LMFDB - Embedded newform 7225.2.a.bp.1.11

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} - 30x^{10} + 343x^{8} - 1860x^{6} + 4823x^{4} - 5230x^{2} + 1681 $ x^12 - 30x^10 + 343x^8 - 1860x^6 + 4823x^4 - 5230*x^2 + 1681
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$3.07592$ 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)

$f(q)$	$=$	\(q+2.38621 q^{2} -3.15462 q^{3} +3.69399 q^{4} -7.52757 q^{6} +0.219993 q^{7} +4.04223 q^{8} +6.95160 q^{9} +0.524950 q^{11} -11.6531 q^{12} +1.96713 q^{13} +0.524950 q^{14} +2.25761 q^{16} +16.5880 q^{18} +4.00000 q^{19} -0.693995 q^{21} +1.25264 q^{22} -0.372668 q^{23} -12.7517 q^{24} +4.69399 q^{26} -12.4658 q^{27} +0.812655 q^{28} +7.00262 q^{29} +2.92062 q^{31} -2.69733 q^{32} -1.65602 q^{33} +25.6792 q^{36} -5.71657 q^{37} +9.54484 q^{38} -6.20555 q^{39} -0.797070 q^{41} -1.65602 q^{42} -2.49417 q^{43} +1.93916 q^{44} -0.889263 q^{46} +6.73955 q^{47} -7.12189 q^{48} -6.95160 q^{49} +7.26658 q^{52} -5.92169 q^{53} -29.7460 q^{54} +0.889263 q^{56} -12.6185 q^{57} +16.7097 q^{58} +6.00000 q^{59} +5.65685 q^{61} +6.96921 q^{62} +1.52931 q^{63} -10.9516 q^{64} -3.95160 q^{66} +11.5120 q^{67} +1.17562 q^{69} -7.16326 q^{71} +28.1000 q^{72} -1.18532 q^{73} -13.6409 q^{74} +14.7760 q^{76} +0.115486 q^{77} -14.8078 q^{78} +6.73050 q^{79} +18.4700 q^{81} -1.90197 q^{82} -6.11732 q^{83} -2.56361 q^{84} -5.95160 q^{86} -22.0906 q^{87} +2.12197 q^{88} +15.9852 q^{89} +0.432757 q^{91} -1.37663 q^{92} -9.21344 q^{93} +16.0820 q^{94} +8.50903 q^{96} +9.21517 q^{97} -16.5880 q^{98} +3.64925 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{4} + 28 q^{9} + 4 q^{16} + 48 q^{19} + 24 q^{21} + 24 q^{26} + 68 q^{36} - 28 q^{49} + 72 q^{59} - 76 q^{64} + 8 q^{66} + 88 q^{69} + 48 q^{76} + 60 q^{81} - 40 q^{84} - 16 q^{86} - 16 q^{89}+ \cdots + 96 q^{94}+O(q^{100}) $ 12 * q + 12 * q^4 + 28 * q^9 + 4 * q^16 + 48 * q^19 + 24 * q^21 + 24 * q^26 + 68 * q^36 - 28 * q^49 + 72 * q^59 - 76 * q^64 + 8 * q^66 + 88 * q^69 + 48 * q^76 + 60 * q^81 - 40 * q^84 - 16 * q^86 - 16 * q^89 + 96 * q^94

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 $

$2$	2.38621	1.68730	0.843652	−	0.536890i	$-0.180401\pi$
$2$	2.38621	1.68730	0.843652	+	0.536890i	$0.180401\pi$
$3$	−3.15462	−1.82132	−0.910659	−	0.413158i	$-0.864426\pi$
$3$	−3.15462	−1.82132	−0.910659	+	0.413158i	$0.864426\pi$
$4$	3.69399	1.84700
$5$	0	0
$5$	0	0
$6$	−7.52757	−3.07312
$7$	0.219993	0.0831497	0.0415749	−	0.999135i	$-0.486762\pi$
$7$	0.219993	0.0831497	0.0415749	+	0.999135i	$0.486762\pi$
$8$	4.04223	1.42914
$9$	6.95160	2.31720
$10$	0	0
$11$	0.524950	0.158278	0.0791392	−	0.996864i	$-0.474783\pi$
$11$	0.524950	0.158278	0.0791392	+	0.996864i	$0.474783\pi$
$12$	−11.6531	−3.36397
$13$	1.96713	0.545585	0.272792	−	0.962073i	$-0.412053\pi$
$13$	1.96713	0.545585	0.272792	+	0.962073i	$0.412053\pi$
$14$	0.524950	0.140299
$15$	0	0
$16$	2.25761	0.564402
$17$	0	0
$17$	0	0
$18$	16.5880	3.90982
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−0.693995	−0.151442
$22$	1.25264	0.267064
$23$	−0.372668	−0.0777066	−0.0388533	−	0.999245i	$-0.512371\pi$
$23$	−0.372668	−0.0777066	−0.0388533	+	0.999245i	$0.512371\pi$
$24$	−12.7517	−2.60292
$25$	0	0
$26$	4.69399	0.920568
$27$	−12.4658	−2.39904
$28$	0.812655	0.153577
$29$	7.00262	1.30035	0.650177	−	0.759783i	$-0.274695\pi$
$29$	7.00262	1.30035	0.650177	+	0.759783i	$0.274695\pi$
$30$	0	0
$31$	2.92062	0.524559	0.262279	−	0.964992i	$-0.415526\pi$
$31$	2.92062	0.524559	0.262279	+	0.964992i	$0.415526\pi$
$32$	−2.69733	−0.476825
$33$	−1.65602	−0.288276
$34$	0	0
$35$	0	0
$36$	25.6792	4.27986
$37$	−5.71657	−0.939798	−0.469899	−	0.882720i	$-0.655710\pi$
$37$	−5.71657	−0.939798	−0.469899	+	0.882720i	$0.655710\pi$
$38$	9.54484	1.54838
$39$	−6.20555	−0.993684
$40$	0	0
$41$	−0.797070	−0.124481	−0.0622407	−	0.998061i	$-0.519825\pi$
$41$	−0.797070	−0.124481	−0.0622407	+	0.998061i	$0.519825\pi$
$42$	−1.65602	−0.255529
$43$	−2.49417	−0.380357	−0.190178	−	0.981750i	$-0.560907\pi$
$43$	−2.49417	−0.380357	−0.190178	+	0.981750i	$0.560907\pi$
$44$	1.93916	0.292340
$45$	0	0
$46$	−0.889263	−0.131115
$47$	6.73955	0.983065	0.491532	−	0.870859i	$-0.336437\pi$
$47$	6.73955	0.983065	0.491532	+	0.870859i	$0.336437\pi$
$48$	−7.12189	−1.02796
$49$	−6.95160	−0.993086
$50$	0	0
$51$	0	0
$52$	7.26658	1.00769
$53$	−5.92169	−0.813406	−0.406703	−	0.913560i	$-0.633322\pi$
$53$	−5.92169	−0.813406	−0.406703	+	0.913560i	$0.633322\pi$
$54$	−29.7460	−4.04792
$55$	0	0
$56$	0.889263	0.118833
$57$	−12.6185	−1.67136
$58$	16.7097	2.19409
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	5.65685	0.724286	0.362143	−	0.932123i	$-0.382045\pi$
$61$	5.65685	0.724286	0.362143	+	0.932123i	$0.382045\pi$
$62$	6.96921	0.885091
$63$	1.52931	0.192675
$64$	−10.9516	−1.36895
$65$	0	0
$66$	−3.95160	−0.486409
$67$	11.5120	1.40641	0.703206	−	0.710987i	$-0.251751\pi$
$67$	11.5120	1.40641	0.703206	+	0.710987i	$0.251751\pi$
$68$	0	0
$69$	1.17562	0.141528
$70$	0	0
$71$	−7.16326	−0.850123	−0.425061	−	0.905165i	$-0.639747\pi$
$71$	−7.16326	−0.850123	−0.425061	+	0.905165i	$0.639747\pi$
$72$	28.1000	3.31161
$73$	−1.18532	−0.138731	−0.0693657	−	0.997591i	$-0.522098\pi$
$73$	−1.18532	−0.138731	−0.0693657	+	0.997591i	$0.522098\pi$
