LMFDB - Embedded newform 1225.4.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$72.2773397570$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1163891200.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 $ x^6 - 2x^5 - 23x^4 + 12x^3 + 154x^2 + 152*x + 28
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 7 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.241849$ 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)

$f(q)$	$=$	$q-1.65606 q^{2} +0.332888 q^{3} -5.25746 q^{4} -0.551283 q^{6} +21.9552 q^{8} -26.8892 q^{9} +O(q^{10})$	\(q-1.65606 q^{2} +0.332888 q^{3} -5.25746 q^{4} -0.551283 q^{6} +21.9552 q^{8} -26.8892 q^{9} +69.5726 q^{11} -1.75014 q^{12} -68.4326 q^{13} +5.70053 q^{16} -104.332 q^{17} +44.5302 q^{18} +71.8929 q^{19} -115.217 q^{22} +101.031 q^{23} +7.30861 q^{24} +113.329 q^{26} -17.9390 q^{27} -114.661 q^{29} -73.6505 q^{31} -185.082 q^{32} +23.1599 q^{33} +172.780 q^{34} +141.369 q^{36} +200.933 q^{37} -119.059 q^{38} -22.7803 q^{39} +417.308 q^{41} -311.175 q^{43} -365.775 q^{44} -167.313 q^{46} -149.697 q^{47} +1.89763 q^{48} -34.7307 q^{51} +359.781 q^{52} -271.474 q^{53} +29.7082 q^{54} +23.9323 q^{57} +189.885 q^{58} -518.028 q^{59} -219.926 q^{61} +121.970 q^{62} +260.903 q^{64} -38.3542 q^{66} -80.6950 q^{67} +548.520 q^{68} +33.6319 q^{69} -91.0463 q^{71} -590.357 q^{72} -882.282 q^{73} -332.758 q^{74} -377.974 q^{76} +37.7257 q^{78} +599.877 q^{79} +720.036 q^{81} -691.087 q^{82} -70.8820 q^{83} +515.325 q^{86} -38.1691 q^{87} +1527.48 q^{88} +802.592 q^{89} -531.165 q^{92} -24.5173 q^{93} +247.908 q^{94} -61.6114 q^{96} -145.648 q^{97} -1870.75 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} - 16 q^{3} + 14 q^{4} + 24 q^{6} + 66 q^{8} + 70 q^{9}+O(q^{10}) $ 6 * q + 2 * q^2 - 16 * q^3 + 14 * q^4 + 24 * q^6 + 66 * q^8 + 70 * q^9	$ 6 q + 2 q^{2} - 16 q^{3} + 14 q^{4} + 24 q^{6} + 66 q^{8} + 70 q^{9} - 16 q^{11} - 160 q^{12} - 168 q^{13} + 298 q^{16} + 4 q^{17} - 354 q^{18} + 308 q^{19} + 236 q^{22} + 336 q^{23} - 92 q^{24} + 56 q^{26} - 964 q^{27} + 176 q^{29} + 392 q^{31} + 770 q^{32} - 188 q^{33} + 812 q^{34} + 230 q^{36} + 140 q^{37} - 20 q^{38} + 140 q^{39} + 656 q^{41} + 388 q^{43} - 160 q^{44} - 388 q^{46} - 628 q^{47} - 1396 q^{48} + 744 q^{51} - 1520 q^{52} + 676 q^{53} + 2284 q^{54} - 1468 q^{57} + 2012 q^{58} + 996 q^{59} + 740 q^{61} + 364 q^{62} + 1426 q^{64} - 3620 q^{66} - 1768 q^{67} + 2940 q^{68} - 1048 q^{69} - 224 q^{71} - 2858 q^{72} - 2640 q^{73} + 928 q^{74} - 1340 q^{76} - 8 q^{78} + 1636 q^{79} + 4442 q^{81} + 1756 q^{82} - 140 q^{83} + 1180 q^{86} - 1940 q^{87} + 5652 q^{88} - 1904 q^{89} + 1952 q^{92} + 1592 q^{93} - 3332 q^{94} - 6460 q^{96} - 516 q^{97} - 2804 q^{99}+O(q^{100}) $ 6 * q + 2 * q^2 - 16 * q^3 + 14 * q^4 + 24 * q^6 + 66 * q^8 + 70 * q^9 - 16 * q^11 - 160 * q^12 - 168 * q^13 + 298 * q^16 + 4 * q^17 - 354 * q^18 + 308 * q^19 + 236 * q^22 + 336 * q^23 - 92 * q^24 + 56 * q^26 - 964 * q^27 + 176 * q^29 + 392 * q^31 + 770 * q^32 - 188 * q^33 + 812 * q^34 + 230 * q^36 + 140 * q^37 - 20 * q^38 + 140 * q^39 + 656 * q^41 + 388 * q^43 - 160 * q^44 - 388 * q^46 - 628 * q^47 - 1396 * q^48 + 744 * q^51 - 1520 * q^52 + 676 * q^53 + 2284 * q^54 - 1468 * q^57 + 2012 * q^58 + 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 - 3620 * q^66 - 1768 * q^67 + 2940 * q^68 - 1048 * q^69 - 224 * q^71 - 2858 * q^72 - 2640 * q^73 + 928 * q^74 - 1340 * q^76 - 8 * q^78 + 1636 * q^79 + 4442 * q^81 + 1756 * q^82 - 140 * q^83 + 1180 * q^86 - 1940 * q^87 + 5652 * q^88 - 1904 * q^89 + 1952 * q^92 + 1592 * q^93 - 3332 * q^94 - 6460 * q^96 - 516 * q^97 - 2804 * 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 $

$2$	−1.65606	−0.585506	−0.292753	−	0.956188i	$-0.594571\pi$
$2$	−1.65606	−0.585506	−0.292753	+	0.956188i	$0.594571\pi$
$3$	0.332888	0.0640643	0.0320321	−	0.999487i	$-0.489802\pi$
$3$	0.332888	0.0640643	0.0320321	+	0.999487i	$0.489802\pi$
$4$	−5.25746	−0.657182
$5$	0	0
$5$	0	0
$6$	−0.551283	−0.0375100
$7$	0	0
$7$	0	0
$8$	21.9552	0.970291
$9$	−26.8892	−0.995896
$10$	0	0
$11$	69.5726	1.90699	0.953497	−	0.301402i	$-0.0974546\pi$
$11$	69.5726	1.90699	0.953497	+	0.301402i	$0.0974546\pi$
$12$	−1.75014	−0.0421019
$13$	−68.4326	−1.45998	−0.729991	−	0.683456i	$-0.760476\pi$
$13$	−68.4326	−1.45998	−0.729991	+	0.683456i	$0.760476\pi$
$14$	0	0
$15$	0	0
$16$	5.70053	0.0890707
$17$	−104.332	−1.48848	−0.744240	−	0.667912i	$-0.767188\pi$
$17$	−104.332	−1.48848	−0.744240	+	0.667912i	$0.767188\pi$
$18$	44.5302	0.583103
$19$	71.8929	0.868072	0.434036	−	0.900896i	$-0.357089\pi$
$19$	71.8929	0.868072	0.434036	+	0.900896i	$0.357089\pi$
$20$	0	0
$21$	0	0
$22$	−115.217	−1.11656
$23$	101.031	0.915929	0.457964	−	0.888970i	$-0.348579\pi$
$23$	101.031	0.915929	0.457964	+	0.888970i	$0.348579\pi$
$24$	7.30861	0.0621610
$25$	0	0
$26$	113.329	0.854829
$27$	−17.9390	−0.127866
$28$	0	0
$29$	−114.661	−0.734205	−0.367102	−	0.930181i	$-0.619650\pi$
$29$	−114.661	−0.734205	−0.367102	+	0.930181i	$0.619650\pi$
$30$	0	0
$31$	−73.6505	−0.426710	−0.213355	−	0.976975i	$-0.568439\pi$
$31$	−73.6505	−0.426710	−0.213355	+	0.976975i	$0.568439\pi$
$32$	−185.082	−1.02244
$33$	23.1599	0.122170
$34$	172.780	0.871514
$35$	0	0
$36$	141.369	0.654485
$37$	200.933	0.892790	0.446395	−	0.894836i	$-0.352708\pi$
$37$	200.933	0.892790	0.446395	+	0.894836i	$0.352708\pi$
$38$	−119.059	−0.508262
$39$	−22.7803	−0.0935327
$40$	0	0
$41$	417.308	1.58957	0.794786	−	0.606889i	$-0.207583\pi$
