.854102	−0.112149
$59$	−13.1893	−1.71710	−0.858550	−	0.512730i	$-0.828634\pi$
$59$	−13.1893	−1.71710	−0.858550	+	0.512730i	$0.828634\pi$
$60$	0	0
$61$	−3.03476	−0.388561	−0.194280	−	0.980946i	$-0.562237\pi$
$61$	−3.03476	−0.388561	−0.194280	+	0.980946i	$0.562237\pi$
$62$	−2.62210	−0.333007
$63$	0	0
$64$	−4.70820	−0.588525
$65$	0	0
$66$	0.746512	0.0918893
$67$	−8.70820	−1.06388	−0.531938	−	0.846783i	$-0.678536\pi$
$67$	−8.70820	−1.06388	−0.531938	+	0.846783i	$0.678536\pi$
$68$	13.7295	1.66494
$69$	3.28980	0.396045
$70$	0	0
$71$	−7.47214	−0.886779	−0.443390	−	0.896329i	$-0.646224\pi$
$71$	−7.47214	−0.886779	−0.443390	+	0.896329i	$0.646224\pi$
$72$	−3.29180	−0.387942
$73$	−2.62210	−0.306893	−0.153447	−	0.988157i	$-0.549037\pi$
$73$	−2.62210	−0.306893	−0.153447	+	0.988157i	$0.549037\pi$
$74$	4.09017	0.475473
$75$	0	0
$76$	−7.86629	−0.902325
$77$	0	0
$78$	1.34752	0.152577
$79$	−4.70820	−0.529714	−0.264857	−	0.964288i	$-0.585325\pi$
$79$	−4.70820	−0.529714	−0.264857	+	0.964288i	$0.585325\pi$
$80$	0	0
$81$	2.70820	0.300912
$82$	1.82688	0.201745
$83$	−6.86474	−0.753503	−0.376751	−	0.926314i	$-0.622959\pi$
$83$	−6.86474	−0.753503	−0.376751	+	0.926314i	$0.622959\pi$
$84$	0	0
$85$	0	0
$86$	1.90983	0.205942
$87$	−1.95440	−0.209533
$88$	3.29180	0.350907
$89$	7.94510	0.842179	0.421089	−	0.907019i	$-0.361648\pi$
$89$	7.94510	0.842179	0.421089	+	0.907019i	$0.361648\pi$
$90$	0	0
$91$	0	0
$92$	6.97871	0.727581
$93$	−6.00000	−0.622171
$94$	−3.62365	−0.373751
$95$	0	0
$96$	3.62365	0.369837
$97$	10.1058	1.02609	0.513046	−	0.858361i	$-0.328517\pi$
$97$	10.1058	1.02609	0.513046	+	0.858361i	$0.328517\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	11.1074	1.10523	0.552613	−	0.833438i	$-0.313631\pi$
$101$	11.1074	1.10523	0.552613	+	0.833438i	$0.313631\pi$
$102$	−2.47214	−0.244778
$103$	−9.48683	−0.934765	−0.467383	−	0.884055i	$-0.654803\pi$
$103$	−9.48683	−0.934765	−0.467383	+	0.884055i	$0.654803\pi$
$104$	5.94200	0.582661
$105$	0	0
$106$	3.70820	0.360173
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−8.48528	−0.816497
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	9.35931	0.888347
$112$	0	0
$113$	−9.76393	−0.918513	−0.459257	−	0.888304i	$-0.651884\pi$
$113$	−9.76393	−0.918513	−0.459257	+	0.888304i	$0.651884\pi$
$114$	1.41641	0.132659
$115$	0	0
$116$	−4.14590	−0.384937
$117$	−9.02546	−0.834404
$118$	5.03786	0.463773
$119$	0	0
$120$	0	0
$121$	−6.00000	−0.545455
$122$	1.15917	0.104947
$123$	4.18034	0.376929
$124$	−12.7279	−1.14300
$125$	0	0
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	10.0902	0.891853
$129$	4.37016	0.384771
$130$	0	0
$131$	−14.8098	−1.29394	−0.646971	−	0.762515i	$-0.723964\pi$
$131$	−14.8098	−1.29394	−0.646971	+	0.762515i	$0.723964\pi$
$132$	3.62365	0.315398
$133$	0	0
$134$	3.32624	0.287343
$135$	0	0
$136$	−10.9010	−0.934757
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	−1.25659	−0.106968
$139$	−19.5927	−1.66183	−0.830914	−	0.556401i	$-0.812182\pi$
$139$	−19.5927	−1.66183	−0.830914	+	0.556401i	$0.812182\pi$
$140$	0	0
$141$	−8.29180	−0.698295
$142$	2.85410	0.239511
$143$	9.02546	0.754747
$144$	−7.03444	−0.586203
$145$	0	0
$146$	1.00155	0.0828890
$147$	0	0
$148$	19.8541	1.63200
$149$	9.65248	0.790762	0.395381	−	0.918517i	$-0.370613\pi$
$149$	9.65248	0.790762	0.395381	+	0.918517i	$0.370613\pi$
$150$	0	0
$151$	8.41641	0.684918	0.342459	−	0.939533i	$-0.388740\pi$
$151$	8.41641	0.684918	0.342459	+	0.939533i	$0.388740\pi$
$152$	6.24574	0.506597
$153$	16.5579	1.33863
$154$	0	0
$155$	0	0
$156$	6.54102	0.523701
$157$	5.45052	0.434999	0.217500	−	0.976060i	$-0.430210\pi$
$157$	5.45052	0.434999	0.217500	+	0.976060i	$0.430210\pi$
$158$	1.79837	0.143071
$159$	8.48528	0.672927
$160$	0	0
$161$	0	0
$162$	−1.03444	−0.0812734
$163$	2.29180	0.179507	0.0897537	−	0.995964i	$-0.471392\pi$
$163$	2.29180	0.179507	0.0897537	+	0.995964i	$0.471392\pi$
$164$	8.86784	0.692462
$165$	0	0
$166$	2.62210	0.203514
$167$	−14.2697	−1.10422	−0.552110	−	0.833772i	$-0.686177\pi$
$167$	−14.2697	−1.10422	−0.552110	+	0.833772i	$0.686177\pi$
$168$	0	0
$169$	3.29180	0.253215
$170$	0	0
$171$	−9.48683	−0.725476
$172$	9.27051	0.706870
$173$	−4.24264	−0.322562	−0.161281	−	0.986909i	$-0.551563\pi$
$173$	−4.24264	−0.322562	−0.161281	+	0.986909i	$0.551563\pi$
$174$	0.746512	0.0565930
$175$	0	0
$176$	7.03444	0.530241
$177$	11.5279	0.866487
$178$	−3.03476	−0.227465
$179$	3.70820	0.277164	0.138582	−	0.990351i	$-0.455746\pi$
