LMFDB - Embedded newform 7225.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 20x^{10} + 135x^{8} - 400x^{6} + 515x^{4} - 222x^{2} + 25 $ x^12 - 20x^10 + 135x^8 - 400x^6 + 515x^4 - 222*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 425)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.85347$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.08389 q^{2} -1.85347 q^{3} +2.34261 q^{4} +3.86244 q^{6} +1.37395 q^{7} -0.713960 q^{8} +0.435362 q^{9} -0.544107 q^{11} -4.34196 q^{12} +5.39918 q^{13} -2.86316 q^{14} -3.19740 q^{16} -0.907247 q^{18} +4.86186 q^{19} -2.54658 q^{21} +1.13386 q^{22} +1.78727 q^{23} +1.32330 q^{24} -11.2513 q^{26} +4.75349 q^{27} +3.21863 q^{28} -1.83415 q^{29} -8.11349 q^{31} +8.09096 q^{32} +1.00849 q^{33} +1.01988 q^{36} +5.97072 q^{37} -10.1316 q^{38} -10.0072 q^{39} +3.82217 q^{41} +5.30680 q^{42} -3.66628 q^{43} -1.27463 q^{44} -3.72447 q^{46} +9.07290 q^{47} +5.92630 q^{48} -5.11226 q^{49} +12.6482 q^{52} +10.5471 q^{53} -9.90576 q^{54} -0.980945 q^{56} -9.01133 q^{57} +3.82217 q^{58} +6.52808 q^{59} +15.4659 q^{61} +16.9076 q^{62} +0.598165 q^{63} -10.4659 q^{64} -2.10158 q^{66} +5.68704 q^{67} -3.31265 q^{69} +1.05995 q^{71} -0.310831 q^{72} -14.6825 q^{73} -12.4423 q^{74} +11.3894 q^{76} -0.747575 q^{77} +20.8540 q^{78} -1.24247 q^{79} -10.1165 q^{81} -7.96500 q^{82} +13.5038 q^{83} -5.96564 q^{84} +7.64014 q^{86} +3.39955 q^{87} +0.388470 q^{88} +0.989860 q^{89} +7.41820 q^{91} +4.18687 q^{92} +15.0381 q^{93} -18.9070 q^{94} -14.9964 q^{96} -11.3905 q^{97} +10.6534 q^{98} -0.236883 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9} + 12 q^{13} + 4 q^{16} + 4 q^{18} + 12 q^{19} - 8 q^{21} + 12 q^{26} + 48 q^{32} - 4 q^{33} - 20 q^{36} - 12 q^{38} + 56 q^{42} + 16 q^{43} + 36 q^{47} + 16 q^{49}+ \cdots + 76 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9 + 12 * q^13 + 4 * q^16 + 4 * q^18 + 12 * q^19 - 8 * q^21 + 12 * q^26 + 48 * q^32 - 4 * q^33 - 20 * q^36 - 12 * q^38 + 56 * q^42 + 16 * q^43 + 36 * q^47 + 16 * q^49 + 64 * q^52 + 48 * q^53 - 32 * q^59 + 44 * q^64 + 40 * q^66 - 20 * q^67 - 72 * q^69 - 92 * q^72 + 36 * q^76 + 24 * q^77 + 8 * q^81 + 52 * q^83 + 36 * q^84 + 84 * q^86 + 48 * q^87 + 24 * q^89 + 60 * q^93 - 8 * q^94 + 76 * 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 $

$2$	−2.08389	−1.47353	−0.736767	−	0.676146i	$-0.763649\pi$
$2$	−2.08389	−1.47353	−0.736767	+	0.676146i	$0.763649\pi$
$3$	−1.85347	−1.07010	−0.535052	−	0.844819i	$-0.679708\pi$
$3$	−1.85347	−1.07010	−0.535052	+	0.844819i	$0.679708\pi$
$4$	2.34261	1.17130
$5$	0	0
$5$	0	0
$6$	3.86244	1.57683
$7$	1.37395	0.519304	0.259652	−	0.965702i	$-0.416392\pi$
$7$	1.37395	0.519304	0.259652	+	0.965702i	$0.416392\pi$
$8$	−0.713960	−0.252423
$9$	0.435362	0.145121
$10$	0	0
$11$	−0.544107	−0.164054	−0.0820272	−	0.996630i	$-0.526139\pi$
$11$	−0.544107	−0.164054	−0.0820272	+	0.996630i	$0.526139\pi$
$12$	−4.34196	−1.25342
$13$	5.39918	1.49746	0.748731	−	0.662874i	$-0.230663\pi$
$13$	5.39918	1.49746	0.748731	+	0.662874i	$0.230663\pi$
$14$	−2.86316	−0.765213
$15$	0	0
$16$	−3.19740	−0.799351
$17$	0	0
$17$	0	0
$18$	−0.907247	−0.213840
$19$	4.86186	1.11539	0.557694	−	0.830047i	$-0.311686\pi$
$19$	4.86186	1.11539	0.557694	+	0.830047i	$0.311686\pi$
$20$	0	0
$21$	−2.54658	−0.555709
$22$	1.13386	0.241740
$23$	1.78727	0.372671	0.186336	−	0.982486i	$-0.440339\pi$
$23$	1.78727	0.372671	0.186336	+	0.982486i	$0.440339\pi$
$24$	1.32330	0.270118
$25$	0	0
$26$	−11.2513	−2.20656
$27$	4.75349	0.914809
$28$	3.21863	0.608263
$29$	−1.83415	−0.340593	−0.170297	−	0.985393i	$-0.554473\pi$
$29$	−1.83415	−0.340593	−0.170297	+	0.985393i	$0.554473\pi$
$30$	0	0
$31$	−8.11349	−1.45723	−0.728613	−	0.684926i	$-0.759835\pi$
$31$	−8.11349	−1.45723	−0.728613	+	0.684926i	$0.759835\pi$
$32$	8.09096	1.43029
$33$	1.00849	0.175555
$34$	0	0
$35$	0	0
$36$	1.01988	0.169980
$37$	5.97072	0.981581	0.490791	−	0.871278i	$-0.336708\pi$
$37$	5.97072	0.981581	0.490791	+	0.871278i	$0.336708\pi$
$38$	−10.1316	−1.64356
$39$	−10.0072	−1.60244
$40$	0	0
$41$	3.82217	0.596923	0.298462	−	0.954422i	$-0.403526\pi$
$41$	3.82217	0.596923	0.298462	+	0.954422i	$0.403526\pi$
$42$	5.30680	0.818857
$43$	−3.66628	−0.559103	−0.279551	−	0.960131i	$-0.590186\pi$
$43$	−3.66628	−0.559103	−0.279551	+	0.960131i	$0.590186\pi$
$44$	−1.27463	−0.192158
$45$	0	0
$46$	−3.72447	−0.549144
$47$	9.07290	1.32342	0.661709	−	0.749760i	$-0.269831\pi$
$47$	9.07290	1.32342	0.661709	+	0.749760i	$0.269831\pi$
$48$	5.92630	0.855387
$49$	−5.11226	−0.730323
$50$	0	0
$51$	0	0
$52$	12.6482	1.75398
$53$	10.5471	1.44875	0.724376	−	0.689405i	$-0.242128\pi$
$53$	10.5471	1.44875	0.724376	+	0.689405i	$0.242128\pi$
$54$	−9.90576	−1.34800
$55$	0	0
$56$	−0.980945	−0.131084
$57$	−9.01133	−1.19358
$58$	3.82217	0.501876
$59$	6.52808	0.849884	0.424942	−	0.905221i	$-0.360295\pi$
$59$	6.52808	0.849884	0.424942	+	0.905221i	$0.360295\pi$
$60$	0	0
$61$	15.4659	1.98020	0.990101	−	0.140358i	$-0.0448254\pi$
$61$	15.4659	1.98020	0.990101	+	0.140358i	$0.0448254\pi$
$62$	16.9076	2.14727
$63$	0.598165	0.0753617
$64$	−10.4659	−1.30824
$65$	0	0
$66$	−2.10158	−0.258686
$67$	5.68704	0.694782	0.347391	−	0.937720i	$-0.387068\pi$
$67$	5.68704	0.694782	0.347391	+	0.937720i	$0.387068\pi$
$68$	0	0
