LMFDB - Embedded newform 5625.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, 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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 7x + 11 $ x^4 - 2x^3 - 6x^2 + 7*x + 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.70636$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70636 q^{2} +0.911672 q^{4} -3.94243 q^{7} -1.85708 q^{8} +O(q^{10})$	$q+1.70636 q^{2} +0.911672 q^{4} -3.94243 q^{7} -1.85708 q^{8} +5.90869 q^{11} +3.29066 q^{13} -6.72721 q^{14} -4.99220 q^{16} -2.70636 q^{17} -2.35813 q^{19} +10.0824 q^{22} +0.584296 q^{23} +5.61505 q^{26} -3.59420 q^{28} +3.91167 q^{29} +2.70934 q^{31} -4.80433 q^{32} -4.61803 q^{34} -0.0208515 q^{37} -4.02383 q^{38} -1.47214 q^{41} +1.27279 q^{43} +5.38679 q^{44} +0.997020 q^{46} -5.43358 q^{47} +8.54276 q^{49} +3.00000 q^{52} -2.81554 q^{53} +7.32142 q^{56} +6.67473 q^{58} +4.69033 q^{59} +5.58132 q^{61} +4.62312 q^{62} +1.78646 q^{64} +6.03076 q^{67} -2.46731 q^{68} +8.10138 q^{71} +13.3166 q^{73} -0.0355801 q^{74} -2.14984 q^{76} -23.2946 q^{77} +16.6648 q^{79} -2.51200 q^{82} -0.781641 q^{83} +2.17183 q^{86} -10.9729 q^{88} +3.47327 q^{89} -12.9732 q^{91} +0.532686 q^{92} -9.27165 q^{94} -2.45443 q^{97} +14.5770 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 8 q^{4} + 2 q^{7} - 15 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 8 * q^4 + 2 * q^7 - 15 * q^8	$ 4 q - 2 q^{2} + 8 q^{4} + 2 q^{7} - 15 q^{8} + 7 q^{11} + q^{13} - 16 q^{14} + 4 q^{16} - 2 q^{17} + 5 q^{19} - 6 q^{22} - q^{23} - 3 q^{26} + 9 q^{28} + 20 q^{29} + 23 q^{31} - 12 q^{32} - 14 q^{34} + 2 q^{37} - 35 q^{38} + 12 q^{41} + 16 q^{43} + 29 q^{44} - 17 q^{46} - 2 q^{47} + 8 q^{49} + 12 q^{52} + 4 q^{53} - 5 q^{56} - 25 q^{58} + 15 q^{59} - 2 q^{61} - 9 q^{62} + 23 q^{64} + 2 q^{67} + 11 q^{68} + 2 q^{71} + 16 q^{73} + 19 q^{74} + 40 q^{76} - 19 q^{77} + 35 q^{79} - 6 q^{82} - 16 q^{83} - 3 q^{86} - 30 q^{88} + 35 q^{89} - 12 q^{91} + 23 q^{92} - 9 q^{94} + 12 q^{97} + q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 8 * q^4 + 2 * q^7 - 15 * q^8 + 7 * q^11 + q^13 - 16 * q^14 + 4 * q^16 - 2 * q^17 + 5 * q^19 - 6 * q^22 - q^23 - 3 * q^26 + 9 * q^28 + 20 * q^29 + 23 * q^31 - 12 * q^32 - 14 * q^34 + 2 * q^37 - 35 * q^38 + 12 * q^41 + 16 * q^43 + 29 * q^44 - 17 * q^46 - 2 * q^47 + 8 * q^49 + 12 * q^52 + 4 * q^53 - 5 * q^56 - 25 * q^58 + 15 * q^59 - 2 * q^61 - 9 * q^62 + 23 * q^64 + 2 * q^67 + 11 * q^68 + 2 * q^71 + 16 * q^73 + 19 * q^74 + 40 * q^76 - 19 * q^77 + 35 * q^79 - 6 * q^82 - 16 * q^83 - 3 * q^86 - 30 * q^88 + 35 * q^89 - 12 * q^91 + 23 * q^92 - 9 * q^94 + 12 * q^97 + q^98

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 $

$2$	1.70636	1.20658	0.603290	−	0.797522i	$-0.293856\pi$
$2$	1.70636	1.20658	0.603290	+	0.797522i	$0.293856\pi$
$3$	0	0
$3$	0	0
$4$	0.911672	0.455836
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.94243	−1.49010	−0.745049	−	0.667009i	$-0.767574\pi$
$7$	−3.94243	−1.49010	−0.745049	+	0.667009i	$0.767574\pi$
$8$	−1.85708	−0.656578
$9$	0	0
$10$	0	0
$11$	5.90869	1.78154	0.890769	−	0.454457i	$-0.150167\pi$
$11$	5.90869	1.78154	0.890769	+	0.454457i	$0.150167\pi$
$12$	0	0
$13$	3.29066	0.912664	0.456332	−	0.889810i	$-0.349163\pi$
$13$	3.29066	0.912664	0.456332	+	0.889810i	$0.349163\pi$
$14$	−6.72721	−1.79792
$15$	0	0
$16$	−4.99220	−1.24805
$17$	−2.70636	−0.656389	−0.328195	−	0.944610i	$-0.606440\pi$
$17$	−2.70636	−0.656389	−0.328195	+	0.944610i	$0.606440\pi$
$18$	0	0
$19$	−2.35813	−0.540993	−0.270497	−	0.962721i	$-0.587188\pi$
$19$	−2.35813	−0.540993	−0.270497	+	0.962721i	$0.587188\pi$
$20$	0	0
$21$	0	0
$22$	10.0824	2.14957
$23$	0.584296	0.121834	0.0609170	−	0.998143i	$-0.480597\pi$
$23$	0.584296	0.121834	0.0609170	+	0.998143i	$0.480597\pi$
$24$	0	0
$25$	0	0
$26$	5.61505	1.10120
$27$	0	0
$28$	−3.59420	−0.679240
$29$	3.91167	0.726379	0.363190	−	0.931715i	$-0.381688\pi$
$29$	3.91167	0.726379	0.363190	+	0.931715i	$0.381688\pi$
$30$	0	0
$31$	2.70934	0.486612	0.243306	−	0.969950i	$-0.421768\pi$
$31$	2.70934	0.486612	0.243306	+	0.969950i	$0.421768\pi$
$32$	−4.80433	−0.849294
$33$	0	0
$34$	−4.61803	−0.791986
$35$	0	0
$36$	0	0
$37$	−0.0208515	−0.00342796	−0.00171398	−	0.999999i	$-0.500546\pi$
$37$	−0.0208515	−0.00342796	−0.00171398	+	0.999999i	$0.500546\pi$
$38$	−4.02383	−0.652752
$39$	0	0
$40$	0	0
$41$	−1.47214	−0.229909	−0.114955	−	0.993371i	$-0.536672\pi$
$41$	−1.47214	−0.229909	−0.114955	+	0.993371i	$0.536672\pi$
$42$	0	0
$43$	1.27279	0.194098	0.0970491	−	0.995280i	$-0.469060\pi$
$43$	1.27279	0.194098	0.0970491	+	0.995280i	$0.469060\pi$
$44$	5.38679	0.812089
$45$	0	0
$46$	0.997020	0.147003
$47$	−5.43358	−0.792568	−0.396284	−	0.918128i	$-0.629701\pi$
$47$	−5.43358	−0.792568	−0.396284	+	0.918128i	$0.629701\pi$
$48$	0	0
$49$	8.54276	1.22039
$50$	0	0
$51$	0	0
$52$	3.00000	0.416025
$53$	−2.81554	−0.386744	−0.193372	−	0.981125i	$-0.561942\pi$
$53$	−2.81554	−0.386744	−0.193372	+	0.981125i	$0.561942\pi$