$41$	417.308	1.58957	0.794786	+	0.606889i	$0.207583\pi$
$42$	0	0
$43$	−311.175	−1.10357	−0.551787	−	0.833985i	$-0.686054\pi$
$43$	−311.175	−1.10357	−0.551787	+	0.833985i	$0.686054\pi$
$44$	−365.775	−1.25324
$45$	0	0
$46$	−167.313	−0.536282
$47$	−149.697	−0.464586	−0.232293	−	0.972646i	$-0.574623\pi$
$47$	−149.697	−0.464586	−0.232293	+	0.972646i	$0.574623\pi$
$48$	1.89763	0.00570625
$49$	0	0
$50$	0	0
$51$	−34.7307	−0.0953583
$52$	359.781	0.959475
$53$	−271.474	−0.703582	−0.351791	−	0.936078i	$-0.614427\pi$
$53$	−271.474	−0.703582	−0.351791	+	0.936078i	$0.614427\pi$
$54$	29.7082	0.0748661
$55$	0	0
$56$	0	0
$57$	23.9323	0.0556124
$58$	189.885	0.429882
$59$	−518.028	−1.14308	−0.571538	−	0.820575i	$-0.693653\pi$
$59$	−518.028	−1.14308	−0.571538	+	0.820575i	$0.693653\pi$
$60$	0	0
$61$	−219.926	−0.461616	−0.230808	−	0.972999i	$-0.574137\pi$
$61$	−219.926	−0.461616	−0.230808	+	0.972999i	$0.574137\pi$
$62$	121.970	0.249842
$63$	0	0
$64$	260.903	0.509576
$65$	0	0
$66$	−38.3542	−0.0715314
$67$	−80.6950	−0.147141	−0.0735706	−	0.997290i	$-0.523439\pi$
$67$	−80.6950	−0.147141	−0.0735706	+	0.997290i	$0.523439\pi$
$68$	548.520	0.978202
$69$	33.6319	0.0586783
$70$	0	0
$71$	−91.0463	−0.152186	−0.0760930	−	0.997101i	$-0.524245\pi$
$71$	−91.0463	−0.152186	−0.0760930	+	0.997101i	$0.524245\pi$
$72$	−590.357	−0.966309
$73$	−882.282	−1.41457	−0.707283	−	0.706931i	$-0.750079\pi$
$73$	−882.282	−1.41457	−0.707283	+	0.706931i	$0.750079\pi$
$74$	−332.758	−0.522734
$75$	0	0
$76$	−377.974	−0.570481
$77$	0	0
$78$	37.7257	0.0547640
$79$	599.877	0.854322	0.427161	−	0.904175i	$-0.359514\pi$
$79$	599.877	0.854322	0.427161	+	0.904175i	$0.359514\pi$
$80$	0	0
$81$	720.036	0.987704
$82$	−691.087	−0.930705
$83$	−70.8820	−0.0937387	−0.0468694	−	0.998901i	$-0.514924\pi$
$83$	−70.8820	−0.0937387	−0.0468694	+	0.998901i	$0.514924\pi$
$84$	0	0
$85$	0	0
$86$	515.325	0.646150
$87$	−38.1691	−0.0470363
$88$	1527.48	1.85034
$89$	802.592	0.955894	0.477947	−	0.878389i	$-0.341381\pi$
$89$	802.592	0.955894	0.477947	+	0.878389i	$0.341381\pi$
$90$	0	0
$91$	0	0
$92$	−531.165	−0.601932
$93$	−24.5173	−0.0273369
$94$	247.908	0.272018
$95$	0	0
$96$	−61.6114	−0.0655020
$97$	−145.648	−0.152457	−0.0762283	−	0.997090i	$-0.524288\pi$
$97$	−145.648	−0.152457	−0.0762283	+	0.997090i	$0.524288\pi$
$98$	0	0
$99$	−1870.75	−1.89917
$100$	0	0
$101$	−619.435	−0.610259	−0.305129	−	0.952311i	$-0.598700\pi$
$101$	−619.435	−0.610259	−0.305129	+	0.952311i	$0.598700\pi$
$102$	57.5163	0.0558329
$103$	1822.08	1.74306	0.871528	−	0.490345i	$-0.163129\pi$
$103$	1822.08	1.74306	0.871528	+	0.490345i	$0.163129\pi$
$104$	−1502.45	−1.41661
$105$	0	0
$106$	449.578	0.411952
$107$	−1089.70	−0.984536	−0.492268	−	0.870444i	$-0.663832\pi$
$107$	−1089.70	−0.984536	−0.492268	+	0.870444i	$0.663832\pi$
$108$	94.3138	0.0840310
$109$	589.667	0.518164	0.259082	−	0.965855i	$-0.416580\pi$
$109$	589.667	0.518164	0.259082	+	0.965855i	$0.416580\pi$
$110$	0	0
$111$	66.8882	0.0571959
$112$	0	0
$113$	900.358	0.749544	0.374772	−	0.927117i	$-0.377721\pi$
$113$	900.358	0.749544	0.374772	+	0.927117i	$0.377721\pi$
$114$	−39.6333	−0.0325614
$115$	0	0
$116$	602.823	0.482506
$117$	1840.10	1.45399
$118$	857.887	0.669279
$119$	0	0
$120$	0	0
$121$	3509.35	2.63663
$122$	364.210	0.270279
$123$	138.917	0.101835
$124$	387.214	0.280426
$125$	0	0
$126$	0	0
$127$	1755.75	1.22676	0.613378	−	0.789790i	$-0.289810\pi$
$127$	1755.75	1.22676	0.613378	+	0.789790i	$0.289810\pi$
$128$	1048.58	0.724082
$129$	−103.586	−0.0706996
$130$	0	0
$131$	1809.10	1.20658	0.603289	−	0.797523i	$-0.293857\pi$
$131$	1809.10	1.20658	0.603289	+	0.797523i	$0.293857\pi$
$132$	−121.762	−0.0802881
$133$	0	0
$134$	133.636	0.0861521
$135$	0	0
$136$	−2290.62	−1.44426
$137$	18.5134	0.0115453	0.00577265	−	0.999983i	$-0.498162\pi$
$137$	18.5134	0.0115453	0.00577265	+	0.999983i	$0.498162\pi$
$138$	−55.6965	−0.0343565
$139$	−625.608	−0.381751	−0.190875	−	0.981614i	$-0.561133\pi$
$139$	−625.608	−0.381751	−0.190875	+	0.981614i	$0.561133\pi$
$140$	0	0
$141$	−49.8323	−0.0297634
$142$	150.778	0.0891059
$143$	−4761.03	−2.78418
$144$	−153.283	−0.0887052
$145$	0	0
$146$	1461.11	0.828237
$147$	0	0
$148$	−1056.40	−0.586725
$149$	−1028.63	−0.565563	−0.282782	−	0.959184i	$-0.591257\pi$
$149$	−1028.63	−0.565563	−0.282782	+	0.959184i	$0.591257\pi$
$150$	0	0
$151$	71.0073	0.0382682	0.0191341	−	0.999817i	$-0.493909\pi$
$151$	71.0073	0.0382682	0.0191341	+	0.999817i	$0.493909\pi$
$152$	1578.42	0.842282
$153$	2805.39	1.48237
$154$	0	0
$155$	0	0
$156$	119.767	0.0614680
$157$	2061.32	1.04784	0.523922	−	0.851767i	$-0.324468\pi$
$157$	2061.32	1.04784	0.523922	+	0.851767i	$0.324468\pi$
$158$	−993.434	−0.500211
$159$	−90.3704	−0.0450745
$160$	0	0
$161$	0	0
$162$	−1192.42	−0.578307
$163$	1963.80	0.943660	0.471830	−	0.881690i	$-0.343594\pi$
$163$	1963.80	0.943660	0.471830	+	0.881690i	$0.343594\pi$
$164$	−2193.98	−1.04464
$165$	0	0
$166$	117.385	0.0548846
$167$	−2855.04	−1.32293	−0.661467	−	0.749974i	$-0.730066\pi$
$167$	−2855.04	−1.32293	−0.661467	+	0.749974i	$0.730066\pi$
$168$	0	0
$169$	2486.01	1.13155
$170$	0	0
$171$	−1933.14	−0.864509
$172$	1635.99	0.725249
$173$	1553.21	0.682590	0.341295	−	0.939956i	$-0.389134\pi$
$173$	1553.21	0.682590	0.341295	+	0.939956i	$0.389134\pi$
$174$	63.2104	0.0275400
$175$	0	0
$176$	396.601	0.169857
$177$	−172.445	−0.0732303
$178$	−1329.14	−0.559682
$179$	269.841	0.112675	0.0563376	−	0.998412i	$-0.482058\pi$