$69$	−3.31265	−0.398797
$70$	0	0
$71$	1.05995	0.125793	0.0628966	−	0.998020i	$-0.479966\pi$
$71$	1.05995	0.125793	0.0628966	+	0.998020i	$0.479966\pi$
$72$	−0.310831	−0.0366317
$73$	−14.6825	−1.71846	−0.859230	−	0.511590i	$-0.829057\pi$
$73$	−14.6825	−1.71846	−0.859230	+	0.511590i	$0.829057\pi$
$74$	−12.4423	−1.44639
$75$	0	0
$76$	11.3894	1.30646
$77$	−0.747575	−0.0851941
$78$	20.8540	2.36125
$79$	−1.24247	−0.139789	−0.0698944	−	0.997554i	$-0.522266\pi$
$79$	−1.24247	−0.139789	−0.0698944	+	0.997554i	$0.522266\pi$
$80$	0	0
$81$	−10.1165	−1.12406
$82$	−7.96500	−0.879587
$83$	13.5038	1.48224	0.741119	−	0.671374i	$-0.234296\pi$
$83$	13.5038	1.48224	0.741119	+	0.671374i	$0.234296\pi$
$84$	−5.96564	−0.650904
$85$	0	0
$86$	7.64014	0.823858
$87$	3.39955	0.364470
$88$	0.388470	0.0414111
$89$	0.989860	0.104925	0.0524625	−	0.998623i	$-0.483293\pi$
$89$	0.989860	0.104925	0.0524625	+	0.998623i	$0.483293\pi$
$90$	0	0
$91$	7.41820	0.777638
$92$	4.18687	0.436511
$93$	15.0381	1.55938
$94$	−18.9070	−1.95010
$95$	0	0
$96$	−14.9964	−1.53056
$97$	−11.3905	−1.15653	−0.578263	−	0.815850i	$-0.696269\pi$
$97$	−11.3905	−1.15653	−0.578263	+	0.815850i	$0.696269\pi$
$98$	10.6534	1.07616
$99$	−0.236883	−0.0238077
$100$	0	0
$101$	6.02268	0.599279	0.299639	−	0.954053i	$-0.403134\pi$
$101$	6.02268	0.599279	0.299639	+	0.954053i	$0.403134\pi$
$102$	0	0
$103$	−18.1789	−1.79122	−0.895612	−	0.444836i	$-0.853262\pi$
$103$	−18.1789	−1.79122	−0.895612	+	0.444836i	$0.853262\pi$
$104$	−3.85479	−0.377994
$105$	0	0
$106$	−21.9790	−2.13479
$107$	9.58605	0.926719	0.463359	−	0.886170i	$-0.346644\pi$
$107$	9.58605	0.926719	0.463359	+	0.886170i	$0.346644\pi$
$108$	11.1356	1.07152
$109$	3.97889	0.381108	0.190554	−	0.981677i	$-0.438972\pi$
$109$	3.97889	0.381108	0.190554	+	0.981677i	$0.438972\pi$
$110$	0	0
$111$	−11.0666	−1.05039
$112$	−4.39307	−0.415106
$113$	−3.55952	−0.334852	−0.167426	−	0.985885i	$-0.553545\pi$
$113$	−3.55952	−0.334852	−0.167426	+	0.985885i	$0.553545\pi$
$114$	18.7786	1.75878
$115$	0	0
$116$	−4.29670	−0.398938
$117$	2.35059	0.217313
$118$	−13.6038	−1.25233
$119$	0	0
$120$	0	0
$121$	−10.7039	−0.973086
$122$	−32.2292	−2.91790
$123$	−7.08430	−0.638770
$124$	−19.0067	−1.70686
$125$	0	0
$126$	−1.24651	−0.111048
$127$	7.76925	0.689409	0.344705	−	0.938711i	$-0.387979\pi$
$127$	7.76925	0.689409	0.344705	+	0.938711i	$0.387979\pi$
$128$	5.62787	0.497438
$129$	6.79536	0.598298
$130$	0	0
$131$	20.1951	1.76445	0.882225	−	0.470828i	$-0.156045\pi$
$131$	20.1951	1.76445	0.882225	+	0.470828i	$0.156045\pi$
$132$	2.36249	0.205628
$133$	6.67996	0.579226
$134$	−11.8512	−1.02379
$135$	0	0
$136$	0	0
$137$	16.6892	1.42585	0.712926	−	0.701239i	$-0.247370\pi$
$137$	16.6892	1.42585	0.712926	+	0.701239i	$0.247370\pi$
$138$	6.90321	0.587641
$139$	6.19810	0.525716	0.262858	−	0.964834i	$-0.415335\pi$
$139$	6.19810	0.525716	0.262858	+	0.964834i	$0.415335\pi$
$140$	0	0
$141$	−16.8164	−1.41619
$142$	−2.20883	−0.185361
$143$	−2.93773	−0.245665
$144$	−1.39203	−0.116002
$145$	0	0
$146$	30.5968	2.53221
$147$	9.47544	0.781521
$148$	13.9871	1.14973
$149$	4.48964	0.367805	0.183903	−	0.982944i	$-0.441127\pi$
$149$	4.48964	0.367805	0.183903	+	0.982944i	$0.441127\pi$
$150$	0	0
$151$	6.35198	0.516917	0.258458	−	0.966022i	$-0.416786\pi$
$151$	6.35198	0.516917	0.258458	+	0.966022i	$0.416786\pi$
$152$	−3.47117	−0.281549
$153$	0	0
$154$	1.55787	0.125536
$155$	0	0
$156$	−23.4430	−1.87694
$157$	6.37303	0.508624	0.254312	−	0.967122i	$-0.418151\pi$
$157$	6.37303	0.508624	0.254312	+	0.967122i	$0.418151\pi$
$158$	2.58917	0.205984
$159$	−19.5487	−1.55031
$160$	0	0
$161$	2.45562	0.193530
$162$	21.0818	1.65634
$163$	4.18879	0.328091	0.164046	−	0.986453i	$-0.447546\pi$
$163$	4.18879	0.328091	0.164046	+	0.986453i	$0.447546\pi$
$164$	8.95386	0.699179
$165$	0	0
$166$	−28.1405	−2.18413
$167$	13.2367	1.02428	0.512142	−	0.858901i	$-0.328852\pi$
$167$	13.2367	1.02428	0.512142	+	0.858901i	$0.328852\pi$
$168$	1.81815	0.140274
$169$	16.1511	1.24239
$170$	0	0
$171$	2.11667	0.161866
$172$	−8.58867	−0.654880
$173$	−8.83779	−0.671925	−0.335963	−	0.941875i	$-0.609062\pi$
$173$	−8.83779	−0.671925	−0.335963	+	0.941875i	$0.609062\pi$
$174$	−7.08430	−0.537059
$175$	0	0
$176$	1.73973	0.131137
$177$	−12.0996	−0.909463
$178$	−2.06276	−0.154611
$179$	−19.2951	−1.44218	−0.721091	−	0.692841i	$-0.756359\pi$
$179$	−19.2951	−1.44218	−0.721091	+	0.692841i	$0.756359\pi$
$180$	0	0
$181$	−4.91297	−0.365178	−0.182589	−	0.983189i	$-0.558448\pi$
$181$	−4.91297	−0.365178	−0.182589	+	0.983189i	$0.558448\pi$
$182$	−15.4587	−1.14588
$183$	−28.6656	−2.11902
$184$	−1.27604	−0.0940707
$185$	0	0
$186$	−31.3379	−2.29780
$187$	0	0
$188$	21.2543	1.55013
$189$	6.53105	0.475064
$190$	0	0
$191$	−5.60780	−0.405766	−0.202883	−	0.979203i	$-0.565031\pi$
$191$	−5.60780	−0.405766	−0.202883	+	0.979203i	$0.565031\pi$
$192$	19.3982	1.39995
$193$	8.34451	0.600651	0.300326	−	0.953837i	$-0.402905\pi$
$193$	8.34451	0.600651	0.300326	+	0.953837i	$0.402905\pi$
$194$	23.7365	1.70418
$195$	0	0
$196$	−11.9760	−0.855431
$197$	−6.07466	−0.432802	−0.216401	−	0.976305i	$-0.569432\pi$
$197$	−6.07466	−0.432802	−0.216401	+	0.976305i	$0.569432\pi$
$198$	0.493639	0.0350814
$199$	−9.49603	−0.673156	−0.336578	−	0.941656i	$-0.609270\pi$