$54$	0	0
$55$	0	0
$56$	7.32142	0.978365
$57$	0	0
$58$	6.67473	0.876435
$59$	4.69033	0.610629	0.305315	−	0.952252i	$-0.401238\pi$
$59$	4.69033	0.610629	0.305315	+	0.952252i	$0.401238\pi$
$60$	0	0
$61$	5.58132	0.714614	0.357307	−	0.933987i	$-0.383695\pi$
$61$	5.58132	0.714614	0.357307	+	0.933987i	$0.383695\pi$
$62$	4.62312	0.587137
$63$	0	0
$64$	1.78646	0.223308
$65$	0	0
$66$	0	0
$67$	6.03076	0.736774	0.368387	−	0.929672i	$-0.379910\pi$
$67$	6.03076	0.736774	0.368387	+	0.929672i	$0.379910\pi$
$68$	−2.46731	−0.299206
$69$	0	0
$70$	0	0
$71$	8.10138	0.961457	0.480728	−	0.876870i	$-0.340372\pi$
$71$	8.10138	0.961457	0.480728	+	0.876870i	$0.340372\pi$
$72$	0	0
$73$	13.3166	1.55859	0.779295	−	0.626658i	$-0.215578\pi$
$73$	13.3166	1.55859	0.779295	+	0.626658i	$0.215578\pi$
$74$	−0.0355801	−0.00413611
$75$	0	0
$76$	−2.14984	−0.246604
$77$	−23.2946	−2.65467
$78$	0	0
$79$	16.6648	1.87494	0.937469	−	0.348067i	$-0.113162\pi$
$79$	16.6648	1.87494	0.937469	+	0.348067i	$0.113162\pi$
$80$	0	0
$81$	0	0
$82$	−2.51200	−0.277404
$83$	−0.781641	−0.0857962	−0.0428981	−	0.999079i	$-0.513659\pi$
$83$	−0.781641	−0.0857962	−0.0428981	+	0.999079i	$0.513659\pi$
$84$	0	0
$85$	0	0
$86$	2.17183	0.234195
$87$	0	0
$88$	−10.9729	−1.16972
$89$	3.47327	0.368166	0.184083	−	0.982911i	$-0.441068\pi$
$89$	3.47327	0.368166	0.184083	+	0.982911i	$0.441068\pi$
$90$	0	0
$91$	−12.9732	−1.35996
$92$	0.532686	0.0555363
$93$	0	0
$94$	−9.27165	−0.956297
$95$	0	0
$96$	0	0
$97$	−2.45443	−0.249209	−0.124605	−	0.992206i	$-0.539766\pi$
$97$	−2.45443	−0.249209	−0.124605	+	0.992206i	$0.539766\pi$
$98$	14.5770	1.47250
$99$	0	0
$100$	0	0
$101$	6.87495	0.684083	0.342042	−	0.939685i	$-0.388882\pi$
$101$	6.87495	0.684083	0.342042	+	0.939685i	$0.388882\pi$
$102$	0	0
$103$	11.7532	1.15807	0.579036	−	0.815302i	$-0.303429\pi$
$103$	11.7532	1.15807	0.579036	+	0.815302i	$0.303429\pi$
$104$	−6.11102	−0.599235
$105$	0	0
$106$	−4.80433	−0.466638
$107$	−5.66780	−0.547927	−0.273964	−	0.961740i	$-0.588335\pi$
$107$	−5.66780	−0.547927	−0.273964	+	0.961740i	$0.588335\pi$
$108$	0	0
$109$	−1.36128	−0.130387	−0.0651934	−	0.997873i	$-0.520766\pi$
$109$	−1.36128	−0.130387	−0.0651934	+	0.997873i	$0.520766\pi$
$110$	0	0
$111$	0	0
$112$	19.6814	1.85972
$113$	10.6857	1.00522	0.502612	−	0.864512i	$-0.332373\pi$
$113$	10.6857	1.00522	0.502612	+	0.864512i	$0.332373\pi$
$114$	0	0
$115$	0	0
$116$	3.56616	0.331110
$117$	0	0
$118$	8.00341	0.736773
$119$	10.6696	0.978085
$120$	0	0
$121$	23.9126	2.17388
$122$	9.52375	0.862239
$123$	0	0
$124$	2.47003	0.221815
$125$	0	0
$126$	0	0
$127$	−13.5428	−1.20173	−0.600863	−	0.799352i	$-0.705176\pi$
$127$	−13.5428	−1.20173	−0.600863	+	0.799352i	$0.705176\pi$
$128$	12.6570	1.11873
$129$	0	0
$130$	0	0
$131$	−4.59236	−0.401236	−0.200618	−	0.979670i	$-0.564295\pi$
$131$	−4.59236	−0.401236	−0.200618	+	0.979670i	$0.564295\pi$
$132$	0	0
$133$	9.29678	0.806133
$134$	10.2907	0.888977
$135$	0	0
$136$	5.02594	0.430970
$137$	−15.2620	−1.30392	−0.651961	−	0.758253i	$-0.726053\pi$
$137$	−15.2620	−1.30392	−0.651961	+	0.758253i	$0.726053\pi$
$138$	0	0
$139$	18.2382	1.54694	0.773471	−	0.633832i	$-0.218519\pi$
$139$	18.2382	1.54694	0.773471	+	0.633832i	$0.218519\pi$
$140$	0	0
$141$	0	0
$142$	13.8239	1.16007
$143$	19.4435	1.62595
$144$	0	0
$145$	0	0
$146$	22.7229	1.88056
$147$	0	0
$148$	−0.0190097	−0.00156259
$149$	−14.7323	−1.20692	−0.603458	−	0.797394i	$-0.706211\pi$
$149$	−14.7323	−1.20692	−0.603458	+	0.797394i	$0.706211\pi$
$150$	0	0
$151$	−17.4354	−1.41887	−0.709437	−	0.704769i	$-0.751051\pi$
$151$	−17.4354	−1.41887	−0.709437	+	0.704769i	$0.751051\pi$
$152$	4.37925	0.355204
$153$	0	0
$154$	−39.7490	−3.20307
$155$	0	0
$156$	0	0
$157$	17.9105	1.42942	0.714708	−	0.699423i	$-0.246560\pi$
$157$	17.9105	1.42942	0.714708	+	0.699423i	$0.246560\pi$
$158$	28.4362	2.26226
$159$	0	0
$160$	0	0
$161$	−2.30354	−0.181545
$162$	0	0
$163$	21.7598	1.70436	0.852180	−	0.523249i	$-0.175280\pi$
$163$	21.7598	1.70436	0.852180	+	0.523249i	$0.175280\pi$
$164$	−1.34210	−0.104801
$165$	0	0
$166$	−1.33376	−0.103520
$167$	22.6030	1.74907	0.874535	−	0.484962i	$-0.161167\pi$
$167$	22.6030	1.74907	0.874535	+	0.484962i	$0.161167\pi$
$168$	0	0
$169$	−2.17157	−0.167044
$170$	0	0
$171$	0	0
$172$	1.16036	0.0884769
$173$	−12.9117	−0.981656	−0.490828	−	0.871256i	$-0.663306\pi$
$173$	−12.9117	−0.981656	−0.490828	+	0.871256i	$0.663306\pi$
$174$	0	0
$175$	0	0
$176$	−29.4974	−2.22345
$177$	0	0
$178$	5.92666	0.444222
$179$	6.43060	0.480645	0.240323	−	0.970693i	$-0.422747\pi$
$179$	6.43060	0.480645	0.240323	+	0.970693i	$0.422747\pi$
$180$	0	0
$181$	14.7071	1.09317	0.546584	−	0.837404i	$-0.315928\pi$
$181$	14.7071	1.09317	0.546584	+	0.837404i	$0.315928\pi$
$182$	−22.1370	−1.64090
$183$	0	0
$184$	−1.08508	−0.0799935
$185$	0	0
$186$	0	0
$187$	−15.9911	−1.16938
$188$	−4.95364	−0.361281
$189$	0	0
$190$	0	0
$191$	−6.71303	−0.485737	−0.242869	−	0.970059i	$-0.578088\pi$
$191$	−6.71303	−0.485737	−0.242869	+	0.970059i	$0.578088\pi$