$74$	−13.6409	−1.58573
$75$	0	0
$76$	14.7760	1.69492
$77$	0.115486	0.0131608
$78$	−14.8078	−1.67665
$79$	6.73050	0.757241	0.378620	−	0.925552i	$-0.376399\pi$
$79$	6.73050	0.757241	0.378620	+	0.925552i	$0.376399\pi$
$80$	0	0
$81$	18.4700	2.05222
$82$	−1.90197	−0.210038
$83$	−6.11732	−0.671463	−0.335731	−	0.941958i	$-0.608983\pi$
$83$	−6.11732	−0.671463	−0.335731	+	0.941958i	$0.608983\pi$
$84$	−2.56361	−0.279713
$85$	0	0
$86$	−5.95160	−0.641778
$87$	−22.0906	−2.36836
$88$	2.12197	0.226203
$89$	15.9852	1.69443	0.847213	−	0.531253i	$-0.178279\pi$
$89$	15.9852	1.69443	0.847213	+	0.531253i	$0.178279\pi$
$90$	0	0
$91$	0.432757	0.0453652
$92$	−1.37663	−0.143524
$93$	−9.21344	−0.955389
$94$	16.0820	1.65873
$95$	0	0
$96$	8.50903	0.868449
$97$	9.21517	0.935659	0.467829	−	0.883819i	$-0.345036\pi$
$97$	9.21517	0.935659	0.467829	+	0.883819i	$0.345036\pi$
$98$	−16.5880	−1.67564
$99$	3.64925	0.366763
$100$	0	0
$101$	−7.20921	−0.717343	−0.358672	−	0.933464i	$-0.616770\pi$
$101$	−7.20921	−0.717343	−0.358672	+	0.933464i	$0.616770\pi$
$102$	0	0
$103$	−9.52456	−0.938482	−0.469241	−	0.883070i	$-0.655472\pi$
$103$	−9.52456	−0.938482	−0.469241	+	0.883070i	$0.655472\pi$
$104$	7.95160	0.779719
$105$	0	0
$106$	−14.1304	−1.37246
$107$	10.7838	1.04251	0.521255	−	0.853401i	$-0.325464\pi$
$107$	10.7838	1.04251	0.521255	+	0.853401i	$0.325464\pi$
$108$	−46.0486	−4.43103
$109$	−14.0737	−1.34802	−0.674008	−	0.738724i	$-0.735429\pi$
$109$	−14.0737	−1.34802	−0.674008	+	0.738724i	$0.735429\pi$
$110$	0	0
$111$	18.0336	1.71167
$112$	0.496659	0.0469299
$113$	7.18921	0.676304	0.338152	−	0.941092i	$-0.390198\pi$
$113$	7.18921	0.676304	0.338152	+	0.941092i	$0.390198\pi$
$114$	−30.1103	−2.82009
$115$	0	0
$116$	25.8677	2.40175
$117$	13.6747	1.26423
$118$	14.3173	1.31801
$119$	0	0
$120$	0	0
$121$	−10.7244	−0.974948
$122$	13.4984	1.22209
$123$	2.51445	0.226720
$124$	10.7888	0.968859
$125$	0	0
$126$	3.64925	0.325101
$127$	9.74047	0.864327	0.432163	−	0.901795i	$-0.357750\pi$
$127$	9.74047	0.864327	0.432163	+	0.901795i	$0.357750\pi$
$128$	−20.7382	−1.83301
$129$	7.86814	0.692751
$130$	0	0
$131$	19.6859	1.71996	0.859980	−	0.510327i	$-0.170476\pi$
$131$	19.6859	1.71996	0.859980	+	0.510327i	$0.170476\pi$
$132$	−6.11732	−0.532444
$133$	0.879974	0.0763034
$134$	27.4700	2.37304
$135$	0	0
$136$	0	0
$137$	−4.65693	−0.397869	−0.198934	−	0.980013i	$-0.563748\pi$
$137$	−4.65693	−0.397869	−0.198934	+	0.980013i	$0.563748\pi$
$138$	2.80528	0.238802
$139$	5.24785	0.445117	0.222558	−	0.974919i	$-0.428559\pi$
$139$	5.24785	0.445117	0.222558	+	0.974919i	$0.428559\pi$
$140$	0	0
$141$	−21.2607	−1.79047
$142$	−17.0930	−1.43442
$143$	1.03265	0.0863544
$144$	15.6940	1.30783
$145$	0	0
$146$	−2.82843	−0.234082
$147$	21.9296	1.80873
$148$	−21.1170	−1.73581
$149$	8.74239	0.716205	0.358102	−	0.933682i	$-0.383424\pi$
$149$	8.74239	0.716205	0.358102	+	0.933682i	$0.383424\pi$
$150$	0	0
$151$	15.9032	1.29418	0.647092	−	0.762412i	$-0.275985\pi$
$151$	15.9032	1.29418	0.647092	+	0.762412i	$0.275985\pi$
$152$	16.1689	1.31147
$153$	0	0
$154$	0.275573	0.0222063
$155$	0	0
$156$	−22.9233	−1.83533
$157$	10.0719	0.803823	0.401911	−	0.915679i	$-0.368346\pi$
$157$	10.0719	0.803823	0.401911	+	0.915679i	$0.368346\pi$
$158$	16.0604	1.27770
$159$	18.6806	1.48147
$160$	0	0
$161$	−0.0819845	−0.00646128
$162$	44.0732	3.46272
$163$	9.58784	0.750978	0.375489	−	0.926827i	$-0.377475\pi$
$163$	9.58784	0.750978	0.375489	+	0.926827i	$0.377475\pi$
$164$	−2.94437	−0.229917
$165$	0	0
$166$	−14.5972	−1.13296
$167$	5.34390	0.413524	0.206762	−	0.978391i	$-0.433707\pi$
$167$	5.34390	0.413524	0.206762	+	0.978391i	$0.433707\pi$
$168$	−2.80528	−0.216432
$169$	−9.13038	−0.702337
$170$	0	0
$171$	27.8064	2.12641
$172$	−9.21344	−0.702518
$173$	−17.4551	−1.32708	−0.663542	−	0.748139i	$-0.730948\pi$
$173$	−17.4551	−1.32708	−0.663542	+	0.748139i	$0.730948\pi$
$174$	−52.7128	−3.99614
$175$	0	0
$176$	1.18513	0.0893327
$177$	−18.9277	−1.42269
$178$	38.1440	2.85901
$179$	−21.3248	−1.59389	−0.796945	−	0.604052i	$-0.793552\pi$
$179$	−21.3248	−1.59389	−0.796945	+	0.604052i	$0.793552\pi$
$180$	0	0
$181$	14.4380	1.07317	0.536584	−	0.843847i	$-0.319714\pi$
$181$	14.4380	1.07317	0.536584	+	0.843847i	$0.319714\pi$
$182$	1.03265	0.0765450
$183$	−17.8452	−1.31916
$184$	−1.50641	−0.111054
$185$	0	0
$186$	−21.9852	−1.61203
$187$	0	0
$188$	24.8959	1.81572
$189$	−2.74239	−0.199480
$190$	0	0
$191$	6.74239	0.487862	0.243931	−	0.969793i	$-0.421563\pi$
$191$	6.74239	0.487862	0.243931	+	0.969793i	$0.421563\pi$
$192$	34.5481	2.49329
$193$	14.6551	1.05490	0.527448	−	0.849587i	$-0.323149\pi$
$193$	14.6551	1.05490	0.527448	+	0.849587i	$0.323149\pi$
$194$	21.9893	1.57874
$195$	0	0
$196$	−25.6792	−1.83423
$197$	0.869327	0.0619370	0.0309685	−	0.999520i	$-0.490141\pi$
$197$	0.869327	0.0619370	0.0309685	+	0.999520i	$0.490141\pi$
$198$	8.70787	0.618841
$199$	13.3004	0.942838	0.471419	−	0.881909i	$-0.343742\pi$
$199$	13.3004	0.942838	0.471419	+	0.881909i	$0.343742\pi$
$200$	0	0
$201$	−36.3159	−2.56152
$202$	−17.2027	−1.21038
$203$	1.54053	0.108124
$204$	0	0
$205$	0	0
$206$	−22.7276	−1.58351
$207$	−2.59064	−0.180062
$208$	4.44102	0.307929
$209$	2.09980	0.145246
$210$	0	0
$211$	17.7914	1.22481	0.612405	−	0.790544i	$-0.290202\pi$