$179$	269.841	0.112675	0.0563376	+	0.998412i	$0.482058\pi$
$180$	0	0
$181$	2229.61	0.915613	0.457806	−	0.889052i	$-0.348635\pi$
$181$	2229.61	0.915613	0.457806	+	0.889052i	$0.348635\pi$
$182$	0	0
$183$	−73.2105	−0.0295731
$184$	2218.15	0.888717
$185$	0	0
$186$	40.6022	0.0160059
$187$	−7258.63	−2.83852
$188$	787.026	0.305318
$189$	0	0
$190$	0	0
$191$	−465.920	−0.176507	−0.0882533	−	0.996098i	$-0.528129\pi$
$191$	−465.920	−0.176507	−0.0882533	+	0.996098i	$0.528129\pi$
$192$	86.8513	0.0326456
$193$	4414.46	1.64642	0.823212	−	0.567734i	$-0.192180\pi$
$193$	4414.46	1.64642	0.823212	+	0.567734i	$0.192180\pi$
$194$	241.202	0.0892643
$195$	0	0
$196$	0	0
$197$	289.812	0.104814	0.0524068	−	0.998626i	$-0.483311\pi$
$197$	289.812	0.104814	0.0524068	+	0.998626i	$0.483311\pi$
$198$	3098.08	1.11197
$199$	4817.73	1.71618	0.858091	−	0.513498i	$-0.171651\pi$
$199$	4817.73	1.71618	0.858091	+	0.513498i	$0.171651\pi$
$200$	0	0
$201$	−26.8624	−0.00942649
$202$	1025.82	0.357310
$203$	0	0
$204$	182.595	0.0626678
$205$	0	0
$206$	−3017.48	−1.02057
$207$	−2716.63	−0.912170
$208$	−390.102	−0.130042
$209$	5001.78	1.65541
$210$	0	0
$211$	2022.01	0.659719	0.329859	−	0.944030i	$-0.392999\pi$
$211$	2022.01	0.659719	0.329859	+	0.944030i	$0.392999\pi$
$212$	1427.26	0.462382
$213$	−30.3082	−0.00974968
$214$	1804.61	0.576452
$215$	0	0
$216$	−393.855	−0.124067
$217$	0	0
$218$	−976.525	−0.303388
$219$	−293.701	−0.0906231
$220$	0	0
$221$	7139.69	2.17315
$222$	−110.771	−0.0334886
$223$	−4343.86	−1.30442	−0.652211	−	0.758037i	$-0.726158\pi$
$223$	−4343.86	−1.30442	−0.652211	+	0.758037i	$0.726158\pi$
$224$	0	0
$225$	0	0
$226$	−1491.05	−0.438863
$227$	2647.59	0.774127	0.387064	−	0.922053i	$-0.373489\pi$
$227$	2647.59	0.774127	0.387064	+	0.922053i	$0.373489\pi$
$228$	−125.823	−0.0365475
$229$	−1445.09	−0.417006	−0.208503	−	0.978022i	$-0.566859\pi$
$229$	−1445.09	−0.417006	−0.208503	+	0.978022i	$0.566859\pi$
$230$	0	0
$231$	0	0
$232$	−2517.39	−0.712392
$233$	6245.91	1.75615	0.878076	−	0.478522i	$-0.158827\pi$
$233$	6245.91	1.75615	0.878076	+	0.478522i	$0.158827\pi$
$234$	−3047.31	−0.851321
$235$	0	0
$236$	2723.51	0.751209
$237$	199.692	0.0547315
$238$	0	0
$239$	1340.24	0.362731	0.181366	−	0.983416i	$-0.441948\pi$
$239$	1340.24	0.362731	0.181366	+	0.983416i	$0.441948\pi$
$240$	0	0
$241$	3369.92	0.900729	0.450364	−	0.892845i	$-0.351294\pi$
$241$	3369.92	0.900729	0.450364	+	0.892845i	$0.351294\pi$
$242$	−5811.70	−1.54376
$243$	724.045	0.191142
$244$	1156.25	0.303366
$245$	0	0
$246$	−230.054	−0.0596249
$247$	−4919.82	−1.26737
$248$	−1617.01	−0.414033
$249$	−23.5958	−0.00600530
$250$	0	0
$251$	−3592.64	−0.903449	−0.451724	−	0.892158i	$-0.649191\pi$
$251$	−3592.64	−0.903449	−0.451724	+	0.892158i	$0.649191\pi$
$252$	0	0
$253$	7028.97	1.74667
$254$	−2907.64	−0.718273
$255$	0	0
$256$	−3823.74	−0.933531
$257$	2.84763	0.000691167 0	0.000345584	−	1.00000i	$-0.499890\pi$
$257$	2.84763	0.000691167 0	0.000345584		1.00000i	$0.499890\pi$
$258$	171.545	0.0413951
$259$	0	0
$260$	0	0
$261$	3083.13	0.731191
$262$	−2995.98	−0.706459
$263$	2817.07	0.660488	0.330244	−	0.943896i	$-0.392869\pi$
$263$	2817.07	0.660488	0.330244	+	0.943896i	$0.392869\pi$
$264$	508.479	0.118541
$265$	0	0
$266$	0	0
$267$	267.173	0.0612386
$268$	424.250	0.0966986
$269$	−1447.43	−0.328072	−0.164036	−	0.986454i	$-0.552451\pi$
$269$	−1447.43	−0.328072	−0.164036	+	0.986454i	$0.552451\pi$
$270$	0	0
$271$	8054.50	1.80545	0.902724	−	0.430221i	$-0.141564\pi$
$271$	8054.50	1.80545	0.902724	+	0.430221i	$0.141564\pi$
$272$	−594.746	−0.132580
$273$	0	0
$274$	−30.6593	−0.00675985
$275$	0	0
$276$	−176.818	−0.0385623
$277$	571.300	0.123921	0.0619604	−	0.998079i	$-0.480265\pi$
$277$	571.300	0.123921	0.0619604	+	0.998079i	$0.480265\pi$
$278$	1036.05	0.223518
$279$	1980.40	0.424959
$280$	0	0
$281$	−1784.48	−0.378837	−0.189418	−	0.981896i	$-0.560660\pi$
$281$	−1784.48	−0.378837	−0.189418	+	0.981896i	$0.560660\pi$
$282$	82.5254	0.0174267
$283$	−3321.05	−0.697582	−0.348791	−	0.937201i	$-0.613408\pi$
$283$	−3321.05	−0.697582	−0.348791	+	0.937201i	$0.613408\pi$
$284$	478.672	0.100014
$285$	0	0
$286$	7884.57	1.63015
$287$	0	0
$288$	4976.70	1.01825
$289$	5972.11	1.21557
$290$	0	0
$291$	−48.4843	−0.00976701
$292$	4638.56	0.929627
$293$	5049.54	1.00682	0.503408	−	0.864049i	$-0.332079\pi$
$293$	5049.54	1.00682	0.503408	+	0.864049i	$0.332079\pi$
$294$	0	0
$295$	0	0
$296$	4411.52	0.866266
$297$	−1248.07	−0.243839
$298$	1703.48	0.331141
$299$	−6913.79	−1.33724
$300$	0	0
$301$	0	0
$302$	−117.593	−0.0224063
$303$	−206.202	−0.0390958
$304$	409.828	0.0773198
$305$	0	0
$306$	−4645.91	−0.867938
$307$	1535.73	0.285500	0.142750	−	0.989759i	$-0.454405\pi$
$307$	1535.73	0.285500	0.142750	+	0.989759i	$0.454405\pi$
$308$	0	0
$309$	606.548	0.111668
$310$	0	0
$311$	−9283.05	−1.69258	−0.846291	−	0.532720i	$-0.821170\pi$
$311$	−9283.05	−1.69258	−0.846291	+	0.532720i	$0.821170\pi$
$312$	−500.147	−0.0907539
$313$	−6025.43	−1.08811	−0.544054	−	0.839050i	$-0.683111\pi$
$313$	−6025.43	−1.08811	−0.544054	+	0.839050i	$0.683111\pi$
$314$	−3413.68	−0.613519
$315$	0	0
$316$	−3153.83	−0.561445
$317$	6977.58	1.23628	0.618139	−	0.786069i	$-0.287887\pi$
$317$	6977.58	1.23628	0.618139	+	0.786069i	$0.287887\pi$
$318$	149.659	0.0263914
$319$	−7977.24	−1.40012
$320$	0	0
$321$	−362.748	−0.0630736
$322$	0	0
$323$	−7500.71	−1.29211
$324$	−3785.56	−0.649102
$325$	0	0
$326$	−3252.17	−0.552519
$327$	196.293	0.0331958
$328$	9162.06	1.54235
$329$	0	0
$330$	0	0