$192$	0	0
$193$	4.82817	0.347539	0.173769	−	0.984786i	$-0.444405\pi$
$193$	4.82817	0.347539	0.173769	+	0.984786i	$0.444405\pi$
$194$	−4.18814	−0.300691
$195$	0	0
$196$	7.78819	0.556299
$197$	14.3841	1.02482	0.512411	−	0.858740i	$-0.328752\pi$
$197$	14.3841	1.02482	0.512411	+	0.858740i	$0.328752\pi$
$198$	0	0
$199$	−8.72608	−0.618575	−0.309288	−	0.950969i	$-0.600091\pi$
$199$	−8.72608	−0.618575	−0.309288	+	0.950969i	$0.600091\pi$
$200$	0	0
$201$	0	0
$202$	11.7312	0.825402
$203$	−15.4215	−1.08238
$204$	0	0
$205$	0	0
$206$	20.0551	1.39731
$207$	0	0
$208$	−16.4276	−1.13905
$209$	−13.9335	−0.963800
$210$	0	0
$211$	2.97617	0.204888	0.102444	−	0.994739i	$-0.467334\pi$
$211$	2.97617	0.204888	0.102444	+	0.994739i	$0.467334\pi$
$212$	−2.56685	−0.176292
$213$	0	0
$214$	−9.67132	−0.661118
$215$	0	0
$216$	0	0
$217$	−10.6814	−0.725100
$218$	−2.32283	−0.157322
$219$	0	0
$220$	0	0
$221$	−8.90571	−0.599063
$222$	0	0
$223$	−0.304683	−0.0204031	−0.0102015	−	0.999948i	$-0.503247\pi$
$223$	−0.304683	−0.0204031	−0.0102015	+	0.999948i	$0.503247\pi$
$224$	18.9408	1.26553
$225$	0	0
$226$	18.2336	1.21288
$227$	9.57221	0.635330	0.317665	−	0.948203i	$-0.397101\pi$
$227$	9.57221	0.635330	0.317665	+	0.948203i	$0.397101\pi$
$228$	0	0
$229$	12.1292	0.801517	0.400759	−	0.916184i	$-0.368747\pi$
$229$	12.1292	0.801517	0.400759	+	0.916184i	$0.368747\pi$
$230$	0	0
$231$	0	0
$232$	−7.26430	−0.476924
$233$	−20.1002	−1.31681	−0.658405	−	0.752664i	$-0.728769\pi$
$233$	−20.1002	−1.31681	−0.658405	+	0.752664i	$0.728769\pi$
$234$	0	0
$235$	0	0
$236$	4.27604	0.278347
$237$	0	0
$238$	18.2063	1.18014
$239$	−17.6637	−1.14257	−0.571284	−	0.820752i	$-0.693555\pi$
$239$	−17.6637	−1.14257	−0.571284	+	0.820752i	$0.693555\pi$
$240$	0	0
$241$	29.5387	1.90276	0.951379	−	0.308024i	$-0.0996676\pi$
$241$	29.5387	1.90276	0.951379	+	0.308024i	$0.0996676\pi$
$242$	40.8036	2.62296
$243$	0	0
$244$	5.08833	0.325747
$245$	0	0
$246$	0	0
$247$	−7.75981	−0.493745
$248$	−5.03147	−0.319499
$249$	0	0
$250$	0	0
$251$	−1.89396	−0.119546	−0.0597729	−	0.998212i	$-0.519038\pi$
$251$	−1.89396	−0.119546	−0.0597729	+	0.998212i	$0.519038\pi$
$252$	0	0
$253$	3.45242	0.217052
$254$	−23.1088	−1.44998
$255$	0	0
$256$	18.0245	1.12653
$257$	−22.1211	−1.37988	−0.689938	−	0.723869i	$-0.742362\pi$
$257$	−22.1211	−1.37988	−0.689938	+	0.723869i	$0.742362\pi$
$258$	0	0
$259$	0.0822054	0.00510799
$260$	0	0
$261$	0	0
$262$	−7.83623	−0.484124
$263$	19.7153	1.21570	0.607849	−	0.794053i	$-0.292033\pi$
$263$	19.7153	1.21570	0.607849	+	0.794053i	$0.292033\pi$
$264$	0	0
$265$	0	0
$266$	15.8637	0.972664
$267$	0	0
$268$	5.49807	0.335848
$269$	−18.1647	−1.10752	−0.553762	−	0.832675i	$-0.686808\pi$
$269$	−18.1647	−1.10752	−0.553762	+	0.832675i	$0.686808\pi$
$270$	0	0
$271$	22.5695	1.37100	0.685500	−	0.728073i	$-0.259584\pi$
$271$	22.5695	1.37100	0.685500	+	0.728073i	$0.259584\pi$
$272$	13.5107	0.819206
$273$	0	0
$274$	−26.0425	−1.57329
$275$	0	0
$276$	0	0
$277$	8.25491	0.495990	0.247995	−	0.968761i	$-0.420228\pi$
$277$	8.25491	0.495990	0.247995	+	0.968761i	$0.420228\pi$
$278$	31.1209	1.86651
$279$	0	0
$280$	0	0
$281$	−1.28129	−0.0764355	−0.0382177	−	0.999269i	$-0.512168\pi$
$281$	−1.28129	−0.0764355	−0.0382177	+	0.999269i	$0.512168\pi$
$282$	0	0
$283$	2.67132	0.158794	0.0793968	−	0.996843i	$-0.474701\pi$
$283$	2.67132	0.158794	0.0793968	+	0.996843i	$0.474701\pi$
$284$	7.38580	0.438266
$285$	0	0
$286$	33.1776	1.96183
$287$	5.80379	0.342587
$288$	0	0
$289$	−9.67560	−0.569153
$290$	0	0
$291$	0	0
$292$	12.1404	0.710461
$293$	−17.6605	−1.03174	−0.515870	−	0.856667i	$-0.672531\pi$
$293$	−17.6605	−1.03174	−0.515870	+	0.856667i	$0.672531\pi$
$294$	0	0
$295$	0	0
$296$	0.0387229	0.00225072
$297$	0	0
$298$	−25.1386	−1.45624
$299$	1.92272	0.111194
$300$	0	0
$301$	−5.01787	−0.289225
$302$	−29.7511	−1.71199
$303$	0	0
$304$	11.7723	0.675186
$305$	0	0
$306$	0	0
$307$	−28.5593	−1.62997	−0.814983	−	0.579484i	$-0.803254\pi$
$307$	−28.5593	−1.62997	−0.814983	+	0.579484i	$0.803254\pi$
$308$	−21.2370	−1.21009
$309$	0	0
$310$	0	0
$311$	29.3283	1.66306	0.831529	−	0.555482i	$-0.187466\pi$
$311$	29.3283	1.66306	0.831529	+	0.555482i	$0.187466\pi$
$312$	0	0
$313$	−17.4933	−0.988777	−0.494388	−	0.869241i	$-0.664608\pi$
$313$	−17.4933	−0.988777	−0.494388	+	0.869241i	$0.664608\pi$
$314$	30.5619	1.72471
$315$	0	0
$316$	15.1928	0.854665
$317$	−3.91763	−0.220036	−0.110018	−	0.993930i	$-0.535091\pi$
$317$	−3.91763	−0.220036	−0.110018	+	0.993930i	$0.535091\pi$
$318$	0	0
$319$	23.1129	1.29407
$320$	0	0
$321$	0	0
$322$	−3.93068	−0.219048
$323$	6.38197	0.355102
$324$	0	0
$325$	0	0
$326$	37.1301	2.05645
$327$	0	0
$328$	2.73388	0.150953
$329$	21.4215	1.18101
$330$	0	0
$331$	−17.5534	−0.964820	−0.482410	−	0.875945i	$-0.660238\pi$
$331$	−17.5534	−0.964820	−0.482410	+	0.875945i	$0.660238\pi$
$332$	−0.712600	−0.0391090
$333$	0	0
$334$	38.5689	2.11039
$335$	0	0
$336$	0	0
$337$	−14.0476	−0.765221	−0.382611	−	0.923910i	$-0.624975\pi$