$199$	−9.49603	−0.673156	−0.336578	+	0.941656i	$0.609270\pi$
$200$	0	0
$201$	−10.5408	−0.743489
$202$	−12.5506	−0.883058
$203$	−2.52003	−0.176872
$204$	0	0
$205$	0	0
$206$	37.8829	2.63943
$207$	0.778108	0.0540822
$208$	−17.2633	−1.19700
$209$	−2.64537	−0.182984
$210$	0	0
$211$	−20.5067	−1.41174	−0.705870	−	0.708341i	$-0.749444\pi$
$211$	−20.5067	−1.41174	−0.705870	+	0.708341i	$0.749444\pi$
$212$	24.7077	1.69693
$213$	−1.96459	−0.134612
$214$	−19.9763	−1.36555
$215$	0	0
$216$	−3.39380	−0.230919
$217$	−11.1475	−0.756744
$218$	−8.29157	−0.561576
$219$	27.2137	1.83893
$220$	0	0
$221$	0	0
$222$	23.0616	1.54779
$223$	6.64902	0.445252	0.222626	−	0.974904i	$-0.428537\pi$
$223$	6.64902	0.445252	0.222626	+	0.974904i	$0.428537\pi$
$224$	11.1166	0.742757
$225$	0	0
$226$	7.41766	0.493415
$227$	−10.5120	−0.697708	−0.348854	−	0.937177i	$-0.613429\pi$
$227$	−10.5120	−0.697708	−0.348854	+	0.937177i	$0.613429\pi$
$228$	−21.1100	−1.39805
$229$	−15.3731	−1.01589	−0.507943	−	0.861391i	$-0.669594\pi$
$229$	−15.3731	−1.01589	−0.507943	+	0.861391i	$0.669594\pi$
$230$	0	0
$231$	1.38561	0.0911665
$232$	1.30951	0.0859735
$233$	−21.0320	−1.37785	−0.688927	−	0.724831i	$-0.741918\pi$
$233$	−21.0320	−1.37785	−0.688927	+	0.724831i	$0.741918\pi$
$234$	−4.89839	−0.320218
$235$	0	0
$236$	15.2927	0.995472
$237$	2.30288	0.149588
$238$	0	0
$239$	−9.10195	−0.588756	−0.294378	−	0.955689i	$-0.595113\pi$
$239$	−9.10195	−0.588756	−0.294378	+	0.955689i	$0.595113\pi$
$240$	0	0
$241$	−12.7279	−0.819878	−0.409939	−	0.912113i	$-0.634450\pi$
$241$	−12.7279	−0.819878	−0.409939	+	0.912113i	$0.634450\pi$
$242$	22.3059	1.43388
$243$	4.49028	0.288052
$244$	36.2305	2.31942
$245$	0	0
$246$	14.7629	0.941249
$247$	26.2501	1.67025
$248$	5.79270	0.367837
$249$	−25.0290	−1.58615
$250$	0	0
$251$	10.1470	0.640475	0.320238	−	0.947337i	$-0.396237\pi$
$251$	10.1470	0.640475	0.320238	+	0.947337i	$0.396237\pi$
$252$	1.40127	0.0882715
$253$	−0.972465	−0.0611383
$254$	−16.1903	−1.01587
$255$	0	0
$256$	9.20390	0.575244
$257$	19.0187	1.18635	0.593176	−	0.805073i	$-0.297874\pi$
$257$	19.0187	1.18635	0.593176	+	0.805073i	$0.297874\pi$
$258$	−14.1608	−0.881612
$259$	8.20348	0.509739
$260$	0	0
$261$	−0.798519	−0.0494271
$262$	−42.0843	−2.59998
$263$	−12.6346	−0.779086	−0.389543	−	0.921008i	$-0.627367\pi$
$263$	−12.6346	−0.779086	−0.389543	+	0.921008i	$0.627367\pi$
$264$	−0.720019	−0.0443141
$265$	0	0
$266$	−13.9203	−0.853509
$267$	−1.83468	−0.112281
$268$	13.3225	0.813802
$269$	−27.9224	−1.70246	−0.851229	−	0.524794i	$-0.824142\pi$
$269$	−27.9224	−1.70246	−0.851229	+	0.524794i	$0.824142\pi$
$270$	0	0
$271$	−7.78171	−0.472705	−0.236353	−	0.971667i	$-0.575952\pi$
$271$	−7.78171	−0.472705	−0.236353	+	0.971667i	$0.575952\pi$
$272$	0	0
$273$	−13.7494	−0.832153
$274$	−34.7784	−2.10104
$275$	0	0
$276$	−7.76025	−0.467112
$277$	30.9745	1.86107	0.930537	−	0.366197i	$-0.119340\pi$
$277$	30.9745	1.86107	0.930537	+	0.366197i	$0.119340\pi$
$278$	−12.9162	−0.774661
$279$	−3.53230	−0.211473
$280$	0	0
$281$	−2.25506	−0.134525	−0.0672627	−	0.997735i	$-0.521427\pi$
$281$	−2.25506	−0.134525	−0.0672627	+	0.997735i	$0.521427\pi$
$282$	35.0435	2.08681
$283$	−26.3006	−1.56341	−0.781704	−	0.623650i	$-0.785649\pi$
$283$	−26.3006	−1.56341	−0.781704	+	0.623650i	$0.785649\pi$
$284$	2.48305	0.147342
$285$	0	0
$286$	6.12191	0.361996
$287$	5.25148	0.309985
$288$	3.52249	0.207565
$289$	0	0
$290$	0	0
$291$	21.1119	1.23760
$292$	−34.3954	−2.01284
$293$	8.22869	0.480725	0.240363	−	0.970683i	$-0.422734\pi$
$293$	8.22869	0.480725	0.240363	+	0.970683i	$0.422734\pi$
$294$	−19.7458	−1.15160
$295$	0	0
$296$	−4.26286	−0.247773
$297$	−2.58640	−0.150078
$298$	−9.35592	−0.541974
$299$	9.64978	0.558061
$300$	0	0
$301$	−5.03729	−0.290345
$302$	−13.2368	−0.761695
$303$	−11.1629	−0.641290
$304$	−15.5453	−0.891586
$305$	0	0
$306$	0	0
$307$	0.620204	0.0353969	0.0176985	−	0.999843i	$-0.494366\pi$
$307$	0.620204	0.0353969	0.0176985	+	0.999843i	$0.494366\pi$
$308$	−1.75128	−0.0997882
$309$	33.6942	1.91679
$310$	0	0
$311$	−23.0834	−1.30894	−0.654469	−	0.756089i	$-0.727108\pi$
$311$	−23.0834	−1.30894	−0.654469	+	0.756089i	$0.727108\pi$
$312$	7.14476	0.404492
$313$	−7.34806	−0.415337	−0.207668	−	0.978199i	$-0.566588\pi$
$313$	−7.34806	−0.415337	−0.207668	+	0.978199i	$0.566588\pi$
$314$	−13.2807	−0.749474
$315$	0	0
$316$	−2.91062	−0.163735
$317$	−26.0638	−1.46389	−0.731946	−	0.681363i	$-0.761387\pi$
$317$	−26.0638	−1.46389	−0.731946	+	0.681363i	$0.761387\pi$
$318$	40.7374	2.28444
$319$	0.997974	0.0558758
$320$	0	0
$321$	−17.7675	−0.991685
$322$	−5.11724	−0.285173
$323$	0	0
$324$	−23.6991	−1.31662
$325$	0	0
$326$	−8.72899	−0.483454
$327$	−7.37476	−0.407825
$328$	−2.72888	−0.150677
$329$	12.4657	0.687257
$330$	0	0
$331$	−0.0195086	−0.00107229	−0.000536145	−	1.00000i	$-0.500171\pi$
$331$	−0.0195086	−0.00107229	−0.000536145		1.00000i	$0.500171\pi$
$332$	31.6342	1.73615
$333$	2.59942	0.142448
$334$	−27.5838	−1.50932
$335$	0	0
$336$	8.14244	0.444206
$337$	−26.3708	−1.43651	−0.718255	−	0.695780i	$-0.755059\pi$
$337$	−26.3708	−1.43651	−0.718255	+	0.695780i	$0.755059\pi$
$338$	−33.6572	−1.83071
$339$	6.59748	0.358326
$340$	0	0
$341$	4.41461	0.239064
$342$	−4.41091	−0.238515
$343$	−16.6416	−0.898564
$344$	2.61758	0.141130
$345$	0	0