$179$	3.70820	0.277164	0.138582	+	0.990351i	$0.455746\pi$
$180$	0	0
$181$	11.1074	0.825605	0.412802	−	0.910821i	$-0.364550\pi$
$181$	11.1074	0.825605	0.412802	+	0.910821i	$0.364550\pi$
$182$	0	0
$183$	2.65248	0.196077
$184$	−5.54102	−0.408489
$185$	0	0
$186$	2.29180	0.168043
$187$	−16.5579	−1.21083
$188$	−17.5896	−1.28285
$189$	0	0
$190$	0	0
$191$	−23.1246	−1.67324	−0.836619	−	0.547785i	$-0.815471\pi$
$191$	−23.1246	−1.67324	−0.836619	+	0.547785i	$0.815471\pi$
$192$	4.11512	0.296983
$193$	10.7082	0.770793	0.385397	−	0.922751i	$-0.374065\pi$
$193$	10.7082	0.770793	0.385397	+	0.922751i	$0.374065\pi$
$194$	−3.86008	−0.277138
$195$	0	0
$196$	0	0
$197$	5.94427	0.423512	0.211756	−	0.977323i	$-0.432082\pi$
$197$	5.94427	0.423512	0.211756	+	0.977323i	$0.432082\pi$
$198$	1.90983	0.135726
$199$	25.2495	1.78989	0.894945	−	0.446176i	$-0.147214\pi$
$199$	25.2495	1.78989	0.894945	+	0.446176i	$0.147214\pi$
$200$	0	0
$201$	7.61125	0.536856
$202$	−4.24264	−0.298511
$203$	0	0
$204$	−12.0000	−0.840168
$205$	0	0
$206$	3.62365	0.252472
$207$	8.41641	0.584981
$208$	12.6978	0.880435
$209$	9.48683	0.656218
$210$	0	0
$211$	7.41641	0.510567	0.255283	−	0.966866i	$-0.417831\pi$
$211$	7.41641	0.510567	0.255283	+	0.966866i	$0.417831\pi$
$212$	18.0000	1.23625
$213$	6.53089	0.447489
$214$	2.29180	0.156664
$215$	0	0
$216$	6.73722	0.458410
$217$	0	0
$218$	0.381966	0.0258700
$219$	2.29180	0.154865
$220$	0	0
$221$	−29.8885	−2.01052
$222$	−3.57494	−0.239934
$223$	5.86319	0.392628	0.196314	−	0.980541i	$-0.437103\pi$
$223$	5.86319	0.392628	0.196314	+	0.980541i	$0.437103\pi$
$224$	0	0
$225$	0	0
$226$	3.72949	0.248082
$227$	26.4574	1.75604	0.878020	−	0.478625i	$-0.158865\pi$
$227$	26.4574	1.75604	0.878020	+	0.478625i	$0.158865\pi$
$228$	6.87539	0.455334
$229$	−7.07107	−0.467269	−0.233635	−	0.972324i	$-0.575062\pi$
$229$	−7.07107	−0.467269	−0.233635	+	0.972324i	$0.575062\pi$
$230$	0	0
$231$	0	0
$232$	3.29180	0.216117
$233$	−4.52786	−0.296630	−0.148315	−	0.988940i	$-0.547385\pi$
$233$	−4.52786	−0.296630	−0.148315	+	0.988940i	$0.547385\pi$
$234$	3.44742	0.225365
$235$	0	0
$236$	24.4543	1.59184
$237$	4.11512	0.267306
$238$	0	0
$239$	29.1246	1.88391	0.941957	−	0.335733i	$-0.108984\pi$
$239$	29.1246	1.88391	0.941957	+	0.335733i	$0.108984\pi$
$240$	0	0
$241$	−14.7310	−0.948909	−0.474454	−	0.880280i	$-0.657355\pi$
$241$	−14.7310	−0.948909	−0.474454	+	0.880280i	$0.657355\pi$
$242$	2.29180	0.147322
$243$	−16.0965	−1.03259
$244$	5.62675	0.360216
$245$	0	0
$246$	−1.59675	−0.101805
$247$	17.1246	1.08961
$248$	10.1058	0.641721
$249$	6.00000	0.380235
$250$	0	0
$251$	−11.5687	−0.730213	−0.365106	−	0.930966i	$-0.618967\pi$
$251$	−11.5687	−0.730213	−0.365106	+	0.930966i	$0.618967\pi$
$252$	0	0
$253$	−8.41641	−0.529135
$254$	2.67376	0.167767
$255$	0	0
$256$	5.56231	0.347644
$257$	−12.1089	−0.755334	−0.377667	−	0.925941i	$-0.623274\pi$
$257$	−12.1089	−0.755334	−0.377667	+	0.925941i	$0.623274\pi$
$258$	−1.66925	−0.103923
$259$	0	0
$260$	0	0
$261$	−5.00000	−0.309492
$262$	5.65685	0.349482
$263$	−29.9443	−1.84644	−0.923221	−	0.384268i	$-0.874454\pi$
$263$	−29.9443	−1.84644	−0.923221	+	0.384268i	$0.874454\pi$
$264$	−2.87714	−0.177075
$265$	0	0
$266$	0	0
$267$	−6.94427	−0.424983
$268$	16.1459	0.986268
$269$	−15.3500	−0.935907	−0.467954	−	0.883753i	$-0.655009\pi$
$269$	−15.3500	−0.935907	−0.467954	+	0.883753i	$0.655009\pi$
$270$	0	0
$271$	−4.44897	−0.270256	−0.135128	−	0.990828i	$-0.543145\pi$
$271$	−4.44897	−0.270256	−0.135128	+	0.990828i	$0.543145\pi$
$272$	−23.2951	−1.41247
$273$	0	0
$274$	−2.29180	−0.138452
$275$	0	0
$276$	−6.09962	−0.367154
$277$	24.0000	1.44202	0.721010	−	0.692925i	$-0.243678\pi$
$277$	24.0000	1.44202	0.721010	+	0.692925i	$0.243678\pi$
$278$	7.48373	0.448844
$279$	−15.3500	−0.918982
$280$	0	0
$281$	−24.5967	−1.46732	−0.733659	−	0.679517i	$-0.762189\pi$
$281$	−24.5967	−1.46732	−0.733659	+	0.679517i	$0.762189\pi$
$282$	3.16718	0.188603
$283$	20.8005	1.23646	0.618232	−	0.785996i	$-0.287849\pi$
$283$	20.8005	1.23646	0.618232	+	0.785996i	$0.287849\pi$
$284$	13.8541	0.822090
$285$	0	0
$286$	−3.44742	−0.203850
$287$	0	0
$288$	9.27051	0.546270
$289$	37.8328	2.22546
$290$	0	0
$291$	−8.83282	−0.517789
$292$	4.86163	0.284506
$293$	−18.4335	−1.07690	−0.538448	−	0.842659i	$-0.680989\pi$
$293$	−18.4335	−1.07690	−0.538448	+	0.842659i	$0.680989\pi$
$294$	0	0
$295$	0	0
$296$	−15.7639	−0.916260
$297$	10.2333	0.593799
$298$	−3.68692	−0.213577
$299$	−15.1924	−0.878599
$300$	0	0
$301$	0	0
$302$	−3.21478	−0.184990