$337$	−14.0476	−0.765221	−0.382611	+	0.923910i	$0.624975\pi$
$338$	−3.70549	−0.201552
$339$	0	0
$340$	0	0
$341$	16.0087	0.866918
$342$	0	0
$343$	−6.08221	−0.328408
$344$	−2.36367	−0.127440
$345$	0	0
$346$	−22.0320	−1.18445
$347$	1.03050	0.0553199	0.0276599	−	0.999617i	$-0.491194\pi$
$347$	1.03050	0.0553199	0.0276599	+	0.999617i	$0.491194\pi$
$348$	0	0
$349$	21.5626	1.15422	0.577109	−	0.816667i	$-0.304181\pi$
$349$	21.5626	1.15422	0.577109	+	0.816667i	$0.304181\pi$
$350$	0	0
$351$	0	0
$352$	−28.3873	−1.51305
$353$	−7.15625	−0.380888	−0.190444	−	0.981698i	$-0.560993\pi$
$353$	−7.15625	−0.380888	−0.190444	+	0.981698i	$0.560993\pi$
$354$	0	0
$355$	0	0
$356$	3.16649	0.167823
$357$	0	0
$358$	10.9729	0.579937
$359$	−27.1518	−1.43302	−0.716510	−	0.697577i	$-0.754262\pi$
$359$	−27.1518	−1.43302	−0.716510	+	0.697577i	$0.754262\pi$
$360$	0	0
$361$	−13.4392	−0.707326
$362$	25.0956	1.31899
$363$	0	0
$364$	−11.8273	−0.619918
$365$	0	0
$366$	0	0
$367$	21.4423	1.11928	0.559641	−	0.828735i	$-0.310939\pi$
$367$	21.4423	1.11928	0.559641	+	0.828735i	$0.310939\pi$
$368$	−2.91692	−0.152055
$369$	0	0
$370$	0	0
$371$	11.1001	0.576287
$372$	0	0
$373$	6.39073	0.330900	0.165450	−	0.986218i	$-0.447092\pi$
$373$	6.39073	0.330900	0.165450	+	0.986218i	$0.447092\pi$
$374$	−27.2865	−1.41095
$375$	0	0
$376$	10.0906	0.520383
$377$	12.8720	0.662940
$378$	0	0
$379$	−0.154667	−0.00794469	−0.00397235	−	0.999992i	$-0.501264\pi$
$379$	−0.154667	−0.00794469	−0.00397235	+	0.999992i	$0.501264\pi$
$380$	0	0
$381$	0	0
$382$	−11.4549	−0.586081
$383$	4.25508	0.217424	0.108712	−	0.994073i	$-0.465327\pi$
$383$	4.25508	0.217424	0.108712	+	0.994073i	$0.465327\pi$
$384$	0	0
$385$	0	0
$386$	8.23860	0.419334
$387$	0	0
$388$	−2.23763	−0.113599
$389$	20.3682	1.03271	0.516354	−	0.856375i	$-0.327289\pi$
$389$	20.3682	1.03271	0.516354	+	0.856375i	$0.327289\pi$
$390$	0	0
$391$	−1.58132	−0.0799706
$392$	−15.8646	−0.801283
$393$	0	0
$394$	24.5444	1.23653
$395$	0	0
$396$	0	0
$397$	−1.95716	−0.0982270	−0.0491135	−	0.998793i	$-0.515640\pi$
$397$	−1.95716	−0.0982270	−0.0491135	+	0.998793i	$0.515640\pi$
$398$	−14.8898	−0.746360
$399$	0	0
$400$	0	0
$401$	32.8337	1.63964	0.819818	−	0.572624i	$-0.194075\pi$
$401$	32.8337	1.63964	0.819818	+	0.572624i	$0.194075\pi$
$402$	0	0
$403$	8.91552	0.444114
$404$	6.26770	0.311830
$405$	0	0
$406$	−26.3147	−1.30597
$407$	−0.123205	−0.00610704
$408$	0	0
$409$	−39.5764	−1.95693	−0.978464	−	0.206417i	$-0.933820\pi$
$409$	−39.5764	−1.95693	−0.978464	+	0.206417i	$0.933820\pi$
$410$	0	0
$411$	0	0
$412$	10.7150	0.527891
$413$	−18.4913	−0.909898
$414$	0	0
$415$	0	0
$416$	−15.8094	−0.775121
$417$	0	0
$418$	−23.7756	−1.16290
$419$	−19.5595	−0.955544	−0.477772	−	0.878484i	$-0.658555\pi$
$419$	−19.5595	−0.955544	−0.477772	+	0.878484i	$0.658555\pi$
$420$	0	0
$421$	−40.3325	−1.96568	−0.982842	−	0.184451i	$-0.940949\pi$
$421$	−40.3325	−1.96568	−0.982842	+	0.184451i	$0.940949\pi$
$422$	5.07842	0.247214
$423$	0	0
$424$	5.22869	0.253928
$425$	0	0
$426$	0	0
$427$	−22.0039	−1.06485
$428$	−5.16718	−0.249765
$429$	0	0
$430$	0	0
$431$	−13.7370	−0.661689	−0.330844	−	0.943685i	$-0.607334\pi$
$431$	−13.7370	−0.661689	−0.330844	+	0.943685i	$0.607334\pi$
$432$	0	0
$433$	−12.4253	−0.597124	−0.298562	−	0.954390i	$-0.596507\pi$
$433$	−12.4253	−0.597124	−0.298562	+	0.954390i	$0.596507\pi$
$434$	−18.2263	−0.874892
$435$	0	0
$436$	−1.24104	−0.0594349
$437$	−1.37785	−0.0659114
$438$	0	0
$439$	10.9654	0.523349	0.261675	−	0.965156i	$-0.415725\pi$
$439$	10.9654	0.523349	0.261675	+	0.965156i	$0.415725\pi$
$440$	0	0
$441$	0	0
$442$	−15.1964	−0.722818
$443$	−8.13187	−0.386357	−0.193178	−	0.981164i	$-0.561880\pi$
$443$	−8.13187	−0.386357	−0.193178	+	0.981164i	$0.561880\pi$
$444$	0	0
$445$	0	0
$446$	−0.519899	−0.0246180
$447$	0	0
$448$	−7.04300	−0.332751
$449$	32.9503	1.55502	0.777511	−	0.628869i	$-0.216482\pi$
$449$	32.9503	1.55502	0.777511	+	0.628869i	$0.216482\pi$
$450$	0	0
$451$	−8.69840	−0.409592
$452$	9.74183	0.458217
$453$	0	0
$454$	16.3337	0.766577
$455$	0	0
$456$	0	0
$457$	−22.8800	−1.07028	−0.535142	−	0.844762i	$-0.679742\pi$
$457$	−22.8800	−1.07028	−0.535142	+	0.844762i	$0.679742\pi$
$458$	20.6967	0.967095
$459$	0	0
$460$	0	0
$461$	−12.0425	−0.560875	−0.280438	−	0.959872i	$-0.590480\pi$
$461$	−12.0425	−0.560875	−0.280438	+	0.959872i	$0.590480\pi$
$462$	0	0
$463$	−27.2002	−1.26410	−0.632049	−	0.774928i	$-0.717786\pi$
$463$	−27.2002	−1.26410	−0.632049	+	0.774928i	$0.717786\pi$
$464$	−19.5278	−0.906557
$465$	0	0
$466$	−34.2983	−1.58884
$467$	3.94243	0.182434	0.0912170	−	0.995831i	$-0.470924\pi$
$467$	3.94243	0.182434	0.0912170	+	0.995831i	$0.470924\pi$
$468$	0	0
$469$	−23.7758	−1.09787
$470$	0	0
$471$	0	0
$472$	−8.71033	−0.400926
$473$	7.52050	0.345793
$474$	0	0
$475$	0	0
$476$	9.72721	0.445846
$477$	0	0
$478$	−30.1406	−1.37860
$479$	−19.9493	−0.911505	−0.455752	−	0.890107i	$-0.650630\pi$
$479$	−19.9493	−0.911505	−0.455752	+	0.890107i	$0.650630\pi$
$480$	0	0
$481$	−0.0686150	−0.00312857