$211$	17.7914	1.22481	0.612405	+	0.790544i	$0.290202\pi$
$212$	−21.8747	−1.50236
$213$	22.5973	1.54834
$214$	25.7324	1.75903
$215$	0	0
$216$	−50.3895	−3.42857
$217$	0.642517	0.0436169
$218$	−33.5828	−2.27451
$219$	3.73924	0.252674
$220$	0	0
$221$	0	0
$222$	43.0319	2.88811
$223$	1.96713	0.131729	0.0658645	−	0.997829i	$-0.479019\pi$
$223$	1.96713	0.131729	0.0658645	+	0.997829i	$0.479019\pi$
$224$	−0.593394	−0.0396478
$225$	0	0
$226$	17.1550	1.14113
$227$	11.9585	0.793712	0.396856	−	0.917881i	$-0.370101\pi$
$227$	11.9585	0.793712	0.396856	+	0.917881i	$0.370101\pi$
$228$	−46.6125	−3.08699
$229$	9.30601	0.614958	0.307479	−	0.951555i	$-0.400515\pi$
$229$	9.30601	0.614958	0.307479	+	0.951555i	$0.400515\pi$
$230$	0	0
$231$	−0.364313	−0.0239700
$232$	28.3062	1.85839
$233$	−25.8009	−1.69027	−0.845137	−	0.534550i	$-0.820481\pi$
$233$	−25.8009	−1.69027	−0.845137	+	0.534550i	$0.820481\pi$
$234$	32.6308	2.13314
$235$	0	0
$236$	22.1640	1.44275
$237$	−21.2322	−1.37918
$238$	0	0
$239$	29.2912	1.89469	0.947345	−	0.320215i	$-0.103755\pi$
$239$	29.2912	1.89469	0.947345	+	0.320215i	$0.103755\pi$
$240$	0	0
$241$	8.48528	0.546585	0.273293	−	0.961931i	$-0.411887\pi$
$241$	8.48528	0.546585	0.273293	+	0.961931i	$0.411887\pi$
$242$	−25.5907	−1.64503
$243$	−20.8683	−1.33870
$244$	20.8964	1.33775
$245$	0	0
$246$	6.00000	0.382546
$247$	7.86854	0.500663
$248$	11.8058	0.749670
$249$	19.2978	1.22295
$250$	0	0
$251$	1.77282	0.111900	0.0559498	−	0.998434i	$-0.482181\pi$
$251$	1.77282	0.111900	0.0559498	+	0.998434i	$0.482181\pi$
$252$	5.64925	0.355869
$253$	−0.195632	−0.0122993
$254$	23.2428	1.45838
$255$	0	0
$256$	−27.5824	−1.72390
$257$	−7.40235	−0.461746	−0.230873	−	0.972984i	$-0.574158\pi$
$257$	−7.40235	−0.461746	−0.230873	+	0.972984i	$0.574158\pi$
$258$	18.7750	1.16888
$259$	−1.25761	−0.0781440
$260$	0	0
$261$	48.6795	3.01318
$262$	46.9746	2.90210
$263$	7.67291	0.473132	0.236566	−	0.971615i	$-0.423978\pi$
$263$	7.67291	0.473132	0.236566	+	0.971615i	$0.423978\pi$
$264$	−6.69399	−0.411987
$265$	0	0
$266$	2.09980	0.128747
$267$	−50.4271	−3.08609
$268$	42.5252	2.59764
$269$	7.87260	0.480001	0.240000	−	0.970773i	$-0.422852\pi$
$269$	7.87260	0.480001	0.240000	+	0.970773i	$0.422852\pi$
$270$	0	0
$271$	18.0672	1.09750	0.548751	−	0.835986i	$-0.315103\pi$
$271$	18.0672	1.09750	0.548751	+	0.835986i	$0.315103\pi$
$272$	0	0
$273$	−1.36518	−0.0826245
$274$	−11.1124	−0.671326
$275$	0	0
$276$	4.34275	0.261403
$277$	−26.5569	−1.59565	−0.797824	−	0.602890i	$-0.794016\pi$
$277$	−26.5569	−1.59565	−0.797824	+	0.602890i	$0.794016\pi$
$278$	12.5225	0.751047
$279$	20.3030	1.21551
$280$	0	0
$281$	13.7728	0.821618	0.410809	−	0.911721i	$-0.365246\pi$
$281$	13.7728	0.821618	0.410809	+	0.911721i	$0.365246\pi$
$282$	−50.7325	−3.02108
$283$	13.7471	0.817181	0.408591	−	0.912718i	$-0.366020\pi$
$283$	13.7471	0.817181	0.408591	+	0.912718i	$0.366020\pi$
$284$	−26.4611	−1.57017
$285$	0	0
$286$	2.46411	0.145706
$287$	−0.175350	−0.0103506
$288$	−18.7507	−1.10490
$289$	0	0
$290$	0	0
$291$	−29.0703	−1.70413
$292$	−4.37857	−0.256237
$293$	19.8873	1.16183	0.580913	−	0.813966i	$-0.302696\pi$
$293$	19.8873	1.16183	0.580913	+	0.813966i	$0.302696\pi$
$294$	52.3287	3.05187
$295$	0	0
$296$	−23.1077	−1.34311
$297$	−6.54392	−0.379717
$298$	20.8612	1.20846
$299$	−0.733088	−0.0423955
$300$	0	0
$301$	−0.548700	−0.0316266
$302$	37.9484	2.18368
$303$	22.7423	1.30651
$304$	9.03043	0.517931
$305$	0	0
$306$	0	0
$307$	−14.9598	−0.853799	−0.426900	−	0.904299i	$-0.640394\pi$
$307$	−14.9598	−0.853799	−0.426900	+	0.904299i	$0.640394\pi$
$308$	0.426603	0.0243080
$309$	30.0463	1.70928
$310$	0	0
$311$	29.6538	1.68151	0.840756	−	0.541414i	$-0.182111\pi$
$311$	29.6538	1.68151	0.840756	+	0.541414i	$0.182111\pi$
$312$	−25.0843	−1.42012
$313$	14.3784	0.812716	0.406358	−	0.913714i	$-0.366799\pi$
$313$	14.3784	0.812716	0.406358	+	0.913714i	$0.366799\pi$
$314$	24.0336	1.35629
$315$	0	0
$316$	24.8624	1.39862
$317$	−13.7751	−0.773687	−0.386843	−	0.922145i	$-0.626435\pi$
$317$	−13.7751	−0.773687	−0.386843	+	0.922145i	$0.626435\pi$
$318$	44.5759	2.49969
$319$	3.67603	0.205818
$320$	0	0
$321$	−34.0188	−1.89874
$322$	−0.195632	−0.0109021
$323$	0	0
$324$	68.2280	3.79044
$325$	0	0
$326$	22.8786	1.26713
$327$	44.3971	2.45516
$328$	−3.22194	−0.177902
$329$	1.48266	0.0817415
$330$	0	0
$331$	−20.1640	−1.10831	−0.554156	−	0.832413i	$-0.686959\pi$
$331$	−20.1640	−1.10831	−0.554156	+	0.832413i	$0.686959\pi$
$332$	−22.5973	−1.24019
$333$	−39.7393	−2.17770
$334$	12.7517	0.697740
$335$	0	0
$336$	−1.56677	−0.0854742
$337$	−28.5829	−1.55701	−0.778504	−	0.627640i	$-0.784021\pi$
$337$	−28.5829	−1.55701	−0.778504	+	0.627640i	$0.784021\pi$
$338$	−21.7870	−1.18506
$339$	−22.6792	−1.23176
$340$	0	0
$341$	1.53318	0.0830264
$342$	66.3519	3.58790
$343$	−3.06926	−0.165725
$344$	−10.0820	−0.543584
$345$	0	0
$346$	−41.6514	−2.23919
$347$	−10.9365	−0.587101	−0.293551	−	0.955944i	$-0.594837\pi$
$347$	−10.9365	−0.587101	−0.293551	+	0.955944i	$0.594837\pi$
$348$	−81.6025	−4.37435
$349$	0.645598	0.0345581	0.0172790	−	0.999851i	$-0.494500\pi$
$349$	0.645598	0.0345581	0.0172790	+	0.999851i	$0.494500\pi$
$350$	0	0
$351$	−24.5219	−1.30888
$352$	−1.41596	−0.0754711
$353$	14.3022	0.761229	0.380615	−	0.924734i	$-0.375712\pi$