$346$	18.4170	0.990105
$347$	9.44465	0.507015	0.253508	−	0.967333i	$-0.418416\pi$
$347$	9.44465	0.507015	0.253508	+	0.967333i	$0.418416\pi$
$348$	7.96382	0.426905
$349$	−13.9171	−0.744964	−0.372482	−	0.928039i	$-0.621493\pi$
$349$	−13.9171	−0.744964	−0.372482	+	0.928039i	$0.621493\pi$
$350$	0	0
$351$	25.6649	1.36989
$352$	−4.40235	−0.234646
$353$	−13.3125	−0.708554	−0.354277	−	0.935140i	$-0.615273\pi$
$353$	−13.3125	−0.708554	−0.354277	+	0.935140i	$0.615273\pi$
$354$	25.2143	1.34013
$355$	0	0
$356$	2.31886	0.122899
$357$	0	0
$358$	40.2088	2.12510
$359$	34.6894	1.83084	0.915419	−	0.402502i	$-0.131859\pi$
$359$	34.6894	1.83084	0.915419	+	0.402502i	$0.131859\pi$
$360$	0	0
$361$	4.63771	0.244090
$362$	10.2381	0.538102
$363$	19.8395	1.04130
$364$	17.3779	0.910851
$365$	0	0
$366$	59.7360	3.12245
$367$	−1.63489	−0.0853406	−0.0426703	−	0.999089i	$-0.513587\pi$
$367$	−1.63489	−0.0853406	−0.0426703	+	0.999089i	$0.513587\pi$
$368$	−5.71462	−0.297895
$369$	1.66403	0.0866259
$370$	0	0
$371$	14.4912	0.752344
$372$	35.2285	1.82651
$373$	−26.2859	−1.36103	−0.680515	−	0.732734i	$-0.738244\pi$
$373$	−26.2859	−1.36103	−0.680515	+	0.732734i	$0.738244\pi$
$374$	0	0
$375$	0	0
$376$	−6.47769	−0.334061
$377$	−9.90291	−0.510026
$378$	−13.6100	−0.700024
$379$	29.1973	1.49976	0.749881	−	0.661572i	$-0.230111\pi$
$379$	29.1973	1.49976	0.749881	+	0.661572i	$0.230111\pi$
$380$	0	0
$381$	−14.4001	−0.737739
$382$	11.6860	0.597910
$383$	−7.23455	−0.369668	−0.184834	−	0.982770i	$-0.559175\pi$
$383$	−7.23455	−0.369668	−0.184834	+	0.982770i	$0.559175\pi$
$384$	−10.4311	−0.532310
$385$	0	0
$386$	−17.3891	−0.885080
$387$	−1.59616	−0.0811373
$388$	−26.6834	−1.35464
$389$	−16.1985	−0.821295	−0.410648	−	0.911794i	$-0.634697\pi$
$389$	−16.1985	−0.821295	−0.410648	+	0.911794i	$0.634697\pi$
$390$	0	0
$391$	0	0
$392$	3.64995	0.184350
$393$	−37.4310	−1.88814
$394$	12.6589	0.637748
$395$	0	0
$396$	−0.554925	−0.0278860
$397$	29.2578	1.46841	0.734205	−	0.678928i	$-0.237555\pi$
$397$	29.2578	1.46841	0.734205	+	0.678928i	$0.237555\pi$
$398$	19.7887	0.991918
$399$	−12.3811	−0.619831
$400$	0	0
$401$	21.0113	1.04926	0.524628	−	0.851332i	$-0.324204\pi$
$401$	21.0113	1.04926	0.524628	+	0.851332i	$0.324204\pi$
$402$	21.9658	1.09556
$403$	−43.8062	−2.18214
$404$	14.1088	0.701938
$405$	0	0
$406$	5.25148	0.260626
$407$	−3.24871	−0.161033
$408$	0	0
$409$	17.5170	0.866161	0.433080	−	0.901355i	$-0.357427\pi$
$409$	17.5170	0.866161	0.433080	+	0.901355i	$0.357427\pi$
$410$	0	0
$411$	−30.9329	−1.52581
$412$	−42.5861	−2.09807
$413$	8.96925	0.441348
$414$	−1.62149	−0.0796921
$415$	0	0
$416$	43.6845	2.14181
$417$	−11.4880	−0.562571
$418$	5.51267	0.269634
$419$	28.8319	1.40853	0.704264	−	0.709938i	$-0.251277\pi$
$419$	28.8319	1.40853	0.704264	+	0.709938i	$0.251277\pi$
$420$	0	0
$421$	25.5578	1.24561	0.622806	−	0.782376i	$-0.285992\pi$
$421$	25.5578	1.24561	0.622806	+	0.782376i	$0.285992\pi$
$422$	42.7338	2.08025
$423$	3.94999	0.192055
$424$	−7.53019	−0.365698
$425$	0	0
$426$	4.09400	0.198355
$427$	21.2493	1.02833
$428$	22.4564	1.08547
$429$	5.44500	0.262887
$430$	0	0
$431$	−22.7806	−1.09730	−0.548651	−	0.836051i	$-0.684859\pi$
$431$	−22.7806	−1.09730	−0.548651	+	0.836051i	$0.684859\pi$
$432$	−15.1988	−0.731253
$433$	33.5441	1.61203	0.806014	−	0.591896i	$-0.201621\pi$
$433$	33.5441	1.61203	0.806014	+	0.591896i	$0.201621\pi$
$434$	23.2303	1.11509
$435$	0	0
$436$	9.32098	0.446394
$437$	8.68945	0.415673
$438$	−56.7103	−2.70973
$439$	−3.90135	−0.186201	−0.0931005	−	0.995657i	$-0.529678\pi$
$439$	−3.90135	−0.186201	−0.0931005	+	0.995657i	$0.529678\pi$
$440$	0	0
$441$	−2.22568	−0.105985
$442$	0	0
$443$	21.1747	1.00604	0.503020	−	0.864275i	$-0.332222\pi$
$443$	21.1747	1.00604	0.503020	+	0.864275i	$0.332222\pi$
$444$	−25.9247	−1.23033
$445$	0	0
$446$	−13.8559	−0.656094
$447$	−8.32142	−0.393590
$448$	−14.3796	−0.679373
$449$	−38.6511	−1.82406	−0.912029	−	0.410125i	$-0.865485\pi$
$449$	−38.6511	−1.82406	−0.912029	+	0.410125i	$0.865485\pi$
$450$	0	0
$451$	−2.07967	−0.0979279
$452$	−8.33857	−0.392213
$453$	−11.7732	−0.553154
$454$	21.9059	1.02810
$455$	0	0
$456$	6.43373	0.301287
$457$	−17.9273	−0.838605	−0.419303	−	0.907847i	$-0.637725\pi$
$457$	−17.9273	−0.838605	−0.419303	+	0.907847i	$0.637725\pi$
$458$	32.0360	1.49694
$459$	0	0
$460$	0	0
$461$	−33.2184	−1.54713	−0.773567	−	0.633715i	$-0.781529\pi$
$461$	−33.2184	−1.54713	−0.773567	+	0.633715i	$0.781529\pi$
$462$	−2.88746	−0.134337
$463$	25.4598	1.18322	0.591609	−	0.806225i	$-0.298493\pi$
$463$	25.4598	1.18322	0.591609	+	0.806225i	$0.298493\pi$
$464$	5.86452	0.272254
$465$	0	0
$466$	43.8285	2.03032
$467$	6.31549	0.292246	0.146123	−	0.989266i	$-0.453320\pi$
$467$	6.31549	0.292246	0.146123	+	0.989266i	$0.453320\pi$
$468$	5.50652	0.254539
$469$	7.81371	0.360803
$470$	0	0
$471$	−11.8122	−0.544280
$472$	−4.66078	−0.214530
$473$	1.99485	0.0917233
$474$	−4.79896	−0.220424
$475$	0	0
$476$	0	0
$477$	4.59179	0.210244
$478$	18.9675	0.867553
$479$	28.6007	1.30680	0.653399	−	0.757013i	$-0.273342\pi$
$479$	28.6007	1.30680	0.653399	+	0.757013i	$0.273342\pi$
$480$	0	0
$481$	32.2370	1.46988
$482$	26.5236	1.20812
$483$	−4.55142	−0.207097
$484$	−25.0752	−1.13978
$485$	0	0
$486$	−9.35726	−0.424454
$487$	5.12923	0.232428	0.116214	−	0.993224i	$-0.462924\pi$