$482$	50.4038	2.29583
$483$	0	0
$484$	21.8005	0.990931
$485$	0	0
$486$	0	0
$487$	−11.0023	−0.498560	−0.249280	−	0.968431i	$-0.580194\pi$
$487$	−11.0023	−0.498560	−0.249280	+	0.968431i	$0.580194\pi$
$488$	−10.3650	−0.469200
$489$	0	0
$490$	0	0
$491$	33.2933	1.50251	0.751253	−	0.660014i	$-0.229450\pi$
$491$	33.2933	1.50251	0.751253	+	0.660014i	$0.229450\pi$
$492$	0	0
$493$	−10.5864	−0.476788
$494$	−13.2411	−0.595743
$495$	0	0
$496$	−13.5256	−0.607316
$497$	−31.9391	−1.43267
$498$	0	0
$499$	−41.1448	−1.84189	−0.920946	−	0.389690i	$-0.872582\pi$
$499$	−41.1448	−1.84189	−0.920946	+	0.389690i	$0.872582\pi$
$500$	0	0
$501$	0	0
$502$	−3.23179	−0.144242
$503$	−32.0761	−1.43020	−0.715102	−	0.699020i	$-0.753620\pi$
$503$	−32.0761	−1.43020	−0.715102	+	0.699020i	$0.753620\pi$
$504$	0	0
$505$	0	0
$506$	5.89108	0.261891
$507$	0	0
$508$	−12.3465	−0.547790
$509$	27.0953	1.20098	0.600489	−	0.799633i	$-0.294972\pi$
$509$	27.0953	1.20098	0.600489	+	0.799633i	$0.294972\pi$
$510$	0	0
$511$	−52.4997	−2.32245
$512$	5.44234	0.240520
$513$	0	0
$514$	−37.7466	−1.66493
$515$	0	0
$516$	0	0
$517$	−32.1053	−1.41199
$518$	0.140272	0.00616321
$519$	0	0
$520$	0	0
$521$	−25.9556	−1.13714	−0.568569	−	0.822636i	$-0.692503\pi$
$521$	−25.9556	−1.13714	−0.568569	+	0.822636i	$0.692503\pi$
$522$	0	0
$523$	−1.79382	−0.0784381	−0.0392190	−	0.999231i	$-0.512487\pi$
$523$	−1.79382	−0.0784381	−0.0392190	+	0.999231i	$0.512487\pi$
$524$	−4.18673	−0.182898
$525$	0	0
$526$	33.6414	1.46684
$527$	−7.33246	−0.319407
$528$	0	0
$529$	−22.6586	−0.985156
$530$	0	0
$531$	0	0
$532$	8.47561	0.367464
$533$	−4.84430	−0.209830
$534$	0	0
$535$	0	0
$536$	−11.1996	−0.483750
$537$	0	0
$538$	−30.9956	−1.33632
$539$	50.4765	2.17418
$540$	0	0
$541$	−7.43542	−0.319674	−0.159837	−	0.987143i	$-0.551097\pi$
$541$	−7.43542	−0.319674	−0.159837	+	0.987143i	$0.551097\pi$
$542$	38.5117	1.65422
$543$	0	0
$544$	13.0023	0.557468
$545$	0	0
$546$	0	0
$547$	19.4868	0.833194	0.416597	−	0.909091i	$-0.363223\pi$
$547$	19.4868	0.833194	0.416597	+	0.909091i	$0.363223\pi$
$548$	−13.9139	−0.594374
$549$	0	0
$550$	0	0
$551$	−9.22425	−0.392966
$552$	0	0
$553$	−65.6999	−2.79384
$554$	14.0859	0.598451
$555$	0	0
$556$	16.6272	0.705152
$557$	26.3285	1.11557	0.557787	−	0.829984i	$-0.311650\pi$
$557$	26.3285	1.11557	0.557787	+	0.829984i	$0.311650\pi$
$558$	0	0
$559$	4.18830	0.177146
$560$	0	0
$561$	0	0
$562$	−2.18635	−0.0922255
$563$	−36.7708	−1.54971	−0.774853	−	0.632141i	$-0.782176\pi$
$563$	−36.7708	−1.54971	−0.774853	+	0.632141i	$0.782176\pi$
$564$	0	0
$565$	0	0
$566$	4.55824	0.191597
$567$	0	0
$568$	−15.0449	−0.631271
$569$	36.2823	1.52103	0.760516	−	0.649320i	$-0.224946\pi$
$569$	36.2823	1.52103	0.760516	+	0.649320i	$0.224946\pi$
$570$	0	0
$571$	−3.36442	−0.140797	−0.0703983	−	0.997519i	$-0.522427\pi$
$571$	−3.36442	−0.140797	−0.0703983	+	0.997519i	$0.522427\pi$
$572$	17.7261	0.741164
$573$	0	0
$574$	9.90337	0.413359
$575$	0	0
$576$	0	0
$577$	−6.88442	−0.286602	−0.143301	−	0.989679i	$-0.545772\pi$
$577$	−6.88442	−0.286602	−0.143301	+	0.989679i	$0.545772\pi$
$578$	−16.5101	−0.686729
$579$	0	0
$580$	0	0
$581$	3.08156	0.127845
$582$	0	0
$583$	−16.6362	−0.689000
$584$	−24.7300	−1.02334
$585$	0	0
$586$	−30.1353	−1.24488
$587$	14.7377	0.608288	0.304144	−	0.952626i	$-0.401630\pi$
$587$	14.7377	0.608288	0.304144	+	0.952626i	$0.401630\pi$
$588$	0	0
$589$	−6.38899	−0.263254
$590$	0	0
$591$	0	0
$592$	0.104095	0.00427826
$593$	−5.82561	−0.239229	−0.119615	−	0.992820i	$-0.538166\pi$
$593$	−5.82561	−0.239229	−0.119615	+	0.992820i	$0.538166\pi$
$594$	0	0
$595$	0	0
$596$	−13.4310	−0.550156
$597$	0	0
$598$	3.28085	0.134164
$599$	27.1527	1.10943	0.554715	−	0.832040i	$-0.312827\pi$
$599$	27.1527	1.10943	0.554715	+	0.832040i	$0.312827\pi$
$600$	0	0
$601$	20.9140	0.853101	0.426551	−	0.904464i	$-0.359729\pi$
$601$	20.9140	0.853101	0.426551	+	0.904464i	$0.359729\pi$
$602$	−8.56231	−0.348974
$603$	0	0
$604$	−15.8954	−0.646774
$605$	0	0
$606$	0	0
$607$	1.46424	0.0594318	0.0297159	−	0.999558i	$-0.490540\pi$
$607$	1.46424	0.0594318	0.0297159	+	0.999558i	$0.490540\pi$
$608$	11.3293	0.459462
$609$	0	0
$610$	0	0
$611$	−17.8800	−0.723349
$612$	0	0
$613$	34.7105	1.40194	0.700971	−	0.713189i	$-0.252750\pi$
$613$	34.7105	1.40194	0.700971	+	0.713189i	$0.252750\pi$
$614$	−48.7326	−1.96669
$615$	0	0
$616$	43.2600	1.74299
$617$	−41.9038	−1.68698	−0.843492	−	0.537142i	$-0.819504\pi$
$617$	−41.9038	−1.68698	−0.843492	+	0.537142i	$0.819504\pi$
$618$	0	0
$619$	−18.7614	−0.754083	−0.377042	−	0.926196i	$-0.623059\pi$
$619$	−18.7614	−0.754083	−0.377042	+	0.926196i	$0.623059\pi$
$620$	0	0
$621$	0	0
$622$	50.0448	2.00661
$623$	−13.6931	−0.548604
$624$	0	0
$625$	0	0
$626$	−29.8498	−1.19304
$627$	0	0
$628$	16.3285	0.651579
$629$	0.0564316	0.00225007
$630$	0	0
$631$	9.88888	0.393670	0.196835	−	0.980437i	$-0.436934\pi$
$631$	9.88888	0.393670	0.196835	+	0.980437i	$0.436934\pi$
$632$	−30.9479	−1.23104
$633$	0	0
$634$	−6.68490	−0.265491