$331$	984.878	0.163546	0.0817731	−	0.996651i	$-0.473942\pi$
$331$	984.878	0.163546	0.0817731	+	0.996651i	$0.473942\pi$
$332$	372.659	0.0616034
$333$	−5402.93	−0.889125
$334$	4728.13	0.774586
$335$	0	0
$336$	0	0
$337$	−51.9653	−0.00839979	−0.00419990	−	0.999991i	$-0.501337\pi$
$337$	−51.9653	−0.00839979	−0.00419990	+	0.999991i	$0.501337\pi$
$338$	−4116.99	−0.662529
$339$	299.718	0.0480190
$340$	0	0
$341$	−5124.06	−0.813734
$342$	3201.40	0.506176
$343$	0	0
$344$	−6831.89	−1.07079
$345$	0	0
$346$	−2572.21	−0.399661
$347$	11300.5	1.74825	0.874123	−	0.485704i	$-0.161437\pi$
$347$	11300.5	1.74825	0.874123	+	0.485704i	$0.161437\pi$
$348$	200.672	0.0309114
$349$	−2016.91	−0.309349	−0.154674	−	0.987966i	$-0.549433\pi$
$349$	−2016.91	−0.309349	−0.154674	+	0.987966i	$0.549433\pi$
$350$	0	0
$351$	1227.61	0.186682
$352$	−12876.6	−1.94979
$353$	7589.41	1.14432	0.572158	−	0.820143i	$-0.306106\pi$
$353$	7589.41	1.14432	0.572158	+	0.820143i	$0.306106\pi$
$354$	285.580	0.0428768
$355$	0	0
$356$	−4219.59	−0.628196
$357$	0	0
$358$	−446.873	−0.0659720
$359$	8734.24	1.28405	0.642027	−	0.766682i	$-0.278094\pi$
$359$	8734.24	1.28405	0.642027	+	0.766682i	$0.278094\pi$
$360$	0	0
$361$	−1690.41	−0.246451
$362$	−3692.38	−0.536097
$363$	1168.22	0.168914
$364$	0	0
$365$	0	0
$366$	121.241	0.0173152
$367$	1890.54	0.268898	0.134449	−	0.990921i	$-0.457074\pi$
$367$	1890.54	0.268898	0.134449	+	0.990921i	$0.457074\pi$
$368$	575.928	0.0815825
$369$	−11221.1	−1.58305
$370$	0	0
$371$	0	0
$372$	128.899	0.0179653
$373$	−2713.88	−0.376728	−0.188364	−	0.982099i	$-0.560318\pi$
$373$	−2713.88	−0.376728	−0.188364	+	0.982099i	$0.560318\pi$
$374$	12020.7	1.66197
$375$	0	0
$376$	−3286.63	−0.450784
$377$	7846.52	1.07193
$378$	0	0
$379$	8941.19	1.21182	0.605908	−	0.795535i	$-0.292810\pi$
$379$	8941.19	1.21182	0.605908	+	0.795535i	$0.292810\pi$
$380$	0	0
$381$	584.469	0.0785912
$382$	771.592	0.103346
$383$	9293.88	1.23994	0.619968	−	0.784627i	$-0.287146\pi$
$383$	9293.88	1.23994	0.619968	+	0.784627i	$0.287146\pi$
$384$	349.060	0.0463878
$385$	0	0
$386$	−7310.62	−0.963992
$387$	8367.23	1.09904
$388$	765.737	0.100192
$389$	10454.8	1.36267	0.681333	−	0.731974i	$-0.261401\pi$
$389$	10454.8	1.36267	0.681333	+	0.731974i	$0.261401\pi$
$390$	0	0
$391$	−10540.7	−1.36334
$392$	0	0
$393$	602.226	0.0772985
$394$	−479.947	−0.0613690
$395$	0	0
$396$	9835.40	1.24810
$397$	3626.03	0.458401	0.229200	−	0.973379i	$-0.426389\pi$
$397$	3626.03	0.458401	0.229200	+	0.973379i	$0.426389\pi$
$398$	−7978.47	−1.00484
$399$	0	0
$400$	0	0
$401$	−8422.52	−1.04888	−0.524440	−	0.851448i	$-0.675725\pi$
$401$	−8422.52	−1.04888	−0.524440	+	0.851448i	$0.675725\pi$
$402$	44.4857	0.00551927
$403$	5040.09	0.622989
$404$	3256.65	0.401051
$405$	0	0
$406$	0	0
$407$	13979.5	1.70254
$408$	−762.519	−0.0925253
$409$	−14580.7	−1.76276	−0.881379	−	0.472409i	$-0.843384\pi$
$409$	−14580.7	−1.76276	−0.881379	+	0.472409i	$0.843384\pi$
$410$	0	0
$411$	6.16288	0.000739641 0
$412$	−9579.51	−1.14551
$413$	0	0
$414$	4498.92	0.534081
$415$	0	0
$416$	12665.6	1.49275
$417$	−208.257	−0.0244566
$418$	−8283.26	−0.969252
$419$	−2537.53	−0.295863	−0.147931	−	0.988998i	$-0.547261\pi$
$419$	−2537.53	−0.295863	−0.147931	+	0.988998i	$0.547261\pi$
$420$	0	0
$421$	9649.52	1.11708	0.558538	−	0.829479i	$-0.311363\pi$
$421$	9649.52	1.11708	0.558538	+	0.829479i	$0.311363\pi$
$422$	−3348.57	−0.386269
$423$	4025.23	0.462680
$424$	−5960.27	−0.682680
$425$	0	0
$426$	50.1922	0.00570850
$427$	0	0
$428$	5729.06	0.647020
$429$	−1584.89	−0.178366
$430$	0	0
$431$	−7262.56	−0.811660	−0.405830	−	0.913949i	$-0.633017\pi$
$431$	−7262.56	−0.811660	−0.405830	+	0.913949i	$0.633017\pi$
$432$	−102.262	−0.0113891
$433$	−11345.0	−1.25914	−0.629570	−	0.776944i	$-0.716769\pi$
$433$	−11345.0	−1.25914	−0.629570	+	0.776944i	$0.716769\pi$
$434$	0	0
$435$	0	0
$436$	−3100.15	−0.340528
$437$	7263.40	0.795092
$438$	486.387	0.0530604
$439$	11705.9	1.27265	0.636323	−	0.771423i	$-0.280455\pi$
$439$	11705.9	1.27265	0.636323	+	0.771423i	$0.280455\pi$
$440$	0	0
$441$	0	0
$442$	−11823.8	−1.27240
$443$	−15078.0	−1.61710	−0.808551	−	0.588426i	$-0.799748\pi$
$443$	−15078.0	−1.61710	−0.808551	+	0.588426i	$0.799748\pi$
$444$	−351.662	−0.0375881
$445$	0	0
$446$	7193.70	0.763747
$447$	−342.419	−0.0362324
$448$	0	0
$449$	1075.45	0.113037	0.0565185	−	0.998402i	$-0.482000\pi$
$449$	1075.45	0.113037	0.0565185	+	0.998402i	$0.482000\pi$
$450$	0	0
$451$	29033.2	3.03131
$452$	−4733.59	−0.492587
$453$	23.6375	0.00245162
$454$	−4384.58	−0.453257
$455$	0	0
$456$	525.437	0.0539602
$457$	10736.9	1.09902	0.549511	−	0.835487i	$-0.314814\pi$
$457$	10736.9	1.09902	0.549511	+	0.835487i	$0.314814\pi$
$458$	2393.16	0.244159
$459$	1871.61	0.190325
$460$	0	0
$461$	452.568	0.0457228	0.0228614	−	0.999739i	$-0.492722\pi$
$461$	452.568	0.0457228	0.0228614	+	0.999739i	$0.492722\pi$
$462$	0	0
$463$	−7118.15	−0.714489	−0.357244	−	0.934011i	$-0.616284\pi$
$463$	−7118.15	−0.714489	−0.357244	+	0.934011i	$0.616284\pi$
$464$	−653.626	−0.0653962
$465$	0	0
$466$	−10343.6	−1.02824
$467$	−973.800	−0.0964927	−0.0482463	−	0.998835i	$-0.515363\pi$
$467$	−973.800	−0.0964927	−0.0482463	+	0.998835i	$0.515363\pi$
$468$	−9674.23	−0.955537
$469$	0	0
$470$	0	0
$471$	686.188	0.0671293
$472$	−11373.4	−1.10912
$473$	−21649.2	−2.10451
$474$	−330.702	−0.0320457
$475$	0	0
$476$	0	0
$477$	7299.72	0.700695
$478$	−2219.52	−0.212381
$479$	9714.00	0.926606	0.463303	−	0.886200i	$-0.346664\pi$