$487$	5.12923	0.232428	0.116214	+	0.993224i	$0.462924\pi$
$488$	−11.0420	−0.499848
$489$	−7.76381	−0.351091
$490$	0	0
$491$	29.5051	1.33155	0.665773	−	0.746155i	$-0.268102\pi$
$491$	29.5051	1.33155	0.665773	+	0.746155i	$0.268102\pi$
$492$	−16.5957	−0.748194
$493$	0	0
$494$	−54.7023	−2.46117
$495$	0	0
$496$	25.9421	1.16483
$497$	1.45632	0.0653249
$498$	52.1577	2.33724
$499$	−3.83354	−0.171613	−0.0858064	−	0.996312i	$-0.527347\pi$
$499$	−3.83354	−0.171613	−0.0858064	+	0.996312i	$0.527347\pi$
$500$	0	0
$501$	−24.5338	−1.09609
$502$	−21.1453	−0.943762
$503$	−28.6376	−1.27689	−0.638445	−	0.769668i	$-0.720422\pi$
$503$	−28.6376	−1.27689	−0.638445	+	0.769668i	$0.720422\pi$
$504$	−0.427066	−0.0190230
$505$	0	0
$506$	2.02651	0.0900894
$507$	−29.9356	−1.32949
$508$	18.2003	0.807508
$509$	23.6043	1.04624	0.523122	−	0.852258i	$-0.324767\pi$
$509$	23.6043	1.04624	0.523122	+	0.852258i	$0.324767\pi$
$510$	0	0
$511$	−20.1731	−0.892403
$512$	−30.4357	−1.34508
$513$	23.1108	1.02037
$514$	−39.6329	−1.74813
$515$	0	0
$516$	15.9189	0.700789
$517$	−4.93663	−0.217113
$518$	−17.0952	−0.751118
$519$	16.3806	0.719029
$520$	0	0
$521$	17.4651	0.765162	0.382581	−	0.923922i	$-0.375035\pi$
$521$	17.4651	0.765162	0.382581	+	0.923922i	$0.375035\pi$
$522$	1.66403	0.0728325
$523$	16.1755	0.707304	0.353652	−	0.935377i	$-0.384940\pi$
$523$	16.1755	0.707304	0.353652	+	0.935377i	$0.384940\pi$
$524$	47.3091	2.06671
$525$	0	0
$526$	26.3293	1.14801
$527$	0	0
$528$	−3.22454	−0.140330
$529$	−19.8057	−0.861116
$530$	0	0
$531$	2.84208	0.123336
$532$	15.6485	0.678449
$533$	20.6366	0.893870
$534$	3.82327	0.165449
$535$	0	0
$536$	−4.06032	−0.175379
$537$	35.7629	1.54328
$538$	58.1873	2.50863
$539$	2.78162	0.119813
$540$	0	0
$541$	8.97397	0.385821	0.192911	−	0.981216i	$-0.438207\pi$
$541$	8.97397	0.385821	0.192911	+	0.981216i	$0.438207\pi$
$542$	16.2163	0.696548
$543$	9.10605	0.390778
$544$	0	0
$545$	0	0
$546$	28.6523	1.22621
$547$	44.0020	1.88139	0.940694	−	0.339256i	$-0.110175\pi$
$547$	44.0020	1.88139	0.940694	+	0.339256i	$0.110175\pi$
$548$	39.0962	1.67011
$549$	6.73325	0.287368
$550$	0	0
$551$	−8.91739	−0.379894
$552$	2.36510	0.100665
$553$	−1.70709	−0.0725929
$554$	−64.5474	−2.74236
$555$	0	0
$556$	14.5197	0.615774
$557$	37.7611	1.59999	0.799995	−	0.600007i	$-0.204835\pi$
$557$	37.7611	1.59999	0.799995	+	0.600007i	$0.204835\pi$
$558$	7.36094	0.311613
$559$	−19.7949	−0.837236
$560$	0	0
$561$	0	0
$562$	4.69929	0.198228
$563$	−32.4625	−1.36813	−0.684065	−	0.729421i	$-0.739789\pi$
$563$	−32.4625	−1.36813	−0.684065	+	0.729421i	$0.739789\pi$
$564$	−39.3942	−1.65879
$565$	0	0
$566$	54.8076	2.30374
$567$	−13.8996	−0.583729
$568$	−0.756763	−0.0317531
$569$	−11.3302	−0.474986	−0.237493	−	0.971389i	$-0.576326\pi$
$569$	−11.3302	−0.474986	−0.237493	+	0.971389i	$0.576326\pi$
$570$	0	0
$571$	11.4874	0.480732	0.240366	−	0.970682i	$-0.422733\pi$
$571$	11.4874	0.480732	0.240366	+	0.970682i	$0.422733\pi$
$572$	−6.88195	−0.287749
$573$	10.3939	0.434211
$574$	−10.9435	−0.456773
$575$	0	0
$576$	−4.55645	−0.189852
$577$	−0.378762	−0.0157681	−0.00788404	−	0.999969i	$-0.502510\pi$
$577$	−0.378762	−0.0157681	−0.00788404	+	0.999969i	$0.502510\pi$
$578$	0	0
$579$	−15.4663	−0.642759
$580$	0	0
$581$	18.5536	0.769732
$582$	−43.9950	−1.82365
$583$	−5.73874	−0.237674
$584$	10.4827	0.433778
$585$	0	0
$586$	−17.1477	−0.708366
$587$	−11.6262	−0.479863	−0.239931	−	0.970790i	$-0.577125\pi$
$587$	−11.6262	−0.479863	−0.239931	+	0.970790i	$0.577125\pi$
$588$	22.1972	0.915399
$589$	−39.4467	−1.62537
$590$	0	0
$591$	11.2592	0.463142
$592$	−19.0908	−0.784627
$593$	24.0394	0.987181	0.493590	−	0.869695i	$-0.335684\pi$
$593$	24.0394	0.987181	0.493590	+	0.869695i	$0.335684\pi$
$594$	5.38979	0.221146
$595$	0	0
$596$	10.5175	0.430812
$597$	17.6006	0.720346
$598$	−20.1091	−0.822322
$599$	41.0811	1.67853	0.839263	−	0.543725i	$-0.182987\pi$
$599$	41.0811	1.67853	0.839263	+	0.543725i	$0.182987\pi$
$600$	0	0
$601$	−24.4701	−0.998157	−0.499079	−	0.866557i	$-0.666328\pi$
$601$	−24.4701	−0.998157	−0.499079	+	0.866557i	$0.666328\pi$
$602$	10.4972	0.427833
$603$	2.47592	0.100827
$604$	14.8802	0.605467
$605$	0	0
$606$	23.2622	0.944963
$607$	−1.37352	−0.0557496	−0.0278748	−	0.999611i	$-0.508874\pi$
$607$	−1.37352	−0.0557496	−0.0278748	+	0.999611i	$0.508874\pi$
$608$	39.3371	1.59533
$609$	4.67081	0.189271
$610$	0	0
$611$	48.9862	1.98177
$612$	0	0
$613$	1.94295	0.0784750	0.0392375	−	0.999230i	$-0.487507\pi$
$613$	1.94295	0.0784750	0.0392375	+	0.999230i	$0.487507\pi$
$614$	−1.29244	−0.0521586
$615$	0	0
$616$	0.533739	0.0215049
$617$	−2.30548	−0.0928151	−0.0464075	−	0.998923i	$-0.514777\pi$
$617$	−2.30548	−0.0928151	−0.0464075	+	0.998923i	$0.514777\pi$
$618$	−70.2150	−2.82446
$619$	−27.6078	−1.10965	−0.554826	−	0.831966i	$-0.687215\pi$
$619$	−27.6078	−1.10965	−0.554826	+	0.831966i	$0.687215\pi$
$620$	0	0
$621$	8.49576	0.340923
$622$	48.1033	1.92877
$623$	1.36002	0.0544880
$624$	31.9971	1.28091
$625$	0	0
$626$	15.3126	0.612013
$627$	4.90313	0.195812
$628$	14.9295	0.595753
$629$	0	0
$630$	0	0
$631$	−10.0851	−0.401480	−0.200740	−	0.979645i	$-0.564335\pi$
$631$	−10.0851	−0.401480	−0.200740	+	0.979645i	$0.564335\pi$
$632$	0.887073	0.0352859
$633$	38.0086	1.51071