$635$	0	0
$636$	0	0
$637$	28.1113	1.11381
$638$	39.4389	1.56140
$639$	0	0
$640$	0	0
$641$	1.81266	0.0715959	0.0357979	−	0.999359i	$-0.488603\pi$
$641$	1.81266	0.0715959	0.0357979	+	0.999359i	$0.488603\pi$
$642$	0	0
$643$	45.7391	1.80377	0.901886	−	0.431974i	$-0.142183\pi$
$643$	45.7391	1.80377	0.901886	+	0.431974i	$0.142183\pi$
$644$	−2.10008	−0.0827546
$645$	0	0
$646$	10.8899	0.428459
$647$	25.6771	1.00947	0.504736	−	0.863274i	$-0.331590\pi$
$647$	25.6771	1.00947	0.504736	+	0.863274i	$0.331590\pi$
$648$	0	0
$649$	27.7137	1.08786
$650$	0	0
$651$	0	0
$652$	19.8378	0.776909
$653$	−35.9684	−1.40755	−0.703775	−	0.710422i	$-0.748504\pi$
$653$	−35.9684	−1.40755	−0.703775	+	0.710422i	$0.748504\pi$
$654$	0	0
$655$	0	0
$656$	7.34919	0.286938
$657$	0	0
$658$	36.5528	1.42498
$659$	32.0924	1.25014	0.625072	−	0.780567i	$-0.285070\pi$
$659$	32.0924	1.25014	0.625072	+	0.780567i	$0.285070\pi$
$660$	0	0
$661$	16.2064	0.630357	0.315179	−	0.949032i	$-0.397936\pi$
$661$	16.2064	0.630357	0.315179	+	0.949032i	$0.397936\pi$
$662$	−29.9524	−1.16413
$663$	0	0
$664$	1.45157	0.0563319
$665$	0	0
$666$	0	0
$667$	2.28557	0.0884977
$668$	20.6065	0.797289
$669$	0	0
$670$	0	0
$671$	32.9783	1.27311
$672$	0	0
$673$	34.1608	1.31680	0.658401	−	0.752667i	$-0.271233\pi$
$673$	34.1608	1.31680	0.658401	+	0.752667i	$0.271233\pi$
$674$	−23.9703	−0.923301
$675$	0	0
$676$	−1.97976	−0.0761446
$677$	−9.17514	−0.352629	−0.176315	−	0.984334i	$-0.556418\pi$
$677$	−9.17514	−0.352629	−0.176315	+	0.984334i	$0.556418\pi$
$678$	0	0
$679$	9.67641	0.371346
$680$	0	0
$681$	0	0
$682$	27.3166	1.04601
$683$	−19.5007	−0.746173	−0.373087	−	0.927797i	$-0.621701\pi$
$683$	−19.5007	−0.746173	−0.373087	+	0.927797i	$0.621701\pi$
$684$	0	0
$685$	0	0
$686$	−10.3784	−0.396251
$687$	0	0
$688$	−6.35400	−0.242244
$689$	−9.26498	−0.352968
$690$	0	0
$691$	29.0458	1.10495	0.552477	−	0.833528i	$-0.313683\pi$
$691$	29.0458	1.10495	0.552477	+	0.833528i	$0.313683\pi$
$692$	−11.7712	−0.447474
$693$	0	0
$694$	1.75840	0.0667479
$695$	0	0
$696$	0	0
$697$	3.98413	0.150910
$698$	36.7936	1.39266
$699$	0	0
$700$	0	0
$701$	20.4085	0.770820	0.385410	−	0.922745i	$-0.374060\pi$
$701$	20.4085	0.770820	0.385410	+	0.922745i	$0.374060\pi$
$702$	0	0
$703$	0.0491705	0.00185450
$704$	10.5557	0.397831
$705$	0	0
$706$	−12.2111	−0.459573
$707$	−27.1040	−1.01935
$708$	0	0
$709$	16.4810	0.618956	0.309478	−	0.950907i	$-0.399846\pi$
$709$	16.4810	0.618956	0.309478	+	0.950907i	$0.399846\pi$
$710$	0	0
$711$	0	0
$712$	−6.45016	−0.241730
$713$	1.58306	0.0592859
$714$	0	0
$715$	0	0
$716$	5.86259	0.219095
$717$	0	0
$718$	−46.3309	−1.72905
$719$	−31.0813	−1.15914	−0.579569	−	0.814923i	$-0.696779\pi$
$719$	−31.0813	−1.15914	−0.579569	+	0.814923i	$0.696779\pi$
$720$	0	0
$721$	−46.3360	−1.72564
$722$	−22.9321	−0.853446
$723$	0	0
$724$	13.4080	0.498305
$725$	0	0
$726$	0	0
$727$	−0.953049	−0.0353466	−0.0176733	−	0.999844i	$-0.505626\pi$
$727$	−0.953049	−0.0353466	−0.0176733	+	0.999844i	$0.505626\pi$
$728$	24.0923	0.892919
$729$	0	0
$730$	0	0
$731$	−3.44462	−0.127404
$732$	0	0
$733$	−15.3751	−0.567893	−0.283947	−	0.958840i	$-0.591644\pi$
$733$	−15.3751	−0.567893	−0.283947	+	0.958840i	$0.591644\pi$
$734$	36.5884	1.35050
$735$	0	0
$736$	−2.80715	−0.103473
$737$	35.6339	1.31259
$738$	0	0
$739$	5.19285	0.191022	0.0955110	−	0.995428i	$-0.469551\pi$
$739$	5.19285	0.191022	0.0955110	+	0.995428i	$0.469551\pi$
$740$	0	0
$741$	0	0
$742$	18.9408	0.695337
$743$	42.3399	1.55330	0.776650	−	0.629932i	$-0.216917\pi$
$743$	42.3399	1.55330	0.776650	+	0.629932i	$0.216917\pi$
$744$	0	0
$745$	0	0
$746$	10.9049	0.399257
$747$	0	0
$748$	−14.5786	−0.533046
$749$	22.3449	0.816465
$750$	0	0
$751$	22.2461	0.811773	0.405887	−	0.913923i	$-0.366963\pi$
$751$	22.2461	0.811773	0.405887	+	0.913923i	$0.366963\pi$
$752$	27.1255	0.989165
$753$	0	0
$754$	21.9642	0.799891
$755$	0	0
$756$	0	0
$757$	−9.27680	−0.337171	−0.168585	−	0.985687i	$-0.553920\pi$
$757$	−9.27680	−0.337171	−0.168585	+	0.985687i	$0.553920\pi$
$758$	−0.263917	−0.00958591
$759$	0	0
$760$	0	0
$761$	17.4122	0.631191	0.315596	−	0.948894i	$-0.397796\pi$
$761$	17.4122	0.631191	0.315596	+	0.948894i	$0.397796\pi$
$762$	0	0
$763$	5.36674	0.194289
$764$	−6.12008	−0.221417
$765$	0	0
$766$	7.26070	0.262340
$767$	15.4343	0.557300
$768$	0	0
$769$	−23.1912	−0.836297	−0.418148	−	0.908379i	$-0.637321\pi$
$769$	−23.1912	−0.836297	−0.418148	+	0.908379i	$0.637321\pi$
$770$	0	0
$771$	0	0
$772$	4.40170	0.158421
$773$	5.50798	0.198108	0.0990541	−	0.995082i	$-0.468418\pi$
$773$	5.50798	0.198108	0.0990541	+	0.995082i	$0.468418\pi$
$774$	0	0
$775$	0	0
$776$	4.55807	0.163625
$777$	0	0
$778$	34.7555	1.24605
$779$	3.47149	0.124379
$780$	0	0
$781$	47.8685	1.71287
$782$	−2.69830	−0.0964909
$783$	0	0
$784$	−42.6471	−1.52311
$785$	0	0
$786$	0	0
$787$	25.7971	0.919568	0.459784	−	0.888031i	$-0.347927\pi$
$787$	25.7971	0.919568	0.459784	+	0.888031i	$0.347927\pi$
$788$	13.1136	0.467151
$789$	0	0
$790$	0	0
$791$	−42.1275	−1.49788