$353$	14.3022	0.761229	0.380615	+	0.924734i	$0.375712\pi$
$354$	−45.1654	−2.40052
$355$	0	0
$356$	59.0492	3.12960
$357$	0	0
$358$	−50.8854	−2.68938
$359$	−15.6760	−0.827349	−0.413675	−	0.910425i	$-0.635755\pi$
$359$	−15.6760	−0.827349	−0.413675	+	0.910425i	$0.635755\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	34.4521	1.81076
$363$	33.8315	1.77569
$364$	1.59860	0.0837895
$365$	0	0
$366$	−42.5824	−2.22582
$367$	−21.9010	−1.14322	−0.571610	−	0.820525i	$-0.693681\pi$
$367$	−21.9010	−1.14322	−0.571610	+	0.820525i	$0.693681\pi$
$368$	−0.841338	−0.0438578
$369$	−5.54091	−0.288448
$370$	0	0
$371$	−1.30273	−0.0676345
$372$	−34.0344	−1.76460
$373$	6.50858	0.337002	0.168501	−	0.985702i	$-0.446107\pi$
$373$	6.50858	0.337002	0.168501	+	0.985702i	$0.446107\pi$
$374$	0	0
$375$	0	0
$376$	27.2428	1.40494
$377$	13.7751	0.709454
$378$	−6.54392	−0.336583
$379$	−7.52757	−0.386666	−0.193333	−	0.981133i	$-0.561930\pi$
$379$	−7.52757	−0.386666	−0.193333	+	0.981133i	$0.561930\pi$
$380$	0	0
$381$	−30.7274	−1.57421
$382$	16.0888	0.823172
$383$	−24.7752	−1.26595	−0.632976	−	0.774172i	$-0.718167\pi$
$383$	−24.7752	−1.26595	−0.632976	+	0.774172i	$0.718167\pi$
$384$	65.4209	3.33850
$385$	0	0
$386$	34.9701	1.77993
$387$	−17.3385	−0.881363
$388$	34.0408	1.72816
$389$	23.6971	1.20149	0.600747	−	0.799440i	$-0.294870\pi$
$389$	23.6971	1.20149	0.600747	+	0.799440i	$0.294870\pi$
$390$	0	0
$391$	0	0
$392$	−28.1000	−1.41926
$393$	−62.1013	−3.13260
$394$	2.07440	0.104507
$395$	0	0
$396$	13.4803	0.677410
$397$	27.7135	1.39090	0.695451	−	0.718573i	$-0.255205\pi$
$397$	27.7135	1.39090	0.695451	+	0.718573i	$0.255205\pi$
$398$	31.7375	1.59086
$399$	−2.77598	−0.138973
$400$	0	0
$401$	30.5431	1.52525	0.762624	−	0.646842i	$-0.223911\pi$
$401$	30.5431	1.52525	0.762624	+	0.646842i	$0.223911\pi$
$402$	−86.6572	−4.32207
$403$	5.74525	0.286191
$404$	−26.6308	−1.32493
$405$	0	0
$406$	3.67603	0.182438
$407$	−3.00092	−0.148750
$408$	0	0
$409$	−5.90321	−0.291895	−0.145947	−	0.989292i	$-0.546623\pi$
$409$	−5.90321	−0.291895	−0.145947	+	0.989292i	$0.546623\pi$
$410$	0	0
$411$	14.6908	0.724645
$412$	−35.1837	−1.73337
$413$	1.31996	0.0649510
$414$	−6.18180	−0.303819
$415$	0	0
$416$	−5.30601	−0.260148
$417$	−16.5549	−0.810699
$418$	5.01057	0.245075
$419$	−32.4867	−1.58708	−0.793539	−	0.608519i	$-0.791764\pi$
$419$	−32.4867	−1.58708	−0.793539	+	0.608519i	$0.791764\pi$
$420$	0	0
$421$	−30.1156	−1.46774	−0.733872	−	0.679288i	$-0.762289\pi$
$421$	−30.1156	−1.46774	−0.733872	+	0.679288i	$0.762289\pi$
$422$	42.4540	2.06663
$423$	46.8507	2.27796
$424$	−23.9368	−1.16247
$425$	0	0
$426$	53.9220	2.61253
$427$	1.24447	0.0602242
$428$	39.8353	1.92551
$429$	−3.25761	−0.157279
$430$	0	0
$431$	−8.32464	−0.400984	−0.200492	−	0.979695i	$-0.564254\pi$
$431$	−8.32464	−0.400984	−0.200492	+	0.979695i	$0.564254\pi$
$432$	−28.1429	−1.35402
$433$	−13.5795	−0.652591	−0.326295	−	0.945268i	$-0.605800\pi$
$433$	−13.5795	−0.652591	−0.326295	+	0.945268i	$0.605800\pi$
$434$	1.53318	0.0735950
$435$	0	0
$436$	−51.9881	−2.48978
$437$	−1.49067	−0.0713085
$438$	8.92260	0.426338
$439$	−20.3714	−0.972276	−0.486138	−	0.873882i	$-0.661595\pi$
$439$	−20.3714	−0.972276	−0.486138	+	0.873882i	$0.661595\pi$
$440$	0	0
$441$	−48.3248	−2.30118
$442$	0	0
$443$	2.45360	0.116574	0.0582871	−	0.998300i	$-0.481436\pi$
$443$	2.45360	0.116574	0.0582871	+	0.998300i	$0.481436\pi$
$444$	66.6160	3.16145
$445$	0	0
$446$	4.69399	0.222267
$447$	−27.5789	−1.30444
$448$	−2.40928	−0.113828
$449$	3.19720	0.150885	0.0754426	−	0.997150i	$-0.475963\pi$
$449$	3.19720	0.150885	0.0754426	+	0.997150i	$0.475963\pi$
$450$	0	0
$451$	−0.418422	−0.0197027
$452$	26.5569	1.24913
$453$	−50.1685	−2.35712
$454$	28.5354	1.33923
$455$	0	0
$456$	−51.0067	−2.38861
$457$	30.1803	1.41177	0.705887	−	0.708325i	$-0.250549\pi$
$457$	30.1803	1.41177	0.705887	+	0.708325i	$0.250549\pi$
$458$	22.2061	1.03762
$459$	0	0
$460$	0	0
$461$	5.41527	0.252214	0.126107	−	0.992017i	$-0.459752\pi$
$461$	5.41527	0.252214	0.126107	+	0.992017i	$0.459752\pi$
$462$	−0.869327	−0.0404447
$463$	−15.7574	−0.732307	−0.366153	−	0.930555i	$-0.619325\pi$
$463$	−15.7574	−0.732307	−0.366153	+	0.930555i	$0.619325\pi$
$464$	15.8092	0.733923
$465$	0	0
$466$	−61.5664	−2.85201
$467$	10.7690	0.498331	0.249166	−	0.968461i	$-0.419844\pi$
$467$	10.7690	0.498331	0.249166	+	0.968461i	$0.419844\pi$
$468$	50.5144	2.33503
$469$	2.53256	0.116943
$470$	0	0
$471$	−31.7729	−1.46402
$472$	24.2534	1.11635
$473$	−1.30931	−0.0602023
$474$	−50.6644	−2.32709
$475$	0	0
$476$	0	0
$477$	−41.1652	−1.88483
$478$	69.8949	3.19692
$479$	−36.9092	−1.68643	−0.843213	−	0.537579i	$-0.819339\pi$
$479$	−36.9092	−1.68643	−0.843213	+	0.537579i	$0.819339\pi$
$480$	0	0
$481$	−11.2453	−0.512740
$482$	20.2477	0.922256
$483$	0.258629	0.0117680
$484$	−39.6160	−1.80073
$485$	0	0
$486$	−49.7961	−2.25880
$487$	38.3234	1.73660	0.868299	−	0.496041i	$-0.165214\pi$
$487$	38.3234	1.73660	0.868299	+	0.496041i	$0.165214\pi$
$488$	22.8663	1.03511
$489$	−30.2460	−1.36777
$490$	0	0
$491$	38.0336	1.71643	0.858216	−	0.513289i	$-0.171573\pi$
$491$	38.0336	1.71643	0.858216	+	0.513289i	$0.171573\pi$
$492$	9.28836	0.418752
$493$	0	0
$494$	18.7760	0.844771
$495$	0	0
$496$	6.59362	0.296062
$497$	−1.57587	−0.0706875