$303$	−9.70820	−0.557722
$304$	13.3469	0.765498
$305$	0	0
$306$	−6.32456	−0.361551
$307$	−14.5548	−0.830686	−0.415343	−	0.909665i	$-0.636338\pi$
$307$	−14.5548	−0.830686	−0.415343	+	0.909665i	$0.636338\pi$
$308$	0	0
$309$	8.29180	0.471704
$310$	0	0
$311$	−9.56564	−0.542418	−0.271209	−	0.962520i	$-0.587423\pi$
$311$	−9.56564	−0.542418	−0.271209	+	0.962520i	$0.587423\pi$
$312$	−5.19350	−0.294024
$313$	−21.2132	−1.19904	−0.599521	−	0.800359i	$-0.704642\pi$
$313$	−21.2132	−1.19904	−0.599521	+	0.800359i	$0.704642\pi$
$314$	−2.08191	−0.117489
$315$	0	0
$316$	8.72949	0.491072
$317$	11.9443	0.670857	0.335429	−	0.942066i	$-0.391119\pi$
$317$	11.9443	0.670857	0.335429	+	0.942066i	$0.391119\pi$
$318$	−3.24109	−0.181751
$319$	5.00000	0.279946
$320$	0	0
$321$	5.24419	0.292702
$322$	0	0
$323$	−31.4164	−1.74806
$324$	−5.02129	−0.278960
$325$	0	0
$326$	−0.875388	−0.0484833
$327$	0.874032	0.0483341
$328$	−7.04096	−0.388772
$329$	0	0
$330$	0	0
$331$	−1.29180	−0.0710035	−0.0355018	−	0.999370i	$-0.511303\pi$
$331$	−1.29180	−0.0710035	−0.0355018	+	0.999370i	$0.511303\pi$
$332$	12.7279	0.698535
$333$	23.9443	1.31214
$334$	5.45052	0.298239
$335$	0	0
$336$	0	0
$337$	23.1246	1.25968	0.629839	−	0.776726i	$-0.283121\pi$
$337$	23.1246	1.25968	0.629839	+	0.776726i	$0.283121\pi$
$338$	−1.25735	−0.0683911
$339$	8.53399	0.463503
$340$	0	0
$341$	15.3500	0.831250
$342$	3.62365	0.195944
$343$	0	0
$344$	−7.36068	−0.396861
$345$	0	0
$346$	1.62054	0.0871210
$347$	−27.6525	−1.48446	−0.742231	−	0.670144i	$-0.766232\pi$
$347$	−27.6525	−1.48446	−0.742231	+	0.670144i	$0.766232\pi$
$348$	3.62365	0.194248
$349$	−1.00155	−0.0536118	−0.0268059	−	0.999641i	$-0.508534\pi$
$349$	−1.00155	−0.0536118	−0.0268059	+	0.999641i	$0.508534\pi$
$350$	0	0
$351$	18.4721	0.985970
$352$	−9.27051	−0.494120
$353$	12.1089	0.644493	0.322247	−	0.946656i	$-0.395562\pi$
$353$	12.1089	0.644493	0.322247	+	0.946656i	$0.395562\pi$
$354$	−4.40325	−0.234030
$355$	0	0
$356$	−14.7310	−0.780743
$357$	0	0
$358$	−1.41641	−0.0748595
$359$	0.0557281	0.00294122	0.00147061	−	0.999999i	$-0.499532\pi$
$359$	0.0557281	0.00294122	0.00147061	+	0.999999i	$0.499532\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	−4.24264	−0.222988
$363$	5.24419	0.275249
$364$	0	0
$365$	0	0
$366$	−1.01316	−0.0529585
$367$	8.69161	0.453698	0.226849	−	0.973930i	$-0.427158\pi$
$367$	8.69161	0.453698	0.226849	+	0.973930i	$0.427158\pi$
$368$	−11.8409	−0.617252
$369$	10.6947	0.556745
$370$	0	0
$371$	0	0
$372$	11.1246	0.576784
$373$	16.1246	0.834901	0.417450	−	0.908700i	$-0.362924\pi$
$373$	16.1246	0.834901	0.417450	+	0.908700i	$0.362924\pi$
$374$	6.32456	0.327035
$375$	0	0
$376$	13.9659	0.720237
$377$	9.02546	0.464835
$378$	0	0
$379$	−19.0000	−0.975964	−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000	−0.975964	−0.487982	+	0.872854i	$0.662267\pi$
$380$	0	0
$381$	6.11822	0.313446
$382$	8.83282	0.451926
$383$	13.1893	0.673941	0.336971	−	0.941515i	$-0.390598\pi$
$383$	13.1893	0.673941	0.336971	+	0.941515i	$0.390598\pi$
$384$	−8.81913	−0.450049
$385$	0	0
$386$	−4.09017	−0.208184
$387$	11.1803	0.568329
$388$	−18.7372	−0.951239
$389$	27.7639	1.40769	0.703844	−	0.710355i	$-0.251466\pi$
$389$	27.7639	1.40769	0.703844	+	0.710355i	$0.251466\pi$
$390$	0	0
$391$	27.8716	1.40953
$392$	0	0
$393$	12.9443	0.652952
$394$	−2.27051	−0.114387
$395$	0	0
$396$	9.27051	0.465861
$397$	−9.89949	−0.496841	−0.248421	−	0.968652i	$-0.579912\pi$
$397$	−9.89949	−0.496841	−0.248421	+	0.968652i	$0.579912\pi$
$398$	−9.64446	−0.483433
$399$	0	0
$400$	0	0
$401$	23.0689	1.15201	0.576003	−	0.817448i	$-0.304612\pi$
$401$	23.0689	1.15201	0.576003	+	0.817448i	$0.304612\pi$
$402$	−2.90724	−0.145000
$403$	27.7082	1.38024
$404$	−20.5942	−1.02460
$405$	0	0
$406$	0	0
$407$	−23.9443	−1.18687
$408$	9.52786	0.471700
$409$	−14.7310	−0.728402	−0.364201	−	0.931320i	$-0.618658\pi$
$409$	−14.7310	−0.728402	−0.364201	+	0.931320i	$0.618658\pi$
$410$	0	0
$411$	−5.24419	−0.258677
$412$	17.5896	0.866575
$413$	0	0
$414$	−3.21478	−0.157998
$415$	0	0
$416$	−16.7341	−0.820458
$417$	17.1246	0.838596
$418$	−3.62365	−0.177238
$419$	4.86163	0.237506	0.118753	−	0.992924i	$-0.462110\pi$
$419$	4.86163	0.237506	0.118753	+	0.992924i	$0.462110\pi$
$420$	0	0
$421$	−9.29180	−0.452854	−0.226427	−	0.974028i	$-0.572705\pi$
$421$	−9.29180	−0.452854	−0.226427	+	0.974028i	$0.572705\pi$
$422$	−2.83282	−0.137899
$423$	−21.2132	−1.03142
$424$	−14.2918	−0.694071
$425$	0	0
$426$	−2.49458	−0.120863