$792$	0	0
$793$	18.3662	0.652203
$794$	−3.33962	−0.118519
$795$	0	0
$796$	−7.95532	−0.281969
$797$	−3.70988	−0.131411	−0.0657054	−	0.997839i	$-0.520930\pi$
$797$	−3.70988	−0.131411	−0.0657054	+	0.997839i	$0.520930\pi$
$798$	0	0
$799$	14.7052	0.520233
$800$	0	0
$801$	0	0
$802$	56.0261	1.97835
$803$	78.6837	2.77669
$804$	0	0
$805$	0	0
$806$	15.2131	0.535859
$807$	0	0
$808$	−12.7674	−0.449154
$809$	−37.5040	−1.31857	−0.659286	−	0.751893i	$-0.729141\pi$
$809$	−37.5040	−1.31857	−0.659286	+	0.751893i	$0.729141\pi$
$810$	0	0
$811$	45.6159	1.60179	0.800896	−	0.598804i	$-0.204357\pi$
$811$	45.6159	1.60179	0.800896	+	0.598804i	$0.204357\pi$
$812$	−14.0593	−0.493386
$813$	0	0
$814$	−0.210232	−0.00736863
$815$	0	0
$816$	0	0
$817$	−3.00140	−0.105006
$818$	−67.5317	−2.36119
$819$	0	0
$820$	0	0
$821$	−45.0511	−1.57229	−0.786147	−	0.618039i	$-0.787927\pi$
$821$	−45.0511	−1.57229	−0.786147	+	0.618039i	$0.787927\pi$
$822$	0	0
$823$	−41.4573	−1.44511	−0.722556	−	0.691312i	$-0.757033\pi$
$823$	−41.4573	−1.44511	−0.722556	+	0.691312i	$0.757033\pi$
$824$	−21.8266	−0.760364
$825$	0	0
$826$	−31.5529	−1.09786
$827$	38.6933	1.34550	0.672749	−	0.739871i	$-0.265113\pi$
$827$	38.6933	1.34550	0.672749	+	0.739871i	$0.265113\pi$
$828$	0	0
$829$	24.1258	0.837922	0.418961	−	0.908004i	$-0.362394\pi$
$829$	24.1258	0.837922	0.418961	+	0.908004i	$0.362394\pi$
$830$	0	0
$831$	0	0
$832$	5.87864	0.203805
$833$	−23.1198	−0.801053
$834$	0	0
$835$	0	0
$836$	−12.7028	−0.439334
$837$	0	0
$838$	−33.3756	−1.15294
$839$	17.7858	0.614032	0.307016	−	0.951704i	$-0.400669\pi$
$839$	17.7858	0.614032	0.307016	+	0.951704i	$0.400669\pi$
$840$	0	0
$841$	−13.6988	−0.472373
$842$	−68.8218	−2.37175
$843$	0	0
$844$	2.71329	0.0933953
$845$	0	0
$846$	0	0
$847$	−94.2739	−3.23929
$848$	14.0557	0.482676
$849$	0	0
$850$	0	0
$851$	−0.0121834	−0.000417642 0
$852$	0	0
$853$	12.8433	0.439745	0.219872	−	0.975529i	$-0.429436\pi$
$853$	12.8433	0.439745	0.219872	+	0.975529i	$0.429436\pi$
$854$	−37.5467	−1.28482
$855$	0	0
$856$	10.5256	0.359757
$857$	15.6015	0.532936	0.266468	−	0.963844i	$-0.414143\pi$
$857$	15.6015	0.532936	0.266468	+	0.963844i	$0.414143\pi$
$858$	0	0
$859$	4.82843	0.164744	0.0823719	−	0.996602i	$-0.473750\pi$
$859$	4.82843	0.164744	0.0823719	+	0.996602i	$0.473750\pi$
$860$	0	0
$861$	0	0
$862$	−23.4403	−0.798381
$863$	13.1548	0.447796	0.223898	−	0.974613i	$-0.428122\pi$
$863$	13.1548	0.447796	0.223898	+	0.974613i	$0.428122\pi$
$864$	0	0
$865$	0	0
$866$	−21.2021	−0.720478
$867$	0	0
$868$	−9.73792	−0.330527
$869$	98.4673	3.34027
$870$	0	0
$871$	19.8452	0.672428
$872$	2.52800	0.0856090
$873$	0	0
$874$	−2.35111	−0.0795274
$875$	0	0
$876$	0	0
$877$	6.25253	0.211133	0.105567	−	0.994412i	$-0.466334\pi$
$877$	6.25253	0.211133	0.105567	+	0.994412i	$0.466334\pi$
$878$	18.7109	0.631463
$879$	0	0
$880$	0	0
$881$	16.7801	0.565335	0.282667	−	0.959218i	$-0.408781\pi$
$881$	16.7801	0.565335	0.282667	+	0.959218i	$0.408781\pi$
$882$	0	0
$883$	16.2367	0.546407	0.273203	−	0.961956i	$-0.411917\pi$
$883$	16.2367	0.546407	0.273203	+	0.961956i	$0.411917\pi$
$884$	−8.11909	−0.273074
$885$	0	0
$886$	−13.8759	−0.466171
$887$	−57.6873	−1.93695	−0.968476	−	0.249108i	$-0.919863\pi$
$887$	−57.6873	−1.93695	−0.968476	+	0.249108i	$0.919863\pi$
$888$	0	0
$889$	53.3914	1.79069
$890$	0	0
$891$	0	0
$892$	−0.277771	−0.00930046
$893$	12.8131	0.428774
$894$	0	0
$895$	0	0
$896$	−49.8994	−1.66702
$897$	0	0
$898$	56.2252	1.87626
$899$	10.5981	0.353465
$900$	0	0
$901$	7.61988	0.253855
$902$	−14.8426	−0.494205
$903$	0	0
$904$	−19.8442	−0.660007
$905$	0	0
$906$	0	0
$907$	9.00465	0.298995	0.149497	−	0.988762i	$-0.452234\pi$
$907$	9.00465	0.298995	0.149497	+	0.988762i	$0.452234\pi$
$908$	8.72672	0.289606
$909$	0	0
$910$	0	0
$911$	−29.9503	−0.992298	−0.496149	−	0.868237i	$-0.665253\pi$
$911$	−29.9503	−0.992298	−0.496149	+	0.868237i	$0.665253\pi$
$912$	0	0
$913$	−4.61847	−0.152849
$914$	−39.0416	−1.29138
$915$	0	0
$916$	11.0578	0.365360
$917$	18.1051	0.597882
$918$	0	0
$919$	8.93560	0.294758	0.147379	−	0.989080i	$-0.452916\pi$
$919$	8.93560	0.294758	0.147379	+	0.989080i	$0.452916\pi$
$920$	0	0
$921$	0	0
$922$	−20.5489	−0.676741
$923$	26.6589	0.877487
$924$	0	0
$925$	0	0
$926$	−46.4133	−1.52524
$927$	0	0
$928$	−18.7930	−0.616910
$929$	41.4596	1.36025	0.680123	−	0.733098i	$-0.261926\pi$
$929$	41.4596	1.36025	0.680123	+	0.733098i	$0.261926\pi$
$930$	0	0
$931$	−20.1450	−0.660225
$932$	−18.3248	−0.600250
$933$	0	0
$934$	6.72721	0.220121
$935$	0	0
$936$	0	0
$937$	20.8585	0.681417	0.340709	−	0.940169i	$-0.389333\pi$
$937$	20.8585	0.681417	0.340709	+	0.940169i	$0.389333\pi$
$938$	−40.5702	−1.32466
$939$	0	0
$940$	0	0
$941$	−3.67382	−0.119763	−0.0598816	−	0.998205i	$-0.519072\pi$
$941$	−3.67382	−0.119763	−0.0598816	+	0.998205i	$0.519072\pi$
$942$	0	0
$943$	−0.860163	−0.0280107
$944$	−23.4151	−0.762096
$945$	0	0
$946$	12.8327	0.417227
$947$	31.6997	1.03010	0.515051	−	0.857160i	$-0.327773\pi$