$479$	9714.00	0.926606	0.463303	+	0.886200i	$0.346664\pi$
$480$	0	0
$481$	−13750.4	−1.30346
$482$	−5580.80	−0.527383
$483$	0	0
$484$	−18450.3	−1.73274
$485$	0	0
$486$	−1199.06	−0.111915
$487$	−923.389	−0.0859194	−0.0429597	−	0.999077i	$-0.513679\pi$
$487$	−923.389	−0.0859194	−0.0429597	+	0.999077i	$0.513679\pi$
$488$	−4828.50	−0.447902
$489$	653.724	0.0604549
$490$	0	0
$491$	1289.11	0.118486	0.0592430	−	0.998244i	$-0.481131\pi$
$491$	1289.11	0.118486	0.0592430	+	0.998244i	$0.481131\pi$
$492$	−730.348	−0.0669240
$493$	11962.7	1.09285
$494$	8147.52	0.742053
$495$	0	0
$496$	−419.846	−0.0380074
$497$	0	0
$498$	39.0760	0.00351614
$499$	−19338.3	−1.73487	−0.867436	−	0.497549i	$-0.834234\pi$
$499$	−19338.3	−1.73487	−0.867436	+	0.497549i	$0.834234\pi$
$500$	0	0
$501$	−950.409	−0.0847528
$502$	5949.64	0.528975
$503$	1772.84	0.157151	0.0785757	−	0.996908i	$-0.474963\pi$
$503$	1772.84	0.157151	0.0785757	+	0.996908i	$0.474963\pi$
$504$	0	0
$505$	0	0
$506$	−11640.4	−1.02269
$507$	827.563	0.0724919
$508$	−9230.80	−0.806202
$509$	3151.75	0.274458	0.137229	−	0.990539i	$-0.456180\pi$
$509$	3151.75	0.274458	0.137229	+	0.990539i	$0.456180\pi$
$510$	0	0
$511$	0	0
$512$	−2056.31	−0.177494
$513$	−1289.69	−0.110997
$514$	−4.71585	−0.000404683 0
$515$	0	0
$516$	544.600	0.0464626
$517$	−10414.8	−0.885964
$518$	0	0
$519$	517.043	0.0437296
$520$	0	0
$521$	−11029.4	−0.927458	−0.463729	−	0.885977i	$-0.653489\pi$
$521$	−11029.4	−0.927458	−0.463729	+	0.885977i	$0.653489\pi$
$522$	−5105.85	−0.428117
$523$	−14448.4	−1.20800	−0.604002	−	0.796983i	$-0.706428\pi$
$523$	−14448.4	−1.20800	−0.604002	+	0.796983i	$0.706428\pi$
$524$	−9511.26	−0.792941
$525$	0	0
$526$	−4665.25	−0.386720
$527$	7684.08	0.635149
$528$	132.023	0.0108818
$529$	−1959.79	−0.161074
$530$	0	0
$531$	13929.4	1.13838
$532$	0	0
$533$	−28557.4	−2.32075
$534$	−442.455	−0.0358556
$535$	0	0
$536$	−1771.67	−0.142770
$537$	89.8266	0.00721845
$538$	2397.04	0.192089
$539$	0	0
$540$	0	0
$541$	−22274.8	−1.77018	−0.885091	−	0.465418i	$-0.845904\pi$
$541$	−22274.8	−1.77018	−0.885091	+	0.465418i	$0.845904\pi$
$542$	−13338.8	−1.05710
$543$	742.211	0.0586580
$544$	19309.9	1.52188
$545$	0	0
$546$	0	0
$547$	−18642.6	−1.45722	−0.728609	−	0.684930i	$-0.759833\pi$
$547$	−18642.6	−1.45722	−0.728609	+	0.684930i	$0.759833\pi$
$548$	−97.3334	−0.00758737
$549$	5913.62	0.459721
$550$	0	0
$551$	−8243.28	−0.637343
$552$	738.394	0.0569350
$553$	0	0
$554$	−946.108	−0.0725565
$555$	0	0
$556$	3289.11	0.250880
$557$	21431.8	1.63033	0.815165	−	0.579229i	$-0.196646\pi$
$557$	21431.8	1.63033	0.815165	+	0.579229i	$0.196646\pi$
$558$	−3279.67	−0.248816
$559$	21294.5	1.61120
$560$	0	0
$561$	−2416.31	−0.181848
$562$	2955.21	0.221811
$563$	6154.85	0.460739	0.230370	−	0.973103i	$-0.426007\pi$
$563$	6154.85	0.460739	0.230370	+	0.973103i	$0.426007\pi$
$564$	261.991	0.0195600
$565$	0	0
$566$	5499.86	0.408439
$567$	0	0
$568$	−1998.94	−0.147665
$569$	−8389.99	−0.618149	−0.309074	−	0.951038i	$-0.600019\pi$
$569$	−8389.99	−0.618149	−0.309074	+	0.951038i	$0.600019\pi$
$570$	0	0
$571$	−600.502	−0.0440109	−0.0220055	−	0.999758i	$-0.507005\pi$
$571$	−600.502	−0.0440109	−0.0220055	+	0.999758i	$0.507005\pi$
$572$	25030.9	1.82971
$573$	−155.099	−0.0113078
$574$	0	0
$575$	0	0
$576$	−7015.46	−0.507484
$577$	−4062.35	−0.293098	−0.146549	−	0.989203i	$-0.546817\pi$
$577$	−4062.35	−0.293098	−0.146549	+	0.989203i	$0.546817\pi$
$578$	−9890.18	−0.711725
$579$	1469.52	0.105477
$580$	0	0
$581$	0	0
$582$	80.2931	0.00571865
$583$	−18887.2	−1.34173
$584$	−19370.6	−1.37254
$585$	0	0
$586$	−8362.35	−0.589498
$587$	8387.50	0.589760	0.294880	−	0.955534i	$-0.404720\pi$
$587$	8387.50	0.589760	0.294880	+	0.955534i	$0.404720\pi$
$588$	0	0
$589$	−5294.95	−0.370415
$590$	0	0
$591$	96.4749	0.00671480
$592$	1145.43	0.0795214
$593$	15342.6	1.06247	0.531236	−	0.847224i	$-0.321728\pi$
$593$	15342.6	1.06247	0.531236	+	0.847224i	$0.321728\pi$
$594$	2066.88	0.142769
$595$	0	0
$596$	5408.00	0.371678
$597$	1603.76	0.109946
$598$	11449.7	0.782963
$599$	−19708.5	−1.34435	−0.672175	−	0.740392i	$-0.734640\pi$
$599$	−19708.5	−1.34435	−0.672175	+	0.740392i	$0.734640\pi$
$600$	0	0
$601$	−19002.5	−1.28973	−0.644866	−	0.764295i	$-0.723087\pi$
$601$	−19002.5	−1.28973	−0.644866	+	0.764295i	$0.723087\pi$
$602$	0	0
$603$	2169.82	0.146537
$604$	−373.318	−0.0251492
$605$	0	0
$606$	341.484	0.0228908
$607$	−29453.1	−1.96946	−0.984732	−	0.174076i	$-0.944306\pi$
$607$	−29453.1	−1.96946	−0.984732	+	0.174076i	$0.944306\pi$
$608$	−13306.1	−0.887553
$609$	0	0
$610$	0	0
$611$	10244.2	0.678288
$612$	−14749.2	−0.974188
$613$	−9987.27	−0.658045	−0.329023	−	0.944322i	$-0.606719\pi$
$613$	−9987.27	−0.658045	−0.329023	+	0.944322i	$0.606719\pi$
$614$	−2543.26	−0.167162
$615$	0	0
$616$	0	0
$617$	21076.1	1.37519	0.687593	−	0.726096i	$-0.258667\pi$
$617$	21076.1	1.37519	0.687593	+	0.726096i	$0.258667\pi$
$618$	−1004.48	−0.0653821
$619$	−314.668	−0.0204323	−0.0102161	−	0.999948i	$-0.503252\pi$
$619$	−314.668	−0.0204323	−0.0102161	+	0.999948i	$0.503252\pi$
$620$	0	0
$621$	−1812.39	−0.117116
$622$	15373.3	0.991018
$623$	0	0
$624$	−129.860	−0.00833103
$625$	0	0
$626$	9978.49	0.637094
$627$	1665.03	0.106052
$628$	−10837.3	−0.688624
$629$	−20963.7	−1.32890
$630$	0	0
$631$	3314.96	0.209138	0.104569	−	0.994518i	$-0.466654\pi$
$631$	3314.96	0.209138	0.104569	+	0.994518i	$0.466654\pi$
$632$	13170.4	0.828941
$633$	673.101	0.0422644
$634$	−11555.3	−0.723849
$635$	0	0
$636$	475.119	0.0296221
$637$	0	0
$638$	13210.8	0.819782