$427$	0	0
$428$	11.1246	0.537728
$429$	−7.88854	−0.380862
$430$	0	0
$431$	6.76393	0.325807	0.162904	−	0.986642i	$-0.447914\pi$
$431$	6.76393	0.325807	0.162904	+	0.986642i	$0.447914\pi$
$432$	14.3972	0.692684
$433$	32.7031	1.57161	0.785806	−	0.618473i	$-0.212248\pi$
$433$	32.7031	1.57161	0.785806	+	0.618473i	$0.212248\pi$
$434$	0	0
$435$	0	0
$436$	1.85410	0.0887954
$437$	−15.9690	−0.763901
$438$	−0.875388	−0.0418277
$439$	17.7658	0.847915	0.423957	−	0.905682i	$-0.360641\pi$
$439$	17.7658	0.847915	0.423957	+	0.905682i	$0.360641\pi$
$440$	0	0
$441$	0	0
$442$	11.4164	0.543023
$443$	−20.1803	−0.958797	−0.479398	−	0.877597i	$-0.659145\pi$
$443$	−20.1803	−0.958797	−0.479398	+	0.877597i	$0.659145\pi$
$444$	−17.3531	−0.823543
$445$	0	0
$446$	−2.23954	−0.106045
$447$	−8.43657	−0.399036
$448$	0	0
$449$	−12.5967	−0.594477	−0.297239	−	0.954803i	$-0.596066\pi$
$449$	−12.5967	−0.594477	−0.297239	+	0.954803i	$0.596066\pi$
$450$	0	0
$451$	−10.6947	−0.503594
$452$	18.1033	0.851509
$453$	−7.35621	−0.345625
$454$	−10.1058	−0.474290
$455$	0	0
$456$	−5.45898	−0.255640
$457$	−7.29180	−0.341096	−0.170548	−	0.985349i	$-0.554554\pi$
$457$	−7.29180	−0.341096	−0.170548	+	0.985349i	$0.554554\pi$
$458$	2.70091	0.126205
$459$	−33.8885	−1.58178
$460$	0	0
$461$	−33.4009	−1.55564	−0.777819	−	0.628489i	$-0.783674\pi$
$461$	−33.4009	−1.55564	−0.777819	+	0.628489i	$0.783674\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	7.03444	0.326566
$465$	0	0
$466$	1.72949	0.0801171
$467$	9.64446	0.446292	0.223146	−	0.974785i	$-0.428367\pi$
$467$	9.64446	0.446292	0.223146	+	0.974785i	$0.428367\pi$
$468$	16.7341	0.773535
$469$	0	0
$470$	0	0
$471$	−4.76393	−0.219510
$472$	−19.4164	−0.893714
$473$	−11.1803	−0.514073
$474$	−1.57184	−0.0721968
$475$	0	0
$476$	0	0
$477$	21.7082	0.993950
$478$	−11.1246	−0.508828
$479$	−9.02546	−0.412384	−0.206192	−	0.978512i	$-0.566107\pi$
$479$	−9.02546	−0.412384	−0.206192	+	0.978512i	$0.566107\pi$
$480$	0	0
$481$	−43.2216	−1.97074
$482$	5.62675	0.256291
$483$	0	0
$484$	11.1246	0.505664
$485$	0	0
$486$	6.14833	0.278894
$487$	−28.7082	−1.30089	−0.650446	−	0.759552i	$-0.725418\pi$
$487$	−28.7082	−1.30089	−0.650446	+	0.759552i	$0.725418\pi$
$488$	−4.46758	−0.202238
$489$	−2.00310	−0.0905835
$490$	0	0
$491$	7.47214	0.337213	0.168606	−	0.985683i	$-0.446073\pi$
$491$	7.47214	0.337213	0.168606	+	0.985683i	$0.446073\pi$
$492$	−7.75078	−0.349432
$493$	−16.5579	−0.745730
$494$	−6.54102	−0.294294
$495$	0	0
$496$	21.5958	0.969678
$497$	0	0
$498$	−2.29180	−0.102698
$499$	−25.4164	−1.13779	−0.568897	−	0.822409i	$-0.692630\pi$
$499$	−25.4164	−1.13779	−0.568897	+	0.822409i	$0.692630\pi$
$500$	0	0
$501$	12.4721	0.557214
$502$	4.41887	0.197224
$503$	−16.8918	−0.753166	−0.376583	−	0.926383i	$-0.622901\pi$
$503$	−16.8918	−0.753166	−0.376583	+	0.926383i	$0.622901\pi$
$504$	0	0
$505$	0	0
$506$	3.21478	0.142914
$507$	−2.87714	−0.127778
$508$	12.9787	0.575837
$509$	5.78437	0.256388	0.128194	−	0.991749i	$-0.459082\pi$
$509$	5.78437	0.256388	0.128194	+	0.991749i	$0.459082\pi$
$510$	0	0
$511$	0	0
$512$	−22.3050	−0.985749
$513$	19.4164	0.857255
$514$	4.62520	0.204009
$515$	0	0
$516$	−8.10272	−0.356702
$517$	21.2132	0.932956
$518$	0	0
$519$	3.70820	0.162772
$520$	0	0
$521$	36.0230	1.57820	0.789099	−	0.614266i	$-0.210548\pi$
$521$	36.0230	1.57820	0.789099	+	0.614266i	$0.210548\pi$
$522$	1.90983	0.0835910
$523$	3.24109	0.141723	0.0708615	−	0.997486i	$-0.477425\pi$
$523$	3.24109	0.141723	0.0708615	+	0.997486i	$0.477425\pi$
$524$	27.4589	1.19955
$525$	0	0
$526$	11.4377	0.498707
$527$	−50.8328	−2.21431
$528$	−6.14833	−0.267572
$529$	−8.83282	−0.384035
$530$	0	0
$531$	29.4922	1.27985
$532$	0	0
$533$	−19.3050	−0.836190
$534$	2.65248	0.114784
$535$	0	0
$536$	−12.8197	−0.553725
$537$	−3.24109	−0.139863
$538$	5.86319	0.252780
$539$	0	0
$540$	0	0
$541$	−8.70820	−0.374395	−0.187197	−	0.982322i	$-0.559940\pi$
$541$	−8.70820	−0.374395	−0.187197	+	0.982322i	$0.559940\pi$
$542$	1.69936	0.0729936
$543$	−9.70820	−0.416619
$544$	30.7000	1.31625
$545$	0	0
$546$	0	0
$547$	−29.0000	−1.23995	−0.619975	−	0.784621i	$-0.712857\pi$
$547$	−29.0000	−1.23995	−0.619975	+	0.784621i	$0.712857\pi$
$548$	−11.1246	−0.475220
$549$	6.78593	0.289616
$550$	0	0
$551$	9.48683	0.404153
$552$	4.84303	0.206133
$553$	0	0
$554$	−9.16718	−0.389476
$555$	0	0
$556$	36.3268	1.54060
$557$	−24.5967	−1.04220	−0.521099	−	0.853496i	$-0.674478\pi$
$557$	−24.5967	−1.04220	−0.521099	+	0.853496i	$0.674478\pi$
$558$	5.86319	0.248208