$498$	46.0486	2.06349
$499$	−9.30610	−0.416598	−0.208299	−	0.978065i	$-0.566793\pi$
$499$	−9.30610	−0.416598	−0.208299	+	0.978065i	$0.566793\pi$
$500$	0	0
$501$	−16.8580	−0.753158
$502$	4.23033	0.188809
$503$	−1.56864	−0.0699421	−0.0349710	−	0.999388i	$-0.511134\pi$
$503$	−1.56864	−0.0699421	−0.0349710	+	0.999388i	$0.511134\pi$
$504$	6.18180	0.275359
$505$	0	0
$506$	−0.466819	−0.0207526
$507$	28.8028	1.27918
$508$	35.9812	1.59641
$509$	24.4247	1.08261	0.541304	−	0.840827i	$-0.317931\pi$
$509$	24.4247	1.08261	0.541304	+	0.840827i	$0.317931\pi$
$510$	0	0
$511$	−0.260763	−0.0115355
$512$	−24.3410	−1.07573
$513$	−49.8632	−2.20151
$514$	−17.6636	−0.779106
$515$	0	0
$516$	29.0649	1.27951
$517$	3.53793	0.155598
$518$	−3.00092	−0.131853
$519$	55.0640	2.41704
$520$	0	0
$521$	−7.50829	−0.328944	−0.164472	−	0.986382i	$-0.552592\pi$
$521$	−7.50829	−0.328944	−0.164472	+	0.986382i	$0.552592\pi$
$522$	116.159	5.08416
$523$	−8.98247	−0.392776	−0.196388	−	0.980526i	$-0.562921\pi$
$523$	−8.98247	−0.392776	−0.196388	+	0.980526i	$0.562921\pi$
$524$	72.7194	3.17676
$525$	0	0
$526$	18.3092	0.798317
$527$	0	0
$528$	−3.73864	−0.162703
$529$	−22.8611	−0.993962
$530$	0	0
$531$	41.7096	1.81004
$532$	3.25062	0.140932
$533$	−1.56794	−0.0679152
$534$	−120.330	−5.20718
$535$	0	0
$536$	46.5340	2.00996
$537$	67.2715	2.90298
$538$	18.7857	0.809908
$539$	−3.64925	−0.157184
$540$	0	0
$541$	−12.7279	−0.547216	−0.273608	−	0.961841i	$-0.588217\pi$
$541$	−12.7279	−0.547216	−0.273608	+	0.961841i	$0.588217\pi$
$542$	43.1121	1.85182
$543$	−45.5464	−1.95458
$544$	0	0
$545$	0	0
$546$	−3.25761	−0.139413
$547$	−2.57260	−0.109997	−0.0549983	−	0.998486i	$-0.517515\pi$
$547$	−2.57260	−0.109997	−0.0549983	+	0.998486i	$0.517515\pi$
$548$	−17.2027	−0.734862
$549$	39.3242	1.67832
$550$	0	0
$551$	28.0105	1.19329
$552$	4.75214	0.202264
$553$	1.48067	0.0629644
$554$	−63.3703	−2.69235
$555$	0	0
$556$	19.3855	0.822129
$557$	−21.0162	−0.890487	−0.445243	−	0.895410i	$-0.646883\pi$
$557$	−21.0162	−0.890487	−0.445243	+	0.895410i	$0.646883\pi$
$558$	48.4472	2.05093
$559$	−4.90636	−0.207517
$560$	0	0
$561$	0	0
$562$	32.8648	1.38632
$563$	−40.5377	−1.70846	−0.854231	−	0.519893i	$-0.825972\pi$
$563$	−40.5377	−1.70846	−0.854231	+	0.519893i	$0.825972\pi$
$564$	−78.5369	−3.30700
$565$	0	0
$566$	32.8035	1.37883
$567$	4.06327	0.170641
$568$	−28.9555	−1.21495
$569$	−0.612010	−0.0256568	−0.0128284	−	0.999918i	$-0.504084\pi$
$569$	−0.612010	−0.0256568	−0.0128284	+	0.999918i	$0.504084\pi$
$570$	0	0
$571$	14.8945	0.623316	0.311658	−	0.950194i	$-0.399116\pi$
$571$	14.8945	0.623316	0.311658	+	0.950194i	$0.399116\pi$
$572$	3.81460	0.159496
$573$	−21.2697	−0.888553
$574$	−0.418422	−0.0174646
$575$	0	0
$576$	−76.1312	−3.17213
$577$	37.2107	1.54910	0.774550	−	0.632513i	$-0.217976\pi$
$577$	37.2107	1.54910	0.774550	+	0.632513i	$0.217976\pi$
$578$	0	0
$579$	−46.2311	−1.92130
$580$	0	0
$581$	−1.34577	−0.0558319
$582$	−69.3679	−2.87539
$583$	−3.10859	−0.128745
$584$	−4.79134	−0.198267
$585$	0	0
$586$	47.4552	1.96035
$587$	9.60470	0.396428	0.198214	−	0.980159i	$-0.436486\pi$
$587$	9.60470	0.396428	0.198214	+	0.980159i	$0.436486\pi$
$588$	81.0080	3.34071
$589$	11.6825	0.481368
$590$	0	0
$591$	−2.74239	−0.112807
$592$	−12.9058	−0.530424
$593$	16.5206	0.678419	0.339210	−	0.940711i	$-0.389840\pi$
$593$	16.5206	0.678419	0.339210	+	0.940711i	$0.389840\pi$
$594$	−15.6152	−0.640698
$595$	0	0
$596$	32.2944	1.32283
$597$	−41.9576	−1.71721
$598$	−1.74930	−0.0715342
$599$	34.8096	1.42228	0.711140	−	0.703050i	$-0.248179\pi$
$599$	34.8096	1.42228	0.711140	+	0.703050i	$0.248179\pi$
$600$	0	0
$601$	−11.6825	−0.476538	−0.238269	−	0.971199i	$-0.576580\pi$
$601$	−11.6825	−0.476538	−0.238269	+	0.971199i	$0.576580\pi$
$602$	−1.30931	−0.0533636
$603$	80.0267	3.25894
$604$	58.7464	2.39036
$605$	0	0
$606$	54.2679	2.20448
$607$	26.9296	1.09304	0.546519	−	0.837447i	$-0.315953\pi$
$607$	26.9296	1.09304	0.546519	+	0.837447i	$0.315953\pi$
$608$	−10.7893	−0.437564
$609$	−4.85978	−0.196928
$610$	0	0
$611$	13.2576	0.536345
$612$	0	0
$613$	−5.43522	−0.219526	−0.109763	−	0.993958i	$-0.535009\pi$
$613$	−5.43522	−0.219526	−0.109763	+	0.993958i	$0.535009\pi$
$614$	−35.6971	−1.44062
$615$	0	0
$616$	0.466819	0.0188087
$617$	3.49860	0.140848	0.0704242	−	0.997517i	$-0.477565\pi$
$617$	3.49860	0.140848	0.0704242	+	0.997517i	$0.477565\pi$
$618$	71.6968	2.88407
$619$	12.1301	0.487549	0.243774	−	0.969832i	$-0.421614\pi$
$619$	12.1301	0.487549	0.243774	+	0.969832i	$0.421614\pi$
$620$	0	0
$621$	4.64560	0.186421
$622$	70.7602	2.83722
$623$	3.51664	0.140891
$624$	−14.0097	−0.560837
$625$	0	0
$626$	34.3099	1.37130
$627$	−6.62407	−0.264540
$628$	37.2054	1.48466
$629$	0	0
$630$	0	0
$631$	12.6183	0.502327	0.251164	−	0.967945i	$-0.419187\pi$
$631$	12.6183	0.502327	0.251164	+	0.967945i	$0.419187\pi$
$632$	27.2062	1.08221
$633$	−56.1250	−2.23077
$634$	−32.8703	−1.30545
$635$	0	0
$636$	69.0062	2.73627
$637$	−13.6747	−0.541813
$638$	8.77178	0.347278
$639$	−49.7961	−1.96991
$640$	0	0
$641$	−30.1997	−1.19282	−0.596408	−	0.802681i	$-0.703406\pi$
$641$	−30.1997	−1.19282	−0.596408	+	0.802681i	$0.703406\pi$
$642$	−81.1759	−3.20376
$643$	19.7117	0.777352	0.388676	−	0.921374i	$-0.372932\pi$
$643$	19.7117	0.777352	0.388676	+	0.921374i	$0.372932\pi$