$639$	2448.16	0.151561
$640$	0	0
$641$	3005.12	0.185172	0.0925858	−	0.995705i	$-0.470487\pi$
$641$	3005.12	0.185172	0.0925858	+	0.995705i	$0.470487\pi$
$642$	600.733	0.0369300
$643$	−21225.7	−1.30180	−0.650902	−	0.759162i	$-0.725609\pi$
$643$	−21225.7	−1.30180	−0.650902	+	0.759162i	$0.725609\pi$
$644$	0	0
$645$	0	0
$646$	12421.6	0.756537
$647$	2740.35	0.166514	0.0832568	−	0.996528i	$-0.473468\pi$
$647$	2740.35	0.166514	0.0832568	+	0.996528i	$0.473468\pi$
$648$	15808.5	0.958360
$649$	−36040.6	−2.17984
$650$	0	0
$651$	0	0
$652$	−10324.6	−0.620157
$653$	−22790.7	−1.36580	−0.682900	−	0.730511i	$-0.739282\pi$
$653$	−22790.7	−1.36580	−0.682900	+	0.730511i	$0.739282\pi$
$654$	−325.073	−0.0194363
$655$	0	0
$656$	2378.87	0.141584
$657$	23723.8	1.40876
$658$	0	0
$659$	19405.1	1.14706	0.573532	−	0.819183i	$-0.305573\pi$
$659$	19405.1	1.14706	0.573532	+	0.819183i	$0.305573\pi$
$660$	0	0
$661$	15637.3	0.920150	0.460075	−	0.887880i	$-0.347823\pi$
$661$	15637.3	0.920150	0.460075	+	0.887880i	$0.347823\pi$
$662$	−1631.02	−0.0957574
$663$	2376.71	0.139222
$664$	−1556.23	−0.0909538
$665$	0	0
$666$	8947.59	0.520589
$667$	−11584.2	−0.672479
$668$	15010.3	0.869409
$669$	−1446.02	−0.0835668
$670$	0	0
$671$	−15300.8	−0.880299
$672$	0	0
$673$	2579.54	0.147747	0.0738735	−	0.997268i	$-0.476464\pi$
$673$	2579.54	0.147747	0.0738735	+	0.997268i	$0.476464\pi$
$674$	86.0578	0.00491813
$675$	0	0
$676$	−13070.1	−0.743634
$677$	−8159.56	−0.463216	−0.231608	−	0.972809i	$-0.574399\pi$
$677$	−8159.56	−0.463216	−0.231608	+	0.972809i	$0.574399\pi$
$678$	−496.351	−0.0281154
$679$	0	0
$680$	0	0
$681$	881.351	0.0495939
$682$	8485.76	0.476446
$683$	20529.3	1.15012	0.575059	−	0.818112i	$-0.304979\pi$
$683$	20529.3	1.15012	0.575059	+	0.818112i	$0.304979\pi$
$684$	10163.4	0.568140
$685$	0	0
$686$	0	0
$687$	−481.053	−0.0267151
$688$	−1773.86	−0.0982961
$689$	18577.7	1.02722
$690$	0	0
$691$	−917.295	−0.0505001	−0.0252500	−	0.999681i	$-0.508038\pi$
$691$	−917.295	−0.0505001	−0.0252500	+	0.999681i	$0.508038\pi$
$692$	−8165.92	−0.448586
$693$	0	0
$694$	−18714.3	−1.02361
$695$	0	0
$696$	−838.009	−0.0456389
$697$	−43538.4	−2.36605
$698$	3340.13	0.181126
$699$	2079.19	0.112507
$700$	0	0
$701$	−10491.3	−0.565266	−0.282633	−	0.959228i	$-0.591208\pi$
$701$	−10491.3	−0.565266	−0.282633	+	0.959228i	$0.591208\pi$
$702$	−2033.01	−0.109303
$703$	14445.7	0.775006
$704$	18151.7	0.971758
$705$	0	0
$706$	−12568.5	−0.670005
$707$	0	0
$708$	906.623	0.0481257
$709$	23837.6	1.26268	0.631339	−	0.775507i	$-0.282506\pi$
$709$	23837.6	1.26268	0.631339	+	0.775507i	$0.282506\pi$
$710$	0	0
$711$	−16130.2	−0.850816
$712$	17621.0	0.927495
$713$	−7440.96	−0.390836
$714$	0	0
$715$	0	0
$716$	−1418.68	−0.0740481
$717$	446.148	0.0232381
$718$	−14464.4	−0.751822
$719$	−3926.19	−0.203647	−0.101824	−	0.994802i	$-0.532468\pi$
$719$	−3926.19	−0.203647	−0.101824	+	0.994802i	$0.532468\pi$
$720$	0	0
$721$	0	0
$722$	2799.42	0.144299
$723$	1121.80	0.0577045
$724$	−11722.1	−0.601724
$725$	0	0
$726$	−1934.64	−0.0989000
$727$	21071.2	1.07495	0.537474	−	0.843281i	$-0.319379\pi$
$727$	21071.2	1.07495	0.537474	+	0.843281i	$0.319379\pi$
$728$	0	0
$729$	−19200.0	−0.975459
$730$	0	0
$731$	32465.4	1.64265
$732$	384.901	0.0194349
$733$	22840.7	1.15094	0.575470	−	0.817823i	$-0.304819\pi$
$733$	22840.7	1.15094	0.575470	+	0.817823i	$0.304819\pi$
$734$	−3130.86	−0.157441
$735$	0	0
$736$	−18699.0	−0.936485
$737$	−5614.16	−0.280597
$738$	18582.8	0.926885
$739$	10837.3	0.539456	0.269728	−	0.962937i	$-0.413066\pi$
$739$	10837.3	0.539456	0.269728	+	0.962937i	$0.413066\pi$
$740$	0	0
$741$	−1637.75	−0.0811931
$742$	0	0
$743$	21631.9	1.06810	0.534050	−	0.845453i	$-0.320669\pi$
$743$	21631.9	1.06810	0.534050	+	0.845453i	$0.320669\pi$
$744$	−538.282	−0.0265247
$745$	0	0
$746$	4494.36	0.220577
$747$	1905.96	0.0933540
$748$	38161.9	1.86543
$749$	0	0
$750$	0	0
$751$	−16179.2	−0.786133	−0.393067	−	0.919510i	$-0.628586\pi$
$751$	−16179.2	−0.786133	−0.393067	+	0.919510i	$0.628586\pi$
$752$	−853.352	−0.0413811
$753$	−1195.95	−0.0578788
$754$	−12994.3	−0.627620
$755$	0	0
$756$	0	0
$757$	40930.9	1.96520	0.982601	−	0.185727i	$-0.0594641\pi$
$757$	40930.9	1.96520	0.982601	+	0.185727i	$0.0594641\pi$
$758$	−14807.2	−0.709526
$759$	2339.86	0.111899
$760$	0	0
$761$	3183.97	0.151667	0.0758337	−	0.997120i	$-0.475838\pi$
$761$	3183.97	0.151667	0.0758337	+	0.997120i	$0.475838\pi$
$762$	−967.917	−0.0460157
$763$	0	0
$764$	2449.55	0.115997
$765$	0	0
$766$	−15391.2	−0.725990
$767$	35450.0	1.66887
$768$	−1272.88	−0.0598059
$769$	33595.8	1.57542	0.787708	−	0.616048i	$-0.211267\pi$
$769$	33595.8	1.57542	0.787708	+	0.616048i	$0.211267\pi$
$770$	0	0
$771$	0.947940	4.42791e−5 0
$772$	−23208.8	−1.08200
$773$	−34386.0	−1.59997	−0.799986	−	0.600019i	$-0.795160\pi$
$773$	−34386.0	−1.59997	−0.799986	+	0.600019i	$0.795160\pi$
$774$	−13856.7	−0.643498
$775$	0	0
$776$	−3197.72	−0.147927
$777$	0	0
$778$	−17313.7	−0.797849
$779$	30001.5	1.37986
$780$	0	0
$781$	−6334.33	−0.290218
$782$	17456.1	0.798245
$783$	2056.90	0.0938795
$784$	0	0
$785$	0	0
$786$	−997.324	−0.0452588
$787$	8212.43	0.371972	0.185986	−	0.982552i	$-0.440452\pi$
$787$	8212.43	0.371972	0.185986	+	0.982552i	$0.440452\pi$
$788$	−1523.68	−0.0688816
$789$	937.769	0.0423136
$790$	0	0
$791$	0	0
$792$	−41072.7	−1.84274
$793$	15050.1	0.673951
$794$	−6004.92	−0.268396
$795$	0	0
$796$	−25329.0	−1.12784
$797$	−36798.3	−1.63546	−0.817732	−	0.575600i	$-0.804769\pi$