$634$	54.3142	2.15709
$635$	0	0
$636$	−45.7950	−1.81589
$637$	−27.6020	−1.09363
$638$	−2.07967	−0.0823350
$639$	0.461463	0.0182552
$640$	0	0
$641$	−3.68916	−0.145713	−0.0728565	−	0.997342i	$-0.523212\pi$
$641$	−3.68916	−0.145713	−0.0728565	+	0.997342i	$0.523212\pi$
$642$	37.0255	1.46128
$643$	−10.1461	−0.400123	−0.200061	−	0.979783i	$-0.564114\pi$
$643$	−10.1461	−0.400123	−0.200061	+	0.979783i	$0.564114\pi$
$644$	5.75255	0.226682
$645$	0	0
$646$	0	0
$647$	40.6109	1.59658	0.798290	−	0.602273i	$-0.205738\pi$
$647$	40.6109	1.59658	0.798290	+	0.602273i	$0.205738\pi$
$648$	7.22280	0.283739
$649$	−3.55197	−0.139427
$650$	0	0
$651$	20.6616	0.809794
$652$	9.81269	0.384295
$653$	6.04491	0.236555	0.118278	−	0.992981i	$-0.462263\pi$
$653$	6.04491	0.236555	0.118278	+	0.992981i	$0.462263\pi$
$654$	15.3682	0.600944
$655$	0	0
$656$	−12.2210	−0.477151
$657$	−6.39221	−0.249384
$658$	−25.9772	−1.01270
$659$	32.7193	1.27456	0.637281	−	0.770631i	$-0.280059\pi$
$659$	32.7193	1.27456	0.637281	+	0.770631i	$0.280059\pi$
$660$	0	0
$661$	0.0904199	0.00351693	0.00175846	−	0.999998i	$-0.499440\pi$
$661$	0.0904199	0.00351693	0.00175846	+	0.999998i	$0.499440\pi$
$662$	0.0406539	0.00158006
$663$	0	0
$664$	−9.64118	−0.374151
$665$	0	0
$666$	−5.41692	−0.209901
$667$	−3.27812	−0.126929
$668$	31.0083	1.19975
$669$	−12.3238	−0.476465
$670$	0	0
$671$	−8.41508	−0.324861
$672$	−20.6043	−0.794827
$673$	11.5667	0.445862	0.222931	−	0.974834i	$-0.428437\pi$
$673$	11.5667	0.445862	0.222931	+	0.974834i	$0.428437\pi$
$674$	54.9540	2.11675
$675$	0	0
$676$	37.8357	1.45522
$677$	27.5593	1.05919	0.529595	−	0.848251i	$-0.322344\pi$
$677$	27.5593	1.05919	0.529595	+	0.848251i	$0.322344\pi$
$678$	−13.7484	−0.528005
$679$	−15.6499	−0.600589
$680$	0	0
$681$	19.4838	0.746619
$682$	−9.19956	−0.352269
$683$	0.203722	0.00779522	0.00389761	−	0.999992i	$-0.498759\pi$
$683$	0.203722	0.00779522	0.00389761	+	0.999992i	$0.498759\pi$
$684$	4.95853	0.189594
$685$	0	0
$686$	34.6794	1.32407
$687$	28.4937	1.08710
$688$	11.7226	0.446919
$689$	56.9455	2.16945
$690$	0	0
$691$	25.7909	0.981131	0.490565	−	0.871404i	$-0.336790\pi$
$691$	25.7909	0.981131	0.490565	+	0.871404i	$0.336790\pi$
$692$	−20.7035	−0.787029
$693$	−0.325466	−0.0123634
$694$	−19.6816	−0.747105
$695$	0	0
$696$	−2.42714	−0.0920005
$697$	0	0
$698$	29.0017	1.09773
$699$	38.9823	1.47445
$700$	0	0
$701$	18.4768	0.697859	0.348930	−	0.937149i	$-0.386545\pi$
$701$	18.4768	0.697859	0.348930	+	0.937149i	$0.386545\pi$
$702$	−53.4829	−2.01858
$703$	29.0288	1.09484
$704$	5.69456	0.214622
$705$	0	0
$706$	27.7419	1.04408
$707$	8.27485	0.311208
$708$	−28.3447	−1.06526
$709$	0.684389	0.0257028	0.0128514	−	0.999917i	$-0.495909\pi$
$709$	0.684389	0.0257028	0.0128514	+	0.999917i	$0.495909\pi$
$710$	0	0
$711$	−0.540924	−0.0202862
$712$	−0.706720	−0.0264855
$713$	−14.5010	−0.543066
$714$	0	0
$715$	0	0
$716$	−45.2008	−1.68923
$717$	16.8702	0.630030
$718$	−72.2890	−2.69780
$719$	−36.2469	−1.35178	−0.675891	−	0.737002i	$-0.736241\pi$
$719$	−36.2469	−1.35178	−0.675891	+	0.737002i	$0.736241\pi$
$720$	0	0
$721$	−24.9769	−0.930190
$722$	−9.66449	−0.359675
$723$	23.5909	0.877353
$724$	−11.5092	−0.427734
$725$	0	0
$726$	−41.3433	−1.53440
$727$	−22.3302	−0.828182	−0.414091	−	0.910235i	$-0.635900\pi$
$727$	−22.3302	−0.828182	−0.414091	+	0.910235i	$0.635900\pi$
$728$	−5.29629	−0.196294
$729$	22.0270	0.815816
$730$	0	0
$731$	0	0
$732$	−67.1522	−2.48202
$733$	17.9985	0.664790	0.332395	−	0.943140i	$-0.392143\pi$
$733$	17.9985	0.664790	0.332395	+	0.943140i	$0.392143\pi$
$734$	3.40694	0.125752
$735$	0	0
$736$	14.4607	0.533029
$737$	−3.09436	−0.113982
$738$	−3.46766	−0.127646
$739$	−8.18227	−0.300990	−0.150495	−	0.988611i	$-0.548087\pi$
$739$	−8.18227	−0.300990	−0.150495	+	0.988611i	$0.548087\pi$
$740$	0	0
$741$	−48.6538	−1.78734
$742$	−30.1980	−1.10860
$743$	−11.8933	−0.436324	−0.218162	−	0.975913i	$-0.570006\pi$
$743$	−11.8933	−0.436324	−0.218162	+	0.975913i	$0.570006\pi$
$744$	−10.7366	−0.393624
$745$	0	0
$746$	54.7769	2.00553
$747$	5.87905	0.215103
$748$	0	0
$749$	13.1708	0.481249
$750$	0	0
$751$	41.5954	1.51784	0.758919	−	0.651185i	$-0.225728\pi$
$751$	41.5954	1.51784	0.758919	+	0.651185i	$0.225728\pi$
$752$	−29.0097	−1.05788
$753$	−18.8073	−0.685375
$754$	20.6366	0.751541
$755$	0	0
$756$	15.2997	0.556445
$757$	4.10700	0.149272	0.0746358	−	0.997211i	$-0.476221\pi$
$757$	4.10700	0.149272	0.0746358	+	0.997211i	$0.476221\pi$
$758$	−60.8440	−2.20995
$759$	1.80244	0.0654243
$760$	0	0
$761$	11.1702	0.404920	0.202460	−	0.979291i	$-0.435106\pi$
$761$	11.1702	0.404920	0.202460	+	0.979291i	$0.435106\pi$
$762$	30.0082	1.08708
$763$	5.46679	0.197911
$764$	−13.1369	−0.475275
$765$	0	0
$766$	15.0760	0.544719
$767$	35.2463	1.27267
$768$	−17.0592	−0.615570
$769$	43.1786	1.55706	0.778530	−	0.627607i	$-0.215966\pi$
$769$	43.1786	1.55706	0.778530	+	0.627607i	$0.215966\pi$
$770$	0	0
$771$	−35.2506	−1.26952
$772$	19.5479	0.703545
$773$	−22.0379	−0.792648	−0.396324	−	0.918111i	$-0.629714\pi$
$773$	−22.0379	−0.792648	−0.396324	+	0.918111i	$0.629714\pi$
$774$	3.32622	0.119559
$775$	0	0
$776$	8.13233	0.291934
$777$	−15.2049	−0.545474
$778$	33.7559	1.21021
$779$	18.5829	0.665801
$780$	0	0
$781$	−0.576727	−0.0206369
$782$	0	0
$783$	−8.71862	−0.311578