$559$	−20.1815	−0.853589
$560$	0	0
$561$	14.4721	0.611014
$562$	9.39512	0.396309
$563$	13.8083	0.581950	0.290975	−	0.956731i	$-0.406020\pi$
$563$	13.8083	0.581950	0.290975	+	0.956731i	$0.406020\pi$
$564$	15.3738	0.647355
$565$	0	0
$566$	−7.94510	−0.333957
$567$	0	0
$568$	−11.0000	−0.461550
$569$	−29.9443	−1.25533	−0.627665	−	0.778484i	$-0.715989\pi$
$569$	−29.9443	−1.25533	−0.627665	+	0.778484i	$0.715989\pi$
$570$	0	0
$571$	26.4164	1.10549	0.552746	−	0.833350i	$-0.313580\pi$
$571$	26.4164	1.10549	0.552746	+	0.833350i	$0.313580\pi$
$572$	−16.7341	−0.699689
$573$	20.2117	0.844354
$574$	0	0
$575$	0	0
$576$	10.5279	0.438661
$577$	−42.1900	−1.75639	−0.878196	−	0.478301i	$-0.841253\pi$
$577$	−42.1900	−1.75639	−0.878196	+	0.478301i	$0.841253\pi$
$578$	−14.4508	−0.601076
$579$	−9.35931	−0.388960
$580$	0	0
$581$	0	0
$582$	3.37384	0.139850
$583$	−21.7082	−0.899062
$584$	−3.86008	−0.159731
$585$	0	0
$586$	7.04096	0.290860
$587$	−5.86319	−0.242000	−0.121000	−	0.992653i	$-0.538610\pi$
$587$	−5.86319	−0.242000	−0.121000	+	0.992653i	$0.538610\pi$
$588$	0	0
$589$	29.1246	1.20006
$590$	0	0
$591$	−5.19548	−0.213714
$592$	−33.6869	−1.38452
$593$	30.7000	1.26070	0.630350	−	0.776311i	$-0.282912\pi$
$593$	30.7000	1.26070	0.630350	+	0.776311i	$0.282912\pi$
$594$	−3.90879	−0.160380
$595$	0	0
$596$	−17.8967	−0.733076
$597$	−22.0689	−0.903219
$598$	5.80298	0.237301
$599$	−6.81966	−0.278644	−0.139322	−	0.990247i	$-0.544492\pi$
$599$	−6.81966	−0.278644	−0.139322	+	0.990247i	$0.544492\pi$
$600$	0	0
$601$	−29.4621	−1.20178	−0.600891	−	0.799331i	$-0.705187\pi$
$601$	−29.4621	−1.20178	−0.600891	+	0.799331i	$0.705187\pi$
$602$	0	0
$603$	19.4721	0.792967
$604$	−15.6049	−0.634953
$605$	0	0
$606$	3.70820	0.150635
$607$	45.2247	1.83562	0.917808	−	0.397025i	$-0.129958\pi$
$607$	45.2247	1.83562	0.917808	+	0.397025i	$0.129958\pi$
$608$	−17.5896	−0.713351
$609$	0	0
$610$	0	0
$611$	38.2918	1.54912
$612$	−30.7000	−1.24098
$613$	−14.4164	−0.582273	−0.291137	−	0.956681i	$-0.594033\pi$
$613$	−14.4164	−0.582273	−0.291137	+	0.956681i	$0.594033\pi$
$614$	5.55944	0.224361
$615$	0	0
$616$	0	0
$617$	−19.3607	−0.779432	−0.389716	−	0.920935i	$-0.627427\pi$
$617$	−19.3607	−0.779432	−0.389716	+	0.920935i	$0.627427\pi$
$618$	−3.16718	−0.127403
$619$	−9.28050	−0.373015	−0.186507	−	0.982454i	$-0.559717\pi$
$619$	−9.28050	−0.373015	−0.186507	+	0.982454i	$0.559717\pi$
$620$	0	0
$621$	−17.2256	−0.691240
$622$	3.65375	0.146502
$623$	0	0
$624$	−11.0983	−0.444288
$625$	0	0
$626$	8.10272	0.323850
$627$	−8.29180	−0.331142
$628$	−10.1058	−0.403266
$629$	79.2934	3.16163
$630$	0	0
$631$	−28.1246	−1.11962	−0.559812	−	0.828620i	$-0.689126\pi$
$631$	−28.1246	−1.11962	−0.559812	+	0.828620i	$0.689126\pi$
$632$	−6.93112	−0.275705
$633$	−6.48218	−0.257643
$634$	−4.56231	−0.181192
$635$	0	0
$636$	−15.7326	−0.623837
$637$	0	0
$638$	−1.90983	−0.0756109
$639$	16.7082	0.660966
$640$	0	0
$641$	−16.5279	−0.652811	−0.326406	−	0.945230i	$-0.605838\pi$
$641$	−16.5279	−0.652811	−0.326406	+	0.945230i	$0.605838\pi$
$642$	−2.00310	−0.0790562
$643$	−22.2148	−0.876064	−0.438032	−	0.898959i	$-0.644324\pi$
$643$	−22.2148	−0.876064	−0.438032	+	0.898959i	$0.644324\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	−19.0525	−0.749030	−0.374515	−	0.927221i	$-0.622191\pi$
$647$	−19.0525	−0.749030	−0.374515	+	0.927221i	$0.622191\pi$
$648$	3.98684	0.156618
$649$	−29.4922	−1.15767
$650$	0	0
$651$	0	0
$652$	−4.24922	−0.166412
$653$	−1.41641	−0.0554283	−0.0277142	−	0.999616i	$-0.508823\pi$
$653$	−1.41641	−0.0554283	−0.0277142	+	0.999616i	$0.508823\pi$
$654$	−0.333851	−0.0130546
$655$	0	0
$656$	−15.0463	−0.587458
$657$	5.86319	0.228745
$658$	0	0
$659$	−8.83282	−0.344078	−0.172039	−	0.985090i	$-0.555035\pi$
$659$	−8.83282	−0.344078	−0.172039	+	0.985090i	$0.555035\pi$
$660$	0	0
$661$	19.9752	0.776946	0.388473	−	0.921460i	$-0.373003\pi$
$661$	19.9752	0.776946	0.388473	+	0.921460i	$0.373003\pi$
$662$	0.493422	0.0191774
$663$	26.1235	1.01455
$664$	−10.1058	−0.392182
$665$	0	0
$666$	−9.14590	−0.354396
$667$	−8.41641	−0.325885
$668$	26.4574	1.02367
$669$	−5.12461	−0.198129
$670$	0	0
$671$	−6.78593	−0.261968
$672$	0	0
$673$	41.1246	1.58524	0.792619	−	0.609718i	$-0.208717\pi$
$673$	41.1246	1.58524	0.792619	+	0.609718i	$0.208717\pi$
$674$	−8.83282	−0.340227
$675$	0	0
$676$	−6.10333	−0.234743
$677$	13.8083	0.530696	0.265348	−	0.964153i	$-0.414513\pi$
$677$	13.8083	0.530696	0.265348	+	0.964153i	$0.414513\pi$
$678$	−3.25969	−0.125188