$644$	−0.302850	−0.0119340
$645$	0	0
$646$	0	0
$647$	21.0418	0.827237	0.413618	−	0.910450i	$-0.364265\pi$
$647$	21.0418	0.827237	0.413618	+	0.910450i	$0.364265\pi$
$648$	74.6598	2.93291
$649$	3.14970	0.123637
$650$	0	0
$651$	−2.02690	−0.0794403
$652$	35.4174	1.38705
$653$	−38.7173	−1.51513	−0.757563	−	0.652762i	$-0.773610\pi$
$653$	−38.7173	−1.51513	−0.757563	+	0.652762i	$0.773610\pi$
$654$	105.941	4.14261
$655$	0	0
$656$	−1.79947	−0.0702575
$657$	−8.23989	−0.321469
$658$	3.53793	0.137923
$659$	−24.1577	−0.941049	−0.470524	−	0.882387i	$-0.655935\pi$
$659$	−24.1577	−0.941049	−0.470524	+	0.882387i	$0.655935\pi$
$660$	0	0
$661$	−25.7432	−1.00129	−0.500647	−	0.865651i	$-0.666905\pi$
$661$	−25.7432	−1.00129	−0.500647	+	0.865651i	$0.666905\pi$
$662$	−48.1155	−1.87006
$663$	0	0
$664$	−24.7276	−0.959616
$665$	0	0
$666$	−94.8264	−3.67445
$667$	−2.60965	−0.101046
$668$	19.7404	0.763777
$669$	−6.20555	−0.239921
$670$	0	0
$671$	2.96957	0.114639
$672$	1.87193	0.0722113
$673$	−10.4958	−0.404583	−0.202292	−	0.979325i	$-0.564839\pi$
$673$	−10.4958	−0.404583	−0.202292	+	0.979325i	$0.564839\pi$
$674$	−68.2047	−2.62715
$675$	0	0
$676$	−33.7276	−1.29721
$677$	8.48787	0.326215	0.163108	−	0.986608i	$-0.447848\pi$
$677$	8.48787	0.326215	0.163108	+	0.986608i	$0.447848\pi$
$678$	−54.1173	−2.07836
$679$	2.02728	0.0777998
$680$	0	0
$681$	−37.7244	−1.44560
$682$	3.65849	0.140091
$683$	14.2898	0.546784	0.273392	−	0.961903i	$-0.411854\pi$
$683$	14.2898	0.546784	0.273392	+	0.961903i	$0.411854\pi$
$684$	102.717	3.92747
$685$	0	0
$686$	−7.32390	−0.279628
$687$	−29.3569	−1.12003
$688$	−5.63085	−0.214674
$689$	−11.6488	−0.443782
$690$	0	0
$691$	−8.82584	−0.335751	−0.167875	−	0.985808i	$-0.553691\pi$
$691$	−8.82584	−0.335751	−0.167875	+	0.985808i	$0.553691\pi$
$692$	−64.4789	−2.45112
$693$	0.802810	0.0304962
$694$	−26.0967	−0.990619
$695$	0	0
$696$	−89.2952	−3.38472
$697$	0	0
$698$	1.54053	0.0583100
$699$	81.3920	3.07853
$700$	0	0
$701$	−18.5340	−0.700019	−0.350010	−	0.936746i	$-0.613822\pi$
$701$	−18.5340	−0.700019	−0.350010	+	0.936746i	$0.613822\pi$
$702$	−58.5144	−2.20848
$703$	−22.8663	−0.862418
$704$	−5.74905	−0.216675
$705$	0	0
$706$	34.1280	1.28443
$707$	−1.58598	−0.0596469
$708$	−69.9188	−2.62771
$709$	−25.4989	−0.957631	−0.478815	−	0.877916i	$-0.658934\pi$
$709$	−25.4989	−0.957631	−0.478815	+	0.877916i	$0.658934\pi$
$710$	0	0
$711$	46.7878	1.75468
$712$	64.6158	2.42158
$713$	−1.08842	−0.0407617
$714$	0	0
$715$	0	0
$716$	−78.7736	−2.94391
$717$	−92.4025	−3.45083
$718$	−37.4063	−1.39599
$719$	4.76313	0.177635	0.0888174	−	0.996048i	$-0.471691\pi$
$719$	4.76313	0.177635	0.0888174	+	0.996048i	$0.471691\pi$
$720$	0	0
$721$	−2.09534	−0.0780345
$722$	−7.15863	−0.266417
$723$	−26.7678	−0.995505
$724$	53.3339	1.98214
$725$	0	0
$726$	80.7289	2.99613
$727$	11.1858	0.414858	0.207429	−	0.978250i	$-0.433490\pi$
$727$	11.1858	0.414858	0.207429	+	0.978250i	$0.433490\pi$
$728$	1.74930	0.0648334
$729$	10.4216	0.385984
$730$	0	0
$731$	0	0
$732$	−65.9201	−2.43648
$733$	26.2804	0.970687	0.485344	−	0.874324i	$-0.338695\pi$
$733$	26.2804	0.970687	0.485344	+	0.874324i	$0.338695\pi$
$734$	−52.2603	−1.92896
$735$	0	0
$736$	1.00521	0.0370524
$737$	6.04321	0.222605
$738$	−13.2218	−0.486700
$739$	6.00000	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$740$	0	0
$741$	−24.8222	−0.911867
$742$	−3.10859	−0.114120
$743$	34.0401	1.24881	0.624405	−	0.781101i	$-0.285341\pi$
$743$	34.0401	1.24881	0.624405	+	0.781101i	$0.285341\pi$
$744$	−37.2428	−1.36539
$745$	0	0
$746$	15.5308	0.568624
$747$	−42.5252	−1.55591
$748$	0	0
$749$	2.37237	0.0866844
$750$	0	0
$751$	−13.0431	−0.475949	−0.237974	−	0.971271i	$-0.576483\pi$
$751$	−13.0431	−0.475949	−0.237974	+	0.971271i	$0.576483\pi$
$752$	15.2153	0.554844
$753$	−5.59258	−0.203805
$754$	32.8703	1.19707
$755$	0	0
$756$	−10.1304	−0.368438
$757$	−31.8716	−1.15839	−0.579197	−	0.815188i	$-0.696634\pi$
$757$	−31.8716	−1.15839	−0.579197	+	0.815188i	$0.696634\pi$
$758$	−17.9624	−0.652423
$759$	0.617144	0.0224009
$760$	0	0
$761$	−18.7549	−0.679863	−0.339932	−	0.940450i	$-0.610404\pi$
$761$	−18.7549	−0.679863	−0.339932	+	0.940450i	$0.610404\pi$
$762$	−73.3221	−2.65618
$763$	−3.09612	−0.112087
$764$	24.9064	0.901081
$765$	0	0
$766$	−59.1187	−2.13605
$767$	11.8028	0.426175
$768$	87.0119	3.13977
$769$	−47.7916	−1.72341	−0.861705	−	0.507410i	$-0.830603\pi$
$769$	−47.7916	−1.72341	−0.861705	+	0.507410i	$0.830603\pi$
$770$	0	0
$771$	23.3516	0.840987
$772$	54.1358	1.94839
$773$	−28.3286	−1.01891	−0.509455	−	0.860497i	$-0.670153\pi$
$773$	−28.3286	−1.01891	−0.509455	+	0.860497i	$0.670153\pi$
$774$	−41.3732	−1.48713
$775$	0	0
$776$	37.2498	1.33719
$777$	3.96727	0.142325
$778$	56.5464	2.02729
$779$	−3.18828	−0.114232
$780$	0	0
$781$	−3.76036	−0.134556
$782$	0	0
$783$	−87.2932	−3.11961
$784$	−15.6940	−0.560500
$785$	0	0
$786$	−148.187	−5.28565
$787$	−42.3119	−1.50826	−0.754129	−	0.656727i	$-0.771941\pi$
$787$	−42.3119	−1.50826	−0.754129	+	0.656727i	$0.771941\pi$
$788$	3.21129	0.114397
$789$	−24.2051	−0.861723
$790$	0	0
$791$	1.58158	0.0562344
$792$	14.7511	0.524157
$793$	11.1278	0.395160
$794$	66.1303	2.34688
$795$	0	0
$796$	49.1315	1.74142
$797$	−54.1578	−1.91837	−0.959185	−	0.282780i	$-0.908743\pi$