$784$	16.3460	0.583784
$785$	0	0
$786$	78.0021	2.78224
$787$	−21.1016	−0.752190	−0.376095	−	0.926581i	$-0.622733\pi$
$787$	−21.1016	−0.752190	−0.376095	+	0.926581i	$0.622733\pi$
$788$	−14.2306	−0.506942
$789$	23.4180	0.833702
$790$	0	0
$791$	−4.89060	−0.173890
$792$	0.169125	0.00600960
$793$	83.5030	2.96528
$794$	−60.9702	−2.16375
$795$	0	0
$796$	−22.2455	−0.788470
$797$	36.8066	1.30376	0.651878	−	0.758324i	$-0.273981\pi$
$797$	36.8066	1.30376	0.651878	+	0.758324i	$0.273981\pi$
$798$	25.8009	0.913343
$799$	0	0
$800$	0	0
$801$	0.430947	0.0152268
$802$	−43.7854	−1.54612
$803$	7.98886	0.281921
$804$	−24.6929	−0.870852
$805$	0	0
$806$	91.2874	3.21546
$807$	51.7534	1.82181
$808$	−4.29995	−0.151272
$809$	21.6608	0.761555	0.380777	−	0.924667i	$-0.375656\pi$
$809$	21.6608	0.761555	0.380777	+	0.924667i	$0.375656\pi$
$810$	0	0
$811$	−32.0893	−1.12681	−0.563404	−	0.826181i	$-0.690509\pi$
$811$	−32.0893	−1.12681	−0.563404	+	0.826181i	$0.690509\pi$
$812$	−5.90345	−0.207170
$813$	14.4232	0.505843
$814$	6.76997	0.237287
$815$	0	0
$816$	0	0
$817$	−17.8250	−0.623617
$818$	−36.5036	−1.27632
$819$	3.22960	0.112851
$820$	0	0
$821$	5.09618	0.177858	0.0889290	−	0.996038i	$-0.471656\pi$
$821$	5.09618	0.177858	0.0889290	+	0.996038i	$0.471656\pi$
$822$	64.4609	2.24833
$823$	33.5174	1.16834	0.584171	−	0.811630i	$-0.301420\pi$
$823$	33.5174	1.16834	0.584171	+	0.811630i	$0.301420\pi$
$824$	12.9790	0.452146
$825$	0	0
$826$	−18.6910	−0.650342
$827$	4.17906	0.145320	0.0726600	−	0.997357i	$-0.476851\pi$
$827$	4.17906	0.145320	0.0726600	+	0.997357i	$0.476851\pi$
$828$	1.82280	0.0633468
$829$	0.834781	0.0289931	0.0144966	−	0.999895i	$-0.495385\pi$
$829$	0.834781	0.0289931	0.0144966	+	0.999895i	$0.495385\pi$
$830$	0	0
$831$	−57.4103	−1.99154
$832$	−56.5072	−1.95903
$833$	0	0
$834$	23.9398	0.828967
$835$	0	0
$836$	−6.19707	−0.214330
$837$	−38.5674	−1.33308
$838$	−60.0825	−2.07552
$839$	−19.1352	−0.660622	−0.330311	−	0.943872i	$-0.607154\pi$
$839$	−19.1352	−0.660622	−0.330311	+	0.943872i	$0.607154\pi$
$840$	0	0
$841$	−25.6359	−0.883996
$842$	−53.2598	−1.83545
$843$	4.17968	0.143956
$844$	−48.0392	−1.65358
$845$	0	0
$846$	−8.23136	−0.283000
$847$	−14.7067	−0.505328
$848$	−33.7233	−1.15806
$849$	48.7474	1.67301
$850$	0	0
$851$	10.6713	0.365807
$852$	−4.60227	−0.157671
$853$	2.89987	0.0992896	0.0496448	−	0.998767i	$-0.484191\pi$
$853$	2.89987	0.0992896	0.0496448	+	0.998767i	$0.484191\pi$
$854$	−44.2813	−1.51528
$855$	0	0
$856$	−6.84405	−0.233925
$857$	−27.8801	−0.952366	−0.476183	−	0.879346i	$-0.657980\pi$
$857$	−27.8801	−0.952366	−0.476183	+	0.879346i	$0.657980\pi$
$858$	−11.3468	−0.387373
$859$	−2.48841	−0.0849034	−0.0424517	−	0.999099i	$-0.513517\pi$
$859$	−2.48841	−0.0849034	−0.0424517	+	0.999099i	$0.513517\pi$
$860$	0	0
$861$	−9.73347	−0.331716
$862$	47.4723	1.61691
$863$	26.8323	0.913381	0.456690	−	0.889626i	$-0.349035\pi$
$863$	26.8323	0.913381	0.456690	+	0.889626i	$0.349035\pi$
$864$	38.4603	1.30845
$865$	0	0
$866$	−69.9024	−2.37538
$867$	0	0
$868$	−26.1143	−0.886377
$869$	0.676036	0.0229330
$870$	0	0
$871$	30.7053	1.04041
$872$	−2.84076	−0.0962004
$873$	−4.95897	−0.167836
$874$	−18.1079	−0.612508
$875$	0	0
$876$	63.7510	2.15395
$877$	33.4680	1.13013	0.565067	−	0.825045i	$-0.308850\pi$
$877$	33.4680	1.13013	0.565067	+	0.825045i	$0.308850\pi$
$878$	8.12999	0.274374
$879$	−15.2517	−0.514426
$880$	0	0
$881$	18.8647	0.635568	0.317784	−	0.948163i	$-0.397061\pi$
$881$	18.8647	0.635568	0.317784	+	0.948163i	$0.397061\pi$
$882$	4.63808	0.156172
$883$	−2.82469	−0.0950584	−0.0475292	−	0.998870i	$-0.515135\pi$
$883$	−2.82469	−0.0950584	−0.0475292	+	0.998870i	$0.515135\pi$
$884$	0	0
$885$	0	0
$886$	−44.1258	−1.48243
$887$	−0.901891	−0.0302825	−0.0151413	−	0.999885i	$-0.504820\pi$
$887$	−0.901891	−0.0302825	−0.0151413	+	0.999885i	$0.504820\pi$
$888$	7.90109	0.265143
$889$	10.6746	0.358013
$890$	0	0
$891$	5.50448	0.184407
$892$	15.5761	0.521525
$893$	44.1112	1.47613
$894$	17.3409	0.579968
$895$	0	0
$896$	7.73241	0.258322
$897$	−17.8856	−0.597183
$898$	80.5448	2.68781
$899$	14.8814	0.496322
$900$	0	0
$901$	0	0
$902$	4.33381	0.144300
$903$	9.33648	0.310699
$904$	2.54135	0.0845242
$905$	0	0
$906$	24.5341	0.815092
$907$	20.9250	0.694803	0.347402	−	0.937716i	$-0.387064\pi$
$907$	20.9250	0.694803	0.347402	+	0.937716i	$0.387064\pi$
$908$	−24.6256	−0.817228
$909$	2.62204	0.0869676
$910$	0	0
$911$	−56.6734	−1.87767	−0.938836	−	0.344364i	$-0.888095\pi$
$911$	−56.6734	−1.87767	−0.938836	+	0.344364i	$0.888095\pi$
$912$	28.8128	0.954089
$913$	−7.34752	−0.243168
$914$	37.3586	1.23571
$915$	0	0
$916$	−36.0132	−1.18991
$917$	27.7470	0.916286
$918$	0	0
$919$	28.1639	0.929040	0.464520	−	0.885563i	$-0.346227\pi$
$919$	28.1639	0.929040	0.464520	+	0.885563i	$0.346227\pi$
$920$	0	0
$921$	−1.14953	−0.0378783
$922$	69.2235	2.27976
$923$	5.72287	0.188371
$924$	3.24594	0.106784
$925$	0	0
$926$	−53.0555	−1.74351
$927$	−7.91441	−0.259943
$928$	−14.8400	−0.487149
$929$	26.8322	0.880336	0.440168	−	0.897916i	$-0.354919\pi$
$929$	26.8322	0.880336	0.440168	+	0.897916i	$0.354919\pi$
$930$	0	0
$931$	−24.8551	−0.814594
$932$	−49.2698	−1.61389
$933$	42.7844	1.40070
$934$	−13.1608	−0.430635
$935$	0	0
$936$	−1.67823	−0.0548546
$937$	−19.9363	−0.651291	−0.325645	−	0.945492i	$-0.605582\pi$