$679$	0	0
$680$	0	0
$681$	−23.1246	−0.886137
$682$	−5.86319	−0.224513
$683$	−41.0689	−1.57146	−0.785729	−	0.618571i	$-0.787712\pi$
$683$	−41.0689	−1.57146	−0.785729	+	0.618571i	$0.787712\pi$
$684$	17.5896	0.672553
$685$	0	0
$686$	0	0
$687$	6.18034	0.235795
$688$	−15.7295	−0.599681
$689$	−39.1853	−1.49284
$690$	0	0
$691$	1.79677	0.0683524	0.0341762	−	0.999416i	$-0.489119\pi$
$691$	1.79677	0.0683524	0.0341762	+	0.999416i	$0.489119\pi$
$692$	7.86629	0.299031
$693$	0	0
$694$	10.5623	0.400940
$695$	0	0
$696$	−2.87714	−0.109058
$697$	35.4164	1.34149
$698$	0.382559	0.0144801
$699$	3.95750	0.149686
$700$	0	0
$701$	31.4164	1.18658	0.593291	−	0.804988i	$-0.297828\pi$
$701$	31.4164	1.18658	0.593291	+	0.804988i	$0.297828\pi$
$702$	−7.05573	−0.266301
$703$	−45.4311	−1.71346
$704$	−10.5279	−0.396784
$705$	0	0
$706$	−4.62520	−0.174072
$707$	0	0
$708$	−21.3738	−0.803278
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	10.5279	0.394826
$712$	11.6963	0.438336
$713$	−25.8384	−0.967656
$714$	0	0
$715$	0	0
$716$	−6.87539	−0.256945
$717$	−25.4558	−0.950666
$718$	−0.0212862	−0.000794395 0
$719$	1.54173	0.0574969	0.0287485	−	0.999587i	$-0.490848\pi$
$719$	1.54173	0.0574969	0.0287485	+	0.999587i	$0.490848\pi$
$720$	0	0
$721$	0	0
$722$	0.381966	0.0142153
$723$	12.8754	0.478841
$724$	−20.5942	−0.765378
$725$	0	0
$726$	−2.00310	−0.0743421
$727$	−5.65685	−0.209801	−0.104901	−	0.994483i	$-0.533452\pi$
$727$	−5.65685	−0.209801	−0.104901	+	0.994483i	$0.533452\pi$
$728$	0	0
$729$	5.94427	0.220158
$730$	0	0
$731$	37.0246	1.36940
$732$	−4.91796	−0.181773
$733$	6.48218	0.239425	0.119712	−	0.992809i	$-0.461803\pi$
$733$	6.48218	0.239425	0.119712	+	0.992809i	$0.461803\pi$
$734$	−3.31990	−0.122540
$735$	0	0
$736$	15.6049	0.575203
$737$	−19.4721	−0.717265
$738$	−4.08502	−0.150372
$739$	−7.29180	−0.268233	−0.134117	−	0.990966i	$-0.542820\pi$
$739$	−7.29180	−0.268233	−0.134117	+	0.990966i	$0.542820\pi$
$740$	0	0
$741$	−14.9675	−0.549843
$742$	0	0
$743$	−33.7082	−1.23663	−0.618317	−	0.785929i	$-0.712185\pi$
$743$	−33.7082	−1.23663	−0.618317	+	0.785929i	$0.712185\pi$
$744$	−8.83282	−0.323827
$745$	0	0
$746$	−6.15905	−0.225499
$747$	15.3500	0.561628
$748$	30.7000	1.12250
$749$	0	0
$750$	0	0
$751$	26.8328	0.979143	0.489572	−	0.871963i	$-0.337153\pi$
$751$	26.8328	0.979143	0.489572	+	0.871963i	$0.337153\pi$
$752$	29.8446	1.08832
$753$	10.1115	0.368482
$754$	−3.44742	−0.125548
$755$	0	0
$756$	0	0
$757$	12.4164	0.451282	0.225641	−	0.974211i	$-0.427552\pi$
$757$	12.4164	0.451282	0.225641	+	0.974211i	$0.427552\pi$
$758$	7.25735	0.263599
$759$	7.35621	0.267014
$760$	0	0
$761$	52.9148	1.91816	0.959080	−	0.283136i	$-0.0913747\pi$
$761$	52.9148	1.91816	0.959080	+	0.283136i	$0.0913747\pi$
$762$	−2.33695	−0.0846589
$763$	0	0
$764$	42.8754	1.55118
$765$	0	0
$766$	−5.03786	−0.182025
$767$	−53.2361	−1.92224
$768$	−4.86163	−0.175429
$769$	8.69161	0.313428	0.156714	−	0.987644i	$-0.449910\pi$
$769$	8.69161	0.313428	0.156714	+	0.987644i	$0.449910\pi$
$770$	0	0
$771$	10.5836	0.381159
$772$	−19.8541	−0.714565
$773$	−15.3500	−0.552102	−0.276051	−	0.961143i	$-0.589026\pi$
$773$	−15.3500	−0.552102	−0.276051	+	0.961143i	$0.589026\pi$
$774$	−4.27051	−0.153500
$775$	0	0
$776$	14.8771	0.534059
$777$	0	0
$778$	−10.6049	−0.380203
$779$	−20.2918	−0.727029
$780$	0	0
$781$	−16.7082	−0.597867
$782$	−10.6460	−0.380700
$783$	10.2333	0.365710
$784$	0	0
$785$	0	0
$786$	−4.94427	−0.176356
$787$	16.3516	0.582871	0.291435	−	0.956591i	$-0.405867\pi$
$787$	16.3516	0.582871	0.291435	+	0.956591i	$0.405867\pi$
$788$	−11.0213	−0.392617
$789$	26.1723	0.931757
$790$	0	0
$791$	0	0
$792$	−7.36068	−0.261550
$793$	−12.2492	−0.434983
$794$	3.78127	0.134192
$795$	0	0
$796$	−46.8152	−1.65932
$797$	39.0277	1.38243	0.691216	−	0.722648i	$-0.257075\pi$
$797$	39.0277	1.38243	0.691216	+	0.722648i	$0.257075\pi$
$798$	0	0
$799$	−70.2492	−2.48524
$800$	0	0
$801$	−17.7658	−0.627723
$802$	−8.81153	−0.311146
$803$	−5.86319	−0.206907
$804$	−14.1120	−0.497693
$805$	0	0
$806$	−10.5836	−0.372791
$807$	13.4164	0.472280
$808$	16.3516	0.575246
$809$	29.0689	1.02201	0.511004	−	0.859578i	$-0.329274\pi$
$809$	29.0689	1.02201	0.511004	+	0.859578i	$0.329274\pi$
$810$	0	0
$811$	7.07107	0.248299	0.124149	−	0.992264i	$-0.460380\pi$
$811$	7.07107	0.248299	0.124149	+	0.992264i	$0.460380\pi$
$812$	0	0
$813$	3.88854	0.136377
$814$	9.14590	0.320564
$815$	0	0
$816$	20.3607	0.712766
$817$	−21.2132	−0.742156
$818$	5.62675	0.196735
$819$	0	0