$797$	−54.1578	−1.91837	−0.959185	+	0.282780i	$0.908743\pi$
$798$	−6.62407	−0.234489
$799$	0	0
$800$	0	0
$801$	111.123	3.92633
$802$	72.8821	2.57356
$803$	−0.622235	−0.0219582
$804$	−134.151	−4.73113
$805$	0	0
$806$	13.7094	0.482892
$807$	−24.8350	−0.874234
$808$	−29.1413	−1.02519
$809$	30.9803	1.08921	0.544604	−	0.838693i	$-0.316680\pi$
$809$	30.9803	1.08921	0.544604	+	0.838693i	$0.316680\pi$
$810$	0	0
$811$	4.15045	0.145742	0.0728710	−	0.997341i	$-0.476784\pi$
$811$	4.15045	0.145742	0.0728710	+	0.997341i	$0.476784\pi$
$812$	5.69071	0.199705
$813$	−56.9950	−1.99890
$814$	−7.16081	−0.250986
$815$	0	0
$816$	0	0
$817$	−9.97667	−0.349039
$818$	−14.0863	−0.492515
$819$	3.00835	0.105120
$820$	0	0
$821$	1.40975	0.0492007	0.0246003	−	0.999697i	$-0.492169\pi$
$821$	1.40975	0.0492007	0.0246003	+	0.999697i	$0.492169\pi$
$822$	35.0554	1.22270
$823$	20.2090	0.704442	0.352221	−	0.935917i	$-0.385427\pi$
$823$	20.2090	0.704442	0.352221	+	0.935917i	$0.385427\pi$
$824$	−38.5004	−1.34123
$825$	0	0
$826$	3.14970	0.109592
$827$	−11.3584	−0.394971	−0.197486	−	0.980306i	$-0.563278\pi$
$827$	−11.3584	−0.394971	−0.197486	+	0.980306i	$0.563278\pi$
$828$	−9.56980	−0.332574
$829$	−10.9336	−0.379741	−0.189870	−	0.981809i	$-0.560807\pi$
$829$	−10.9336	−0.379741	−0.189870	+	0.981809i	$0.560807\pi$
$830$	0	0
$831$	83.7768	2.90618
$832$	−21.5433	−0.746879
$833$	0	0
$834$	−39.5036	−1.36790
$835$	0	0
$836$	7.75666	0.268270
$837$	−36.4078	−1.25844
$838$	−77.5200	−2.67788
$839$	−38.8677	−1.34186	−0.670931	−	0.741520i	$-0.734105\pi$
$839$	−38.8677	−1.34186	−0.670931	+	0.741520i	$0.734105\pi$
$840$	0	0
$841$	20.0367	0.690922
$842$	−71.8621	−2.47653
$843$	−43.4480	−1.49643
$844$	65.7213	2.26222
$845$	0	0
$846$	111.796	3.84361
$847$	−2.35930	−0.0810666
$848$	−13.3688	−0.459088
$849$	−43.3669	−1.48835
$850$	0	0
$851$	2.13038	0.0730285
$852$	83.4745	2.85979
$853$	−32.0255	−1.09653	−0.548266	−	0.836304i	$-0.684712\pi$
$853$	−32.0255	−1.09653	−0.548266	+	0.836304i	$0.684712\pi$
$854$	2.96957	0.101617
$855$	0	0
$856$	43.5906	1.48990
$857$	25.6769	0.877107	0.438553	−	0.898705i	$-0.355491\pi$
$857$	25.6769	0.877107	0.438553	+	0.898705i	$0.355491\pi$
$858$	−7.77333	−0.265377
$859$	−38.7065	−1.32065	−0.660324	−	0.750981i	$-0.729581\pi$
$859$	−38.7065	−1.32065	−0.660324	+	0.750981i	$0.729581\pi$
$860$	0	0
$861$	0.553162	0.0188517
$862$	−19.8643	−0.676582
$863$	−29.5881	−1.00719	−0.503596	−	0.863939i	$-0.667990\pi$
$863$	−29.5881	−1.00719	−0.503596	+	0.863939i	$0.667990\pi$
$864$	33.6243	1.14392
$865$	0	0
$866$	−32.4036	−1.10112
$867$	0	0
$868$	2.37346	0.0805603
$869$	3.53318	0.119855
$870$	0	0
$871$	22.6456	0.767317
$872$	−56.8890	−1.92651
$873$	64.0602	2.16811
$874$	−3.55705	−0.120319
$875$	0	0
$876$	13.8127	0.466689
$877$	−26.4995	−0.894825	−0.447413	−	0.894328i	$-0.647654\pi$
$877$	−26.4995	−0.894825	−0.447413	+	0.894328i	$0.647654\pi$
$878$	−48.6105	−1.64053
$879$	−62.7367	−2.11606
$880$	0	0
$881$	−10.9924	−0.370344	−0.185172	−	0.982706i	$-0.559284\pi$
$881$	−10.9924	−0.370344	−0.185172	+	0.982706i	$0.559284\pi$
$882$	−115.313	−3.88279
$883$	−56.2202	−1.89196	−0.945980	−	0.324225i	$-0.894896\pi$
$883$	−56.2202	−1.89196	−0.945980	+	0.324225i	$0.894896\pi$
$884$	0	0
$885$	0	0
$886$	5.85481	0.196696
$887$	−6.90120	−0.231720	−0.115860	−	0.993266i	$-0.536962\pi$
$887$	−6.90120	−0.231720	−0.115860	+	0.993266i	$0.536962\pi$
$888$	72.8958	2.44622
$889$	2.14284	0.0718685
$890$	0	0
$891$	9.69582	0.324822
$892$	7.26658	0.243303
$893$	26.9582	0.902122
$894$	−65.8090	−2.20098
$895$	0	0
$896$	−4.56226	−0.152414
$897$	2.31261	0.0772158
$898$	7.62919	0.254589
$899$	20.4520	0.682113
$900$	0	0
$901$	0	0
$902$	−0.998442	−0.0332445
$903$	1.73094	0.0576020
$904$	29.0604	0.966534
$905$	0	0
$906$	−119.713	−3.97718
$907$	−31.2128	−1.03640	−0.518202	−	0.855258i	$-0.673399\pi$
$907$	−31.2128	−1.03640	−0.518202	+	0.855258i	$0.673399\pi$
$908$	44.1746	1.46598
$909$	−50.1156	−1.66223
$910$	0	0
$911$	−42.4932	−1.40786	−0.703931	−	0.710268i	$-0.748574\pi$
$911$	−42.4932	−1.40786	−0.703931	+	0.710268i	$0.748574\pi$
$912$	−28.4875	−0.943317
$913$	−3.21129	−0.106278
$914$	72.0164	2.38209
$915$	0	0
$916$	34.3763	1.13583
$917$	4.33076	0.143014
$918$	0	0
$919$	29.1312	0.960949	0.480475	−	0.877009i	$-0.340464\pi$
$919$	29.1312	0.960949	0.480475	+	0.877009i	$0.340464\pi$
$920$	0	0
$921$	47.1923	1.55504
$922$	12.9220	0.425562
$923$	−14.0911	−0.463814
$924$	−1.34577	−0.0442726
$925$	0	0
$926$	−37.6004	−1.23562
$927$	−66.2109	−2.17465
$928$	−18.8884	−0.620041
$929$	−1.30273	−0.0427412	−0.0213706	−	0.999772i	$-0.506803\pi$
$929$	−1.30273	−0.0427412	−0.0213706	+	0.999772i	$0.506803\pi$
$930$	0	0
$931$	−27.8064	−0.911318
$932$	−95.3084	−3.12193
$933$	−93.5463	−3.06257
$934$	25.6971	0.840836
$935$	0	0
$936$	55.2764	1.80677
$937$	−9.43458	−0.308214	−0.154107	−	0.988054i	$-0.549250\pi$
$937$	−9.43458	−0.308214	−0.154107	+	0.988054i	$0.549250\pi$
$938$	6.04321	0.197318
$939$	−45.3584	−1.48021
$940$	0	0
$941$	−25.1776	−0.820767	−0.410383	−	0.911913i	$-0.634605\pi$
$941$	−25.1776	−0.820767	−0.410383	+	0.911913i	$0.634605\pi$
$942$	−75.8167	−2.47024
$943$	0.297042	0.00967302
$944$	13.5456	0.440873
$945$	0	0
$946$	−3.12430	−0.101580
$947$	27.6569	0.898727	0.449364	−	0.893349i	$-0.351651\pi$