$937$	−19.9363	−0.651291	−0.325645	+	0.945492i	$0.605582\pi$
$938$	−16.2829	−0.531656
$939$	13.6194	0.444453
$940$	0	0
$941$	−2.38853	−0.0778638	−0.0389319	−	0.999242i	$-0.512396\pi$
$941$	−2.38853	−0.0778638	−0.0389319	+	0.999242i	$0.512396\pi$
$942$	24.6155	0.802015
$943$	6.83125	0.222456
$944$	−20.8729	−0.679355
$945$	0	0
$946$	−4.15705	−0.135157
$947$	58.0237	1.88552	0.942758	−	0.333476i	$-0.108222\pi$
$947$	58.0237	1.88552	0.942758	+	0.333476i	$0.108222\pi$
$948$	5.39475	0.175213
$949$	−79.2735	−2.57333
$950$	0	0
$951$	48.3086	1.56651
$952$	0	0
$953$	−10.4939	−0.339930	−0.169965	−	0.985450i	$-0.554365\pi$
$953$	−10.4939	−0.339930	−0.169965	+	0.985450i	$0.554365\pi$
$954$	−9.56881	−0.309802
$955$	0	0
$956$	−21.3223	−0.689613
$957$	−1.84972	−0.0597929
$958$	−59.6008	−1.92561
$959$	22.9301	0.740451
$960$	0	0
$961$	34.8287	1.12351
$962$	−67.1784	−2.16592
$963$	4.17340	0.134486
$964$	−29.8165	−0.960326
$965$	0	0
$966$	9.48467	0.305164
$967$	−10.5850	−0.340391	−0.170195	−	0.985410i	$-0.554440\pi$
$967$	−10.5850	−0.340391	−0.170195	+	0.985410i	$0.554440\pi$
$968$	7.64219	0.245629
$969$	0	0
$970$	0	0
$971$	9.38889	0.301304	0.150652	−	0.988587i	$-0.451863\pi$
$971$	9.38889	0.301304	0.150652	+	0.988587i	$0.451863\pi$
$972$	10.5190	0.337396
$973$	8.51588	0.273007
$974$	−10.6888	−0.342490
$975$	0	0
$976$	−49.4506	−1.58288
$977$	−20.1085	−0.643329	−0.321664	−	0.946854i	$-0.604242\pi$
$977$	−20.1085	−0.643329	−0.321664	+	0.946854i	$0.604242\pi$
$978$	16.1789	0.517345
$979$	−0.538590	−0.0172134
$980$	0	0
$981$	1.73225	0.0553066
$982$	−61.4854	−1.96208
$983$	−14.7205	−0.469511	−0.234755	−	0.972055i	$-0.575429\pi$
$983$	−14.7205	−0.469511	−0.234755	+	0.972055i	$0.575429\pi$
$984$	5.05790	0.161240
$985$	0	0
$986$	0	0
$987$	−23.1049	−0.735436
$988$	61.4936	1.95637
$989$	−6.55263	−0.208362
$990$	0	0
$991$	30.1230	0.956890	0.478445	−	0.878118i	$-0.341201\pi$
$991$	30.1230	0.956890	0.478445	+	0.878118i	$0.341201\pi$
$992$	−65.6459	−2.08426
$993$	0.0361587	0.00114746
$994$	−3.03482	−0.0962586
$995$	0	0
$996$	−58.6331	−1.85786
$997$	−47.1540	−1.49338	−0.746691	−	0.665171i	$-0.768359\pi$
$997$	−47.1540	−1.49338	−0.746691	+	0.665171i	$0.768359\pi$
$998$	7.98868	0.252877
$999$	28.3818	0.897959

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.br.1.1		12
5.4	even	2		7225.2.a.bm.1.12		12
17.2	even	8		425.2.e.e.276.1	yes	12
17.9	even	8		425.2.e.e.251.6	yes	12
17.16	even	2	inner	7225.2.a.br.1.2		12
85.2	odd	8		425.2.j.a.174.6		12
85.9	even	8		425.2.e.c.251.1	✓	12
85.19	even	8		425.2.e.c.276.6	yes	12
85.43	odd	8		425.2.j.a.149.6		12
85.53	odd	8		425.2.j.d.174.1		12
85.77	odd	8		425.2.j.d.149.1		12
85.84	even	2		7225.2.a.bm.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.e.c.251.1	✓	12	85.9	even	8
425.2.e.c.276.6	yes	12	85.19	even	8
425.2.e.e.251.6	yes	12	17.9	even	8
425.2.e.e.276.1	yes	12	17.2	even	8
425.2.j.a.149.6		12	85.43	odd	8
425.2.j.a.174.6		12	85.2	odd	8
425.2.j.d.149.1		12	85.77	odd	8
425.2.j.d.174.1		12	85.53	odd	8
7225.2.a.bm.1.11		12	85.84	even	2
7225.2.a.bm.1.12		12	5.4	even	2
7225.2.a.br.1.1		12	1.1	even	1	trivial
7225.2.a.br.1.2		12	17.16	even	2	inner

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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q-2.08389 q^{2} -1.85347 q^{3} +2.34261 q^{4} +3.86244 q^{6} +1.37395 q^{7} -0.713960 q^{8} +0.435362 q^{9} -0.544107 q^{11} -4.34196 q^{12} +5.39918 q^{13} -2.86316 q^{14} -3.19740 q^{16} -0.907247 q^{18} +4.86186 q^{19} -2.54658 q^{21} +1.13386 q^{22} +1.78727 q^{23} +1.32330 q^{24} -11.2513 q^{26} +4.75349 q^{27} +3.21863 q^{28} -1.83415 q^{29} -8.11349 q^{31} +8.09096 q^{32} +1.00849 q^{33} +1.01988 q^{36} +5.97072 q^{37} -10.1316 q^{38} -10.0072 q^{39} +3.82217 q^{41} +5.30680 q^{42} -3.66628 q^{43} -1.27463 q^{44} -3.72447 q^{46} +9.07290 q^{47} +5.92630 q^{48} -5.11226 q^{49} +12.6482 q^{52} +10.5471 q^{53} -9.90576 q^{54} -0.980945 q^{56} -9.01133 q^{57} +3.82217 q^{58} +6.52808 q^{59} +15.4659 q^{61} +16.9076 q^{62} +0.598165 q^{63} -10.4659 q^{64} -2.10158 q^{66} +5.68704 q^{67} -3.31265 q^{69} +1.05995 q^{71} -0.310831 q^{72} -14.6825 q^{73} -12.4423 q^{74} +11.3894 q^{76} -0.747575 q^{77} +20.8540 q^{78} -1.24247 q^{79} -10.1165 q^{81} -7.96500 q^{82} +13.5038 q^{83} -5.96564 q^{84} +7.64014 q^{86} +3.39955 q^{87} +0.388470 q^{88} +0.989860 q^{89} +7.41820 q^{91} +4.18687 q^{92} +15.0381 q^{93} -18.9070 q^{94} -14.9964 q^{96} -11.3905 q^{97} +10.6534 q^{98} -0.236883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9} + 12 q^{13} + 4 q^{16} + 4 q^{18} + 12 q^{19} - 8 q^{21} + 12 q^{26} + 48 q^{32} - 4 q^{33} - 20 q^{36} - 12 q^{38} + 56 q^{42} + 16 q^{43} + 36 q^{47} + 16 q^{49}+ \cdots + 76 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9 + 12 * q^13 + 4 * q^16 + 4 * q^18 + 12 * q^19 - 8 * q^21 + 12 * q^26 + 48 * q^32 - 4 * q^33 - 20 * q^36 - 12 * q^38 + 56 * q^42 + 16 * q^43 + 36 * q^47 + 16 * q^49 + 64 * q^52 + 48 * q^53 - 32 * q^59 + 44 * q^64 + 40 * q^66 - 20 * q^67 - 72 * q^69 - 92 * q^72 + 36 * q^76 + 24 * q^77 + 8 * q^81 + 52 * q^83 + 36 * q^84 + 84 * q^86 + 48 * q^87 + 24 * q^89 + 60 * q^93 - 8 * q^94 + 76 * q^98

Introduction

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