$820$	0	0
$821$	10.5836	0.369370	0.184685	−	0.982798i	$-0.440874\pi$
$821$	10.5836	0.369370	0.184685	+	0.982798i	$0.440874\pi$
$822$	2.00310	0.0698662
$823$	29.5410	1.02974	0.514868	−	0.857270i	$-0.327841\pi$
$823$	29.5410	1.02974	0.514868	+	0.857270i	$0.327841\pi$
$824$	−13.9659	−0.486525
$825$	0	0
$826$	0	0
$827$	1.47214	0.0511912	0.0255956	−	0.999672i	$-0.491852\pi$
$827$	1.47214	0.0511912	0.0255956	+	0.999672i	$0.491852\pi$
$828$	−15.6049	−0.542307
$829$	10.0757	0.349944	0.174972	−	0.984573i	$-0.444016\pi$
$829$	10.0757	0.349944	0.174972	+	0.984573i	$0.444016\pi$
$830$	0	0
$831$	−20.9768	−0.727676
$832$	−19.0038	−0.658837
$833$	0	0
$834$	−6.54102	−0.226497
$835$	0	0
$836$	−17.5896	−0.608348
$837$	31.4164	1.08591
$838$	−1.85698	−0.0641483
$839$	−31.3190	−1.08125	−0.540626	−	0.841263i	$-0.681813\pi$
$839$	−31.3190	−1.08125	−0.540626	+	0.841263i	$0.681813\pi$
$840$	0	0
$841$	−24.0000	−0.827586
$842$	3.54915	0.122312
$843$	21.4983	0.740442
$844$	−13.7508	−0.473321
$845$	0	0
$846$	8.10272	0.278577
$847$	0	0
$848$	−30.5410	−1.04878
$849$	−18.1803	−0.623948
$850$	0	0
$851$	40.3050	1.38164
$852$	−12.1089	−0.414845
$853$	22.2148	0.760619	0.380309	−	0.924859i	$-0.375818\pi$
$853$	22.2148	0.760619	0.380309	+	0.924859i	$0.375818\pi$
$854$	0	0
$855$	0	0
$856$	−8.83282	−0.301899
$857$	−20.6730	−0.706177	−0.353088	−	0.935590i	$-0.614869\pi$
$857$	−20.6730	−0.706177	−0.353088	+	0.935590i	$0.614869\pi$
$858$	3.01316	0.102867
$859$	39.7742	1.35708	0.678539	−	0.734564i	$-0.262613\pi$
$859$	39.7742	1.35708	0.678539	+	0.734564i	$0.262613\pi$
$860$	0	0
$861$	0	0
$862$	−2.58359	−0.0879975
$863$	1.36068	0.0463181	0.0231590	−	0.999732i	$-0.492628\pi$
$863$	1.36068	0.0463181	0.0231590	+	0.999732i	$0.492628\pi$
$864$	−18.9737	−0.645497
$865$	0	0
$866$	−12.4915	−0.424478
$867$	−33.0671	−1.12302
$868$	0	0
$869$	−10.5279	−0.357133
$870$	0	0
$871$	−35.1490	−1.19098
$872$	−1.47214	−0.0498528
$873$	−22.5973	−0.764803
$874$	6.09962	0.206323
$875$	0	0
$876$	−4.24922	−0.143568
$877$	38.2918	1.29302	0.646511	−	0.762905i	$-0.276227\pi$
$877$	38.2918	1.29302	0.646511	+	0.762905i	$0.276227\pi$
$878$	−6.78593	−0.229014
$879$	16.1115	0.543426
$880$	0	0
$881$	−1.62054	−0.0545975	−0.0272988	−	0.999627i	$-0.508691\pi$
$881$	−1.62054	−0.0545975	−0.0272988	+	0.999627i	$0.508691\pi$
$882$	0	0
$883$	32.7082	1.10072	0.550359	−	0.834928i	$-0.314491\pi$
$883$	32.7082	1.10072	0.550359	+	0.834928i	$0.314491\pi$
$884$	55.4164	1.86386
$885$	0	0
$886$	7.70820	0.258962
$887$	24.9945	0.839232	0.419616	−	0.907702i	$-0.362165\pi$
$887$	24.9945	0.839232	0.419616	+	0.907702i	$0.362165\pi$
$888$	13.7782	0.462366
$889$	0	0
$890$	0	0
$891$	6.05573	0.202875
$892$	−10.8709	−0.363986
$893$	40.2492	1.34689
$894$	3.22248	0.107776
$895$	0	0
$896$	0	0
$897$	13.2786	0.443361
$898$	4.81153	0.160563
$899$	15.3500	0.511952
$900$	0	0
$901$	71.8885	2.39495
$902$	4.08502	0.136016
$903$	0	0
$904$	−14.3738	−0.478067
$905$	0	0
$906$	2.80982	0.0933501
$907$	14.8328	0.492516	0.246258	−	0.969204i	$-0.420799\pi$
$907$	14.8328	0.492516	0.246258	+	0.969204i	$0.420799\pi$
$908$	−49.0547	−1.62794
$909$	−24.8369	−0.823786
$910$	0	0
$911$	21.7639	0.721071	0.360536	−	0.932745i	$-0.382594\pi$
$911$	21.7639	0.721071	0.360536	+	0.932745i	$0.382594\pi$
$912$	−11.6656	−0.386288
$913$	−15.3500	−0.508011
$914$	2.78522	0.0921268
$915$	0	0
$916$	13.1105	0.433182
$917$	0	0
$918$	12.9443	0.427225
$919$	31.8328	1.05007	0.525034	−	0.851081i	$-0.324053\pi$
$919$	31.8328	1.05007	0.525034	+	0.851081i	$0.324053\pi$
$920$	0	0
$921$	12.7214	0.419183
$922$	12.7580	0.420163
$923$	−30.1599	−0.992724
$924$	0	0
$925$	0	0
$926$	−5.34752	−0.175731
$927$	21.2132	0.696733
$928$	−9.27051	−0.304319
$929$	−47.0516	−1.54371	−0.771857	−	0.635797i	$-0.780672\pi$
$929$	−47.0516	−1.54371	−0.771857	+	0.635797i	$0.780672\pi$
$930$	0	0
$931$	0	0
$932$	8.39512	0.274991
$933$	8.36068	0.273716
$934$	−3.68385	−0.120539
$935$	0	0
$936$	−13.2867	−0.434290
$937$	28.4906	0.930747	0.465374	−	0.885114i	$-0.345920\pi$
$937$	28.4906	0.930747	0.465374	+	0.885114i	$0.345920\pi$
$938$	0	0
$939$	18.5410	0.605063
$940$	0	0
$941$	3.62365	0.118128	0.0590638	−	0.998254i	$-0.481188\pi$
$941$	3.62365	0.118128	0.0590638	+	0.998254i	$0.481188\pi$
$942$	1.81966	0.0592877
$943$	18.0022	0.586233
$944$	−41.4922	−1.35046
$945$	0	0
$946$	4.27051	0.138846
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−7.62985	−0.247806
$949$	−10.5836	−0.343558
$950$	0	0
$951$	−10.4397	−0.338530