$797$	−36798.3	−1.63546	−0.817732	+	0.575600i	$0.804769\pi$
$798$	0	0
$799$	15618.2	0.691528
$800$	0	0
$801$	−21581.0	−0.951971
$802$	13948.2	0.614125
$803$	−61382.7	−2.69757
$804$	141.228	0.00619492
$805$	0	0
$806$	−8346.70	−0.364764
$807$	−481.832	−0.0210177
$808$	−13599.8	−0.592128
$809$	10186.2	0.442678	0.221339	−	0.975197i	$-0.428957\pi$
$809$	10186.2	0.442678	0.221339	+	0.975197i	$0.428957\pi$
$810$	0	0
$811$	21196.9	0.917786	0.458893	−	0.888492i	$-0.348246\pi$
$811$	21196.9	0.917786	0.458893	+	0.888492i	$0.348246\pi$
$812$	0	0
$813$	2681.24	0.115665
$814$	−23150.8	−0.996851
$815$	0	0
$816$	−197.983	−0.00849364
$817$	−22371.3	−0.957982
$818$	24146.5	1.03211
$819$	0	0
$820$	0	0
$821$	7563.94	0.321539	0.160769	−	0.986992i	$-0.448602\pi$
$821$	7563.94	0.321539	0.160769	+	0.986992i	$0.448602\pi$
$822$	−10.2061	−0.000433065 0
$823$	−8399.80	−0.355770	−0.177885	−	0.984051i	$-0.556925\pi$
$823$	−8399.80	−0.355770	−0.177885	+	0.984051i	$0.556925\pi$
$824$	40004.1	1.69127
$825$	0	0
$826$	0	0
$827$	−5479.54	−0.230402	−0.115201	−	0.993342i	$-0.536751\pi$
$827$	−5479.54	−0.230402	−0.115201	+	0.993342i	$0.536751\pi$
$828$	14282.6	0.599462
$829$	9252.56	0.387641	0.193821	−	0.981037i	$-0.437912\pi$
$829$	9252.56	0.387641	0.193821	+	0.981037i	$0.437912\pi$
$830$	0	0
$831$	190.179	0.00793890
$832$	−17854.2	−0.743972
$833$	0	0
$834$	344.887	0.0143195
$835$	0	0
$836$	−26296.6	−1.08790
$837$	1321.22	0.0545615
$838$	4202.31	0.173230
$839$	−34517.6	−1.42036	−0.710178	−	0.704022i	$-0.751386\pi$
$839$	−34517.6	−1.42036	−0.710178	+	0.704022i	$0.751386\pi$
$840$	0	0
$841$	−11241.9	−0.460943
$842$	−15980.2	−0.654055
$843$	−594.031	−0.0242699
$844$	−10630.6	−0.433555
$845$	0	0
$846$	−6666.04	−0.270902
$847$	0	0
$848$	−1547.55	−0.0626686
$849$	−1105.54	−0.0446901
$850$	0	0
$851$	20300.4	0.817732
$852$	159.344	0.00640732
$853$	1498.57	0.0601525	0.0300763	−	0.999548i	$-0.490425\pi$
$853$	1498.57	0.0601525	0.0300763	+	0.999548i	$0.490425\pi$
$854$	0	0
$855$	0	0
$856$	−23924.6	−0.955287
$857$	9357.02	0.372963	0.186482	−	0.982458i	$-0.440291\pi$
$857$	9357.02	0.372963	0.186482	+	0.982458i	$0.440291\pi$
$858$	2624.67	0.104435
$859$	−29960.9	−1.19005	−0.595025	−	0.803707i	$-0.702858\pi$
$859$	−29960.9	−1.19005	−0.595025	+	0.803707i	$0.702858\pi$
$860$	0	0
$861$	0	0
$862$	12027.3	0.475232
$863$	33941.0	1.33878	0.669389	−	0.742912i	$-0.266556\pi$
$863$	33941.0	1.33878	0.669389	+	0.742912i	$0.266556\pi$
$864$	3320.19	0.130735
$865$	0	0
$866$	18788.1	0.737235
$867$	1988.04	0.0778747
$868$	0	0
$869$	41735.0	1.62919
$870$	0	0
$871$	5522.16	0.214824
$872$	12946.2	0.502769
$873$	3916.35	0.151831
$874$	−12028.6	−0.465532
$875$	0	0
$876$	1544.12	0.0595559
$877$	39436.6	1.51845	0.759224	−	0.650830i	$-0.225579\pi$
$877$	39436.6	1.51845	0.759224	+	0.650830i	$0.225579\pi$
$878$	−19385.7	−0.745142
$879$	1680.93	0.0645010
$880$	0	0
$881$	32411.4	1.23946	0.619732	−	0.784813i	$-0.287241\pi$
$881$	32411.4	1.23946	0.619732	+	0.784813i	$0.287241\pi$
$882$	0	0
$883$	17121.4	0.652526	0.326263	−	0.945279i	$-0.394210\pi$
$883$	17121.4	0.652526	0.326263	+	0.945279i	$0.394210\pi$
$884$	−37536.6	−1.42816
$885$	0	0
$886$	24970.1	0.946824
$887$	687.797	0.0260360	0.0130180	−	0.999915i	$-0.495856\pi$
$887$	687.797	0.0260360	0.0130180	+	0.999915i	$0.495856\pi$
$888$	1468.54	0.0554967
$889$	0	0
$890$	0	0
$891$	50094.8	1.88355
$892$	22837.6	0.857243
$893$	−10762.2	−0.403294
$894$	567.068	0.0212143
$895$	0	0
$896$	0	0
$897$	−2301.52	−0.0856693
$898$	−1781.01	−0.0661839
$899$	8444.81	0.313293
$900$	0	0
$901$	28323.4	1.04727
$902$	−48080.8	−1.77485
$903$	0	0
$904$	19767.5	0.727276
$905$	0	0
$906$	−39.1451	−0.00143544
$907$	−19104.2	−0.699388	−0.349694	−	0.936864i	$-0.613714\pi$
$907$	−19104.2	−0.699388	−0.349694	+	0.936864i	$0.613714\pi$
$908$	−13919.6	−0.508743
$909$	16656.1	0.607754
$910$	0	0
$911$	23135.3	0.841390	0.420695	−	0.907202i	$-0.361786\pi$
$911$	23135.3	0.841390	0.420695	+	0.907202i	$0.361786\pi$
$912$	136.427	0.00495344
$913$	−4931.45	−0.178759
$914$	−17781.0	−0.643484
$915$	0	0
$916$	7597.50	0.274049
$917$	0	0
$918$	−3099.50	−0.111437
$919$	47373.9	1.70046	0.850230	−	0.526412i	$-0.176463\pi$
$919$	47373.9	1.70046	0.850230	+	0.526412i	$0.176463\pi$
$920$	0	0
$921$	511.224	0.0182903
$922$	−749.481	−0.0267710
$923$	6230.53	0.222189
$924$	0	0
$925$	0	0
$926$	11788.1	0.418338
$927$	−48994.2	−1.73590
$928$	21221.6	0.750682
$929$	−7617.72	−0.269031	−0.134515	−	0.990912i	$-0.542948\pi$
$929$	−7617.72	−0.269031	−0.134515	+	0.990912i	$0.542948\pi$
$930$	0	0
$931$	0	0
$932$	−32837.6	−1.15411
$933$	−3090.21	−0.108434
$934$	1612.67	0.0564971
$935$	0	0
$936$	40399.6	1.41079
$937$	52091.2	1.81616	0.908082	−	0.418792i	$-0.137546\pi$
$937$	52091.2	1.81616	0.908082	+	0.418792i	$0.137546\pi$
$938$	0	0
$939$	−2005.79	−0.0697088
$940$	0	0
$941$	−56765.1	−1.96651	−0.983257	−	0.182225i	$-0.941670\pi$
$941$	−56765.1	−1.96651	−0.983257	+	0.182225i	$0.941670\pi$
$942$	−1136.37	−0.0393046
$943$	42160.9	1.45594
$944$	−2953.03	−0.101815
$945$	0	0
$946$	35852.5	1.23220
$947$	47629.2	1.63436	0.817181	−	0.576381i	$-0.195535\pi$
$947$	47629.2	1.63436	0.817181	+	0.576381i	$0.195535\pi$
$948$	−1049.87	−0.0359686
$949$	60376.8	2.06524
$950$	0	0
$951$	2322.75	0.0792012
$952$	0	0
$953$	−12893.5	−0.438259	−0.219129	−	0.975696i	$-0.570322\pi$
$953$	−12893.5	−0.438259	−0.219129	+	0.975696i	$0.570322\pi$
$954$	−12088.8	−0.410261
$955$	0	0
$956$	−7046.24	−0.238380
$957$	−2655.52	−0.0896979