$947$	27.6569	0.898727	0.449364	+	0.893349i	$0.351651\pi$
$948$	−78.4315	−2.54734
$949$	−2.33169	−0.0756898
$950$	0	0
$951$	43.4552	1.40913
$952$	0	0
$953$	−16.0279	−0.519195	−0.259598	−	0.965717i	$-0.583590\pi$
$953$	−16.0279	−0.519195	−0.259598	+	0.965717i	$0.583590\pi$
$954$	−98.2288	−3.18027
$955$	0	0
$956$	108.202	3.49949
$957$	−11.5965	−0.374860
$958$	−88.0732	−2.84552
$959$	−1.02449	−0.0330827
$960$	0	0
$961$	−22.4700	−0.724838
$962$	−26.8336	−0.865149
$963$	74.9648	2.41571
$964$	31.3446	1.00954
$965$	0	0
$966$	0.617144	0.0198563
$967$	10.9849	0.353252	0.176626	−	0.984278i	$-0.443482\pi$
$967$	10.9849	0.353252	0.176626	+	0.984278i	$0.443482\pi$
$968$	−43.3506	−1.39334
$969$	0	0
$970$	0	0
$971$	32.2944	1.03638	0.518188	−	0.855267i	$-0.326607\pi$
$971$	32.2944	1.03638	0.518188	+	0.855267i	$0.326607\pi$
$972$	−77.0874	−2.47258
$973$	1.15449	0.0370113
$974$	91.4476	2.93017
$975$	0	0
$976$	12.7710	0.408788
$977$	−2.59459	−0.0830084	−0.0415042	−	0.999138i	$-0.513215\pi$
$977$	−2.59459	−0.0830084	−0.0415042	+	0.999138i	$0.513215\pi$
$978$	−72.1732	−2.30784
$979$	8.39143	0.268191
$980$	0	0
$981$	−97.8347	−3.12362
$982$	90.7561	2.89614
$983$	−9.61652	−0.306719	−0.153360	−	0.988170i	$-0.549009\pi$
$983$	−9.61652	−0.306719	−0.153360	+	0.988170i	$0.549009\pi$
$984$	10.1640	0.324015
$985$	0	0
$986$	0	0
$987$	−4.67721	−0.148877
$988$	29.0663	0.924723
$989$	0.929495	0.0295562
$990$	0	0
$991$	−28.2650	−0.897867	−0.448933	−	0.893565i	$-0.648196\pi$
$991$	−28.2650	−0.897867	−0.448933	+	0.893565i	$0.648196\pi$
$992$	−7.87787	−0.250123
$993$	63.6096	2.01859
$994$	−3.76036	−0.119271
$995$	0	0
$996$	71.2859	2.25878
$997$	−3.34593	−0.105967	−0.0529833	−	0.998595i	$-0.516873\pi$
$997$	−3.34593	−0.105967	−0.0529833	+	0.998595i	$0.516873\pi$
$998$	−22.2063	−0.702928
$999$	71.2616	2.25462

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bp.1.11		12
5.2	odd	4		1445.2.b.f.579.12		12
5.3	odd	4		1445.2.b.f.579.1		12
5.4	even	2	inner	7225.2.a.bp.1.2		12
17.2	even	8		425.2.e.d.276.6		12
17.9	even	8		425.2.e.d.251.1		12
17.16	even	2	inner	7225.2.a.bp.1.12		12
85.2	odd	8		85.2.j.c.4.1	✓	12
85.9	even	8		425.2.e.d.251.6		12
85.19	even	8		425.2.e.d.276.1		12
85.33	odd	4		1445.2.b.f.579.2		12
85.43	odd	8		85.2.j.c.64.1	yes	12
85.53	odd	8		85.2.j.c.4.6	yes	12
85.67	odd	4		1445.2.b.f.579.11		12
85.77	odd	8		85.2.j.c.64.6	yes	12
85.84	even	2	inner	7225.2.a.bp.1.1		12
255.2	even	8		765.2.t.e.514.6		12
255.53	even	8		765.2.t.e.514.1		12
255.77	even	8		765.2.t.e.64.1		12
255.128	even	8		765.2.t.e.64.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.j.c.4.1	✓	12	85.2	odd	8
85.2.j.c.4.6	yes	12	85.53	odd	8
85.2.j.c.64.1	yes	12	85.43	odd	8
85.2.j.c.64.6	yes	12	85.77	odd	8
425.2.e.d.251.1		12	17.9	even	8
425.2.e.d.251.6		12	85.9	even	8
425.2.e.d.276.1		12	85.19	even	8
425.2.e.d.276.6		12	17.2	even	8
765.2.t.e.64.1		12	255.77	even	8
765.2.t.e.64.6		12	255.128	even	8
765.2.t.e.514.1		12	255.53	even	8
765.2.t.e.514.6		12	255.2	even	8
1445.2.b.f.579.1		12	5.3	odd	4
1445.2.b.f.579.2		12	85.33	odd	4
1445.2.b.f.579.11		12	85.67	odd	4
1445.2.b.f.579.12		12	5.2	odd	4
7225.2.a.bp.1.1		12	85.84	even	2	inner
7225.2.a.bp.1.2		12	5.4	even	2	inner
7225.2.a.bp.1.11		12	1.1	even	1	trivial
7225.2.a.bp.1.12		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)

\(f(q)\)	\(=\)	\(q+2.38621 q^{2} -3.15462 q^{3} +3.69399 q^{4} -7.52757 q^{6} +0.219993 q^{7} +4.04223 q^{8} +6.95160 q^{9} +0.524950 q^{11} -11.6531 q^{12} +1.96713 q^{13} +0.524950 q^{14} +2.25761 q^{16} +16.5880 q^{18} +4.00000 q^{19} -0.693995 q^{21} +1.25264 q^{22} -0.372668 q^{23} -12.7517 q^{24} +4.69399 q^{26} -12.4658 q^{27} +0.812655 q^{28} +7.00262 q^{29} +2.92062 q^{31} -2.69733 q^{32} -1.65602 q^{33} +25.6792 q^{36} -5.71657 q^{37} +9.54484 q^{38} -6.20555 q^{39} -0.797070 q^{41} -1.65602 q^{42} -2.49417 q^{43} +1.93916 q^{44} -0.889263 q^{46} +6.73955 q^{47} -7.12189 q^{48} -6.95160 q^{49} +7.26658 q^{52} -5.92169 q^{53} -29.7460 q^{54} +0.889263 q^{56} -12.6185 q^{57} +16.7097 q^{58} +6.00000 q^{59} +5.65685 q^{61} +6.96921 q^{62} +1.52931 q^{63} -10.9516 q^{64} -3.95160 q^{66} +11.5120 q^{67} +1.17562 q^{69} -7.16326 q^{71} +28.1000 q^{72} -1.18532 q^{73} -13.6409 q^{74} +14.7760 q^{76} +0.115486 q^{77} -14.8078 q^{78} +6.73050 q^{79} +18.4700 q^{81} -1.90197 q^{82} -6.11732 q^{83} -2.56361 q^{84} -5.95160 q^{86} -22.0906 q^{87} +2.12197 q^{88} +15.9852 q^{89} +0.432757 q^{91} -1.37663 q^{92} -9.21344 q^{93} +16.0820 q^{94} +8.50903 q^{96} +9.21517 q^{97} -16.5880 q^{98} +3.64925 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{4} + 28 q^{9} + 4 q^{16} + 48 q^{19} + 24 q^{21} + 24 q^{26} + 68 q^{36} - 28 q^{49} + 72 q^{59} - 76 q^{64} + 8 q^{66} + 88 q^{69} + 48 q^{76} + 60 q^{81} - 40 q^{84} - 16 q^{86} - 16 q^{89}+ \cdots + 96 q^{94}+O(q^{100}) \) 12 * q + 12 * q^4 + 28 * q^9 + 4 * q^16 + 48 * q^19 + 24 * q^21 + 24 * q^26 + 68 * q^36 - 28 * q^49 + 72 * q^59 - 76 * q^64 + 8 * q^66 + 88 * q^69 + 48 * q^76 + 60 * q^81 - 40 * q^84 - 16 * q^86 - 16 * q^89 + 96 * q^94

Introduction

L-functions

Modular forms

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Database

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