$952$	0	0
$953$	36.5967	1.18548	0.592742	−	0.805392i	$-0.298045\pi$
$953$	36.5967	1.18548	0.592742	+	0.805392i	$0.298045\pi$
$954$	−8.29180	−0.268457
$955$	0	0
$956$	−54.0000	−1.74648
$957$	−4.37016	−0.141267
$958$	3.44742	0.111381
$959$	0	0
$960$	0	0
$961$	16.1246	0.520149
$962$	16.5092	0.532278
$963$	13.4164	0.432338
$964$	27.3128	0.879687
$965$	0	0
$966$	0	0
$967$	46.2492	1.48727	0.743637	−	0.668583i	$-0.233099\pi$
$967$	46.2492	1.48727	0.743637	+	0.668583i	$0.233099\pi$
$968$	−8.83282	−0.283897
$969$	27.4589	0.882108
$970$	0	0
$971$	−12.7279	−0.408458	−0.204229	−	0.978923i	$-0.565469\pi$
$971$	−12.7279	−0.408458	−0.204229	+	0.978923i	$0.565469\pi$
$972$	29.8446	0.957266
$973$	0	0
$974$	10.9656	0.351359
$975$	0	0
$976$	−9.54704	−0.305593
$977$	27.7639	0.888247	0.444123	−	0.895966i	$-0.353515\pi$
$977$	27.7639	0.888247	0.444123	+	0.895966i	$0.353515\pi$
$978$	0.765117	0.0244658
$979$	17.7658	0.567797
$980$	0	0
$981$	2.23607	0.0713922
$982$	−2.85410	−0.0910781
$983$	−8.10272	−0.258437	−0.129218	−	0.991616i	$-0.541247\pi$
$983$	−8.10272	−0.258437	−0.129218	+	0.991616i	$0.541247\pi$
$984$	6.15403	0.196183
$985$	0	0
$986$	6.32456	0.201415
$987$	0	0
$988$	−31.7508	−1.01013
$989$	18.8197	0.598430
$990$	0	0
$991$	−5.00000	−0.158830	−0.0794151	−	0.996842i	$-0.525305\pi$
$991$	−5.00000	−0.158830	−0.0794151	+	0.996842i	$0.525305\pi$
$992$	−28.4605	−0.903622
$993$	1.12907	0.0358300
$994$	0	0
$995$	0	0
$996$	−11.1246	−0.352497
$997$	48.6722	1.54146	0.770731	−	0.637160i	$-0.219891\pi$
$997$	48.6722	1.54146	0.770731	+	0.637160i	$0.219891\pi$
$998$	9.70820	0.307308
$999$	−49.0060	−1.55048

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.a.ba.1.3	✓	4
5.2	odd	4		1225.2.b.n.99.4		8
5.3	odd	4		1225.2.b.n.99.5		8
5.4	even	2		1225.2.a.bc.1.2	yes	4
7.6	odd	2	inner	1225.2.a.ba.1.4	yes	4
35.13	even	4		1225.2.b.n.99.6		8
35.27	even	4		1225.2.b.n.99.3		8
35.34	odd	2		1225.2.a.bc.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1225.2.a.ba.1.3	✓	4	1.1	even	1	trivial
1225.2.a.ba.1.4	yes	4	7.6	odd	2	inner
1225.2.a.bc.1.1	yes	4	35.34	odd	2
1225.2.a.bc.1.2	yes	4	5.4	even	2
1225.2.b.n.99.3		8	35.27	even	4
1225.2.b.n.99.4		8	5.2	odd	4
1225.2.b.n.99.5		8	5.3	odd	4
1225.2.b.n.99.6		8	35.13	even	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381966 q^{2} -0.874032 q^{3} -1.85410 q^{4} +0.333851 q^{6} +1.47214 q^{8} -2.23607 q^{9} +O(q^{10})\)	\(q-0.381966 q^{2} -0.874032 q^{3} -1.85410 q^{4} +0.333851 q^{6} +1.47214 q^{8} -2.23607 q^{9} +2.23607 q^{11} +1.62054 q^{12} +4.03631 q^{13} +3.14590 q^{16} -7.40492 q^{17} +0.854102 q^{18} +4.24264 q^{19} -0.854102 q^{22} -3.76393 q^{23} -1.28669 q^{24} -1.54173 q^{26} +4.57649 q^{27} +2.23607 q^{29} +6.86474 q^{31} -4.14590 q^{32} -1.95440 q^{33} +2.82843 q^{34} +4.14590 q^{36} -10.7082 q^{37} -1.62054 q^{38} -3.52786 q^{39} -4.78282 q^{41} -5.00000 q^{43} -4.14590 q^{44} +1.43769 q^{46} +9.48683 q^{47} -2.74962 q^{48} +6.47214 q^{51} -7.48373 q^{52} -9.70820 q^{53} -1.74806 q^{54} -3.70820 q^{57} -0.854102 q^{58} -13.1893 q^{59} -3.03476 q^{61} -2.62210 q^{62} -4.70820 q^{64} +0.746512 q^{66} -8.70820 q^{67} +13.7295 q^{68} +3.28980 q^{69} -7.47214 q^{71} -3.29180 q^{72} -2.62210 q^{73} +4.09017 q^{74} -7.86629 q^{76} +1.34752 q^{78} -4.70820 q^{79} +2.70820 q^{81} +1.82688 q^{82} -6.86474 q^{83} +1.90983 q^{86} -1.95440 q^{87} +3.29180 q^{88} +7.94510 q^{89} +6.97871 q^{92} -6.00000 q^{93} -3.62365 q^{94} +3.62365 q^{96} +10.1058 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} + 6 q^{4} - 12 q^{8}+O(q^{10}) \) 4 * q - 6 * q^2 + 6 * q^4 - 12 * q^8	\( 4 q - 6 q^{2} + 6 q^{4} - 12 q^{8} + 26 q^{16} - 10 q^{18} + 10 q^{22} - 24 q^{23} - 30 q^{32} + 30 q^{36} - 16 q^{37} - 32 q^{39} - 20 q^{43} - 30 q^{44} + 46 q^{46} + 8 q^{51} - 12 q^{53} + 12 q^{57} + 10 q^{58} + 8 q^{64} - 8 q^{67} - 12 q^{71} - 40 q^{72} - 6 q^{74} + 68 q^{78} + 8 q^{79} - 16 q^{81} + 30 q^{86} + 40 q^{88} - 66 q^{92} - 24 q^{93} - 20 q^{99}+O(q^{100}) \) 4 * q - 6 * q^2 + 6 * q^4 - 12 * q^8 + 26 * q^16 - 10 * q^18 + 10 * q^22 - 24 * q^23 - 30 * q^32 + 30 * q^36 - 16 * q^37 - 32 * q^39 - 20 * q^43 - 30 * q^44 + 46 * q^46 + 8 * q^51 - 12 * q^53 + 12 * q^57 + 10 * q^58 + 8 * q^64 - 8 * q^67 - 12 * q^71 - 40 * q^72 - 6 * q^74 + 68 * q^78 + 8 * q^79 - 16 * q^81 + 30 * q^86 + 40 * q^88 - 66 * q^92 - 24 * q^93 - 20 * q^99

Introduction

L-functions

Modular forms

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