$958$	−16087.0	−0.542534
$959$	0	0
$960$	0	0
$961$	−24366.6	−0.817918
$962$	22771.5	0.763183
$963$	29301.2	0.980496
$964$	−17717.2	−0.591943
$965$	0	0
$966$	0	0
$967$	28420.0	0.945114	0.472557	−	0.881300i	$-0.343331\pi$
$967$	28420.0	0.945114	0.472557	+	0.881300i	$0.343331\pi$
$968$	77048.4	2.55830
$969$	−2496.89	−0.0827779
$970$	0	0
$971$	29273.8	0.967497	0.483749	−	0.875207i	$-0.339275\pi$
$971$	29273.8	0.967497	0.483749	+	0.875207i	$0.339275\pi$
$972$	−3806.64	−0.125615
$973$	0	0
$974$	1529.19	0.0503063
$975$	0	0
$976$	−1253.69	−0.0411165
$977$	−12055.0	−0.394753	−0.197377	−	0.980328i	$-0.563242\pi$
$977$	−12055.0	−0.394753	−0.197377	+	0.980328i	$0.563242\pi$
$978$	−1082.61	−0.0353967
$979$	55838.4	1.82288
$980$	0	0
$981$	−15855.7	−0.516037
$982$	−2134.84	−0.0693743
$983$	−10605.1	−0.344099	−0.172049	−	0.985088i	$-0.555039\pi$
$983$	−10605.1	−0.344099	−0.172049	+	0.985088i	$0.555039\pi$
$984$	3049.94	0.0988094
$985$	0	0
$986$	−19811.0	−0.639870
$987$	0	0
$988$	25865.7	0.832893
$989$	−31438.2	−1.01080
$990$	0	0
$991$	45229.1	1.44980	0.724898	−	0.688856i	$-0.241887\pi$
$991$	45229.1	1.44980	0.724898	+	0.688856i	$0.241887\pi$
$992$	13631.4	0.436287
$993$	327.854	0.0104775
$994$	0	0
$995$	0	0
$996$	124.054	0.00394658
$997$	49676.8	1.57802	0.789008	−	0.614383i	$-0.210595\pi$
$997$	49676.8	1.57802	0.789008	+	0.614383i	$0.210595\pi$
$998$	32025.4	1.01578
$999$	−3604.55	−0.114157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.bi.1.2		6
5.4	even	2		245.4.a.p.1.5	yes	6
7.6	odd	2		1225.4.a.bj.1.2		6
15.14	odd	2		2205.4.a.ca.1.2		6
35.4	even	6		245.4.e.p.226.2		12
35.9	even	6		245.4.e.p.116.2		12
35.19	odd	6		245.4.e.q.116.2		12
35.24	odd	6		245.4.e.q.226.2		12
35.34	odd	2		245.4.a.o.1.5	✓	6
105.104	even	2		2205.4.a.bz.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.o.1.5	✓	6	35.34	odd	2
245.4.a.p.1.5	yes	6	5.4	even	2
245.4.e.p.116.2		12	35.9	even	6
245.4.e.p.226.2		12	35.4	even	6
245.4.e.q.116.2		12	35.19	odd	6
245.4.e.q.226.2		12	35.24	odd	6
1225.4.a.bi.1.2		6	1.1	even	1	trivial
1225.4.a.bj.1.2		6	7.6	odd	2
2205.4.a.bz.1.2		6	105.104	even	2
2205.4.a.ca.1.2		6	15.14	odd	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q-1.65606 q^{2} +0.332888 q^{3} -5.25746 q^{4} -0.551283 q^{6} +21.9552 q^{8} -26.8892 q^{9} +O(q^{10})\)	\(q-1.65606 q^{2} +0.332888 q^{3} -5.25746 q^{4} -0.551283 q^{6} +21.9552 q^{8} -26.8892 q^{9} +69.5726 q^{11} -1.75014 q^{12} -68.4326 q^{13} +5.70053 q^{16} -104.332 q^{17} +44.5302 q^{18} +71.8929 q^{19} -115.217 q^{22} +101.031 q^{23} +7.30861 q^{24} +113.329 q^{26} -17.9390 q^{27} -114.661 q^{29} -73.6505 q^{31} -185.082 q^{32} +23.1599 q^{33} +172.780 q^{34} +141.369 q^{36} +200.933 q^{37} -119.059 q^{38} -22.7803 q^{39} +417.308 q^{41} -311.175 q^{43} -365.775 q^{44} -167.313 q^{46} -149.697 q^{47} +1.89763 q^{48} -34.7307 q^{51} +359.781 q^{52} -271.474 q^{53} +29.7082 q^{54} +23.9323 q^{57} +189.885 q^{58} -518.028 q^{59} -219.926 q^{61} +121.970 q^{62} +260.903 q^{64} -38.3542 q^{66} -80.6950 q^{67} +548.520 q^{68} +33.6319 q^{69} -91.0463 q^{71} -590.357 q^{72} -882.282 q^{73} -332.758 q^{74} -377.974 q^{76} +37.7257 q^{78} +599.877 q^{79} +720.036 q^{81} -691.087 q^{82} -70.8820 q^{83} +515.325 q^{86} -38.1691 q^{87} +1527.48 q^{88} +802.592 q^{89} -531.165 q^{92} -24.5173 q^{93} +247.908 q^{94} -61.6114 q^{96} -145.648 q^{97} -1870.75 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} - 16 q^{3} + 14 q^{4} + 24 q^{6} + 66 q^{8} + 70 q^{9}+O(q^{10}) \) 6 * q + 2 * q^2 - 16 * q^3 + 14 * q^4 + 24 * q^6 + 66 * q^8 + 70 * q^9	\( 6 q + 2 q^{2} - 16 q^{3} + 14 q^{4} + 24 q^{6} + 66 q^{8} + 70 q^{9} - 16 q^{11} - 160 q^{12} - 168 q^{13} + 298 q^{16} + 4 q^{17} - 354 q^{18} + 308 q^{19} + 236 q^{22} + 336 q^{23} - 92 q^{24} + 56 q^{26} - 964 q^{27} + 176 q^{29} + 392 q^{31} + 770 q^{32} - 188 q^{33} + 812 q^{34} + 230 q^{36} + 140 q^{37} - 20 q^{38} + 140 q^{39} + 656 q^{41} + 388 q^{43} - 160 q^{44} - 388 q^{46} - 628 q^{47} - 1396 q^{48} + 744 q^{51} - 1520 q^{52} + 676 q^{53} + 2284 q^{54} - 1468 q^{57} + 2012 q^{58} + 996 q^{59} + 740 q^{61} + 364 q^{62} + 1426 q^{64} - 3620 q^{66} - 1768 q^{67} + 2940 q^{68} - 1048 q^{69} - 224 q^{71} - 2858 q^{72} - 2640 q^{73} + 928 q^{74} - 1340 q^{76} - 8 q^{78} + 1636 q^{79} + 4442 q^{81} + 1756 q^{82} - 140 q^{83} + 1180 q^{86} - 1940 q^{87} + 5652 q^{88} - 1904 q^{89} + 1952 q^{92} + 1592 q^{93} - 3332 q^{94} - 6460 q^{96} - 516 q^{97} - 2804 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 - 16 * q^3 + 14 * q^4 + 24 * q^6 + 66 * q^8 + 70 * q^9 - 16 * q^11 - 160 * q^12 - 168 * q^13 + 298 * q^16 + 4 * q^17 - 354 * q^18 + 308 * q^19 + 236 * q^22 + 336 * q^23 - 92 * q^24 + 56 * q^26 - 964 * q^27 + 176 * q^29 + 392 * q^31 + 770 * q^32 - 188 * q^33 + 812 * q^34 + 230 * q^36 + 140 * q^37 - 20 * q^38 + 140 * q^39 + 656 * q^41 + 388 * q^43 - 160 * q^44 - 388 * q^46 - 628 * q^47 - 1396 * q^48 + 744 * q^51 - 1520 * q^52 + 676 * q^53 + 2284 * q^54 - 1468 * q^57 + 2012 * q^58 + 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 - 3620 * q^66 - 1768 * q^67 + 2940 * q^68 - 1048 * q^69 - 224 * q^71 - 2858 * q^72 - 2640 * q^73 + 928 * q^74 - 1340 * q^76 - 8 * q^78 + 1636 * q^79 + 4442 * q^81 + 1756 * q^82 - 140 * q^83 + 1180 * q^86 - 1940 * q^87 + 5652 * q^88 - 1904 * q^89 + 1952 * q^92 + 1592 * q^93 - 3332 * q^94 - 6460 * q^96 - 516 * q^97 - 2804 * q^99

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