$947$	31.6997	1.03010	0.515051	+	0.857160i	$0.327773\pi$
$948$	0	0
$949$	43.8204	1.42247
$950$	0	0
$951$	0	0
$952$	−19.8144	−0.642189
$953$	−25.6214	−0.829960	−0.414980	−	0.909831i	$-0.636211\pi$
$953$	−25.6214	−0.829960	−0.414980	+	0.909831i	$0.636211\pi$
$954$	0	0
$955$	0	0
$956$	−16.1035	−0.520824
$957$	0	0
$958$	−34.0407	−1.09980
$959$	60.1694	1.94297
$960$	0	0
$961$	−23.6595	−0.763209
$962$	−0.117082	−0.00377488
$963$	0	0
$964$	26.9296	0.867345
$965$	0	0
$966$	0	0
$967$	−29.0166	−0.933112	−0.466556	−	0.884492i	$-0.654505\pi$
$967$	−29.0166	−0.933112	−0.466556	+	0.884492i	$0.654505\pi$
$968$	−44.4077	−1.42732
$969$	0	0
$970$	0	0
$971$	3.77287	0.121077	0.0605386	−	0.998166i	$-0.480718\pi$
$971$	3.77287	0.121077	0.0605386	+	0.998166i	$0.480718\pi$
$972$	0	0
$973$	−71.9027	−2.30510
$974$	−18.7739	−0.601553
$975$	0	0
$976$	−27.8630	−0.891874
$977$	−36.7167	−1.17467	−0.587336	−	0.809343i	$-0.699823\pi$
$977$	−36.7167	−1.17467	−0.587336	+	0.809343i	$0.699823\pi$
$978$	0	0
$979$	20.5225	0.655902
$980$	0	0
$981$	0	0
$982$	56.8104	1.81289
$983$	−23.5627	−0.751534	−0.375767	−	0.926714i	$-0.622621\pi$
$983$	−23.5627	−0.751534	−0.375767	+	0.926714i	$0.622621\pi$
$984$	0	0
$985$	0	0
$986$	−18.0642	−0.575282
$987$	0	0
$988$	−7.07440	−0.225067
$989$	0.743684	0.0236478
$990$	0	0
$991$	29.1375	0.925583	0.462791	−	0.886467i	$-0.346848\pi$
$991$	29.1375	0.925583	0.462791	+	0.886467i	$0.346848\pi$
$992$	−13.0166	−0.413277
$993$	0	0
$994$	−54.4997	−1.72863
$995$	0	0
$996$	0	0
$997$	43.9240	1.39109	0.695544	−	0.718484i	$-0.255164\pi$
$997$	43.9240	1.39109	0.695544	+	0.718484i	$0.255164\pi$
$998$	−70.2078	−2.22239
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.i.1.4		4
3.2	odd	2		1875.2.a.h.1.1		4
5.4	even	2		5625.2.a.n.1.1		4
15.2	even	4		1875.2.b.c.1249.3		8
15.8	even	4		1875.2.b.c.1249.6		8
15.14	odd	2		1875.2.a.e.1.4		4
25.6	even	5		225.2.h.c.136.1		8
25.21	even	5		225.2.h.c.91.1		8
75.8	even	20		375.2.i.b.199.2		16
75.17	even	20		375.2.i.b.199.3		16
75.29	odd	10		375.2.g.b.76.1		8
75.44	odd	10		375.2.g.b.301.1		8
75.47	even	20		375.2.i.b.49.2		16
75.53	even	20		375.2.i.b.49.3		16
75.56	odd	10		75.2.g.b.61.2	yes	8
75.71	odd	10		75.2.g.b.16.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.b.16.2	✓	8	75.71	odd	10
75.2.g.b.61.2	yes	8	75.56	odd	10
225.2.h.c.91.1		8	25.21	even	5
225.2.h.c.136.1		8	25.6	even	5
375.2.g.b.76.1		8	75.29	odd	10
375.2.g.b.301.1		8	75.44	odd	10
375.2.i.b.49.2		16	75.47	even	20
375.2.i.b.49.3		16	75.53	even	20
375.2.i.b.199.2		16	75.8	even	20
375.2.i.b.199.3		16	75.17	even	20
1875.2.a.e.1.4		4	15.14	odd	2
1875.2.a.h.1.1		4	3.2	odd	2
1875.2.b.c.1249.3		8	15.2	even	4
1875.2.b.c.1249.6		8	15.8	even	4
5625.2.a.i.1.4		4	1.1	even	1	trivial
5625.2.a.n.1.1		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q+1.70636 q^{2} +0.911672 q^{4} -3.94243 q^{7} -1.85708 q^{8} +O(q^{10})\)	\(q+1.70636 q^{2} +0.911672 q^{4} -3.94243 q^{7} -1.85708 q^{8} +5.90869 q^{11} +3.29066 q^{13} -6.72721 q^{14} -4.99220 q^{16} -2.70636 q^{17} -2.35813 q^{19} +10.0824 q^{22} +0.584296 q^{23} +5.61505 q^{26} -3.59420 q^{28} +3.91167 q^{29} +2.70934 q^{31} -4.80433 q^{32} -4.61803 q^{34} -0.0208515 q^{37} -4.02383 q^{38} -1.47214 q^{41} +1.27279 q^{43} +5.38679 q^{44} +0.997020 q^{46} -5.43358 q^{47} +8.54276 q^{49} +3.00000 q^{52} -2.81554 q^{53} +7.32142 q^{56} +6.67473 q^{58} +4.69033 q^{59} +5.58132 q^{61} +4.62312 q^{62} +1.78646 q^{64} +6.03076 q^{67} -2.46731 q^{68} +8.10138 q^{71} +13.3166 q^{73} -0.0355801 q^{74} -2.14984 q^{76} -23.2946 q^{77} +16.6648 q^{79} -2.51200 q^{82} -0.781641 q^{83} +2.17183 q^{86} -10.9729 q^{88} +3.47327 q^{89} -12.9732 q^{91} +0.532686 q^{92} -9.27165 q^{94} -2.45443 q^{97} +14.5770 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 8 q^{4} + 2 q^{7} - 15 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 8 * q^4 + 2 * q^7 - 15 * q^8	\( 4 q - 2 q^{2} + 8 q^{4} + 2 q^{7} - 15 q^{8} + 7 q^{11} + q^{13} - 16 q^{14} + 4 q^{16} - 2 q^{17} + 5 q^{19} - 6 q^{22} - q^{23} - 3 q^{26} + 9 q^{28} + 20 q^{29} + 23 q^{31} - 12 q^{32} - 14 q^{34} + 2 q^{37} - 35 q^{38} + 12 q^{41} + 16 q^{43} + 29 q^{44} - 17 q^{46} - 2 q^{47} + 8 q^{49} + 12 q^{52} + 4 q^{53} - 5 q^{56} - 25 q^{58} + 15 q^{59} - 2 q^{61} - 9 q^{62} + 23 q^{64} + 2 q^{67} + 11 q^{68} + 2 q^{71} + 16 q^{73} + 19 q^{74} + 40 q^{76} - 19 q^{77} + 35 q^{79} - 6 q^{82} - 16 q^{83} - 3 q^{86} - 30 q^{88} + 35 q^{89} - 12 q^{91} + 23 q^{92} - 9 q^{94} + 12 q^{97} + q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 8 * q^4 + 2 * q^7 - 15 * q^8 + 7 * q^11 + q^13 - 16 * q^14 + 4 * q^16 - 2 * q^17 + 5 * q^19 - 6 * q^22 - q^23 - 3 * q^26 + 9 * q^28 + 20 * q^29 + 23 * q^31 - 12 * q^32 - 14 * q^34 + 2 * q^37 - 35 * q^38 + 12 * q^41 + 16 * q^43 + 29 * q^44 - 17 * q^46 - 2 * q^47 + 8 * q^49 + 12 * q^52 + 4 * q^53 - 5 * q^56 - 25 * q^58 + 15 * q^59 - 2 * q^61 - 9 * q^62 + 23 * q^64 + 2 * q^67 + 11 * q^68 + 2 * q^71 + 16 * q^73 + 19 * q^74 + 40 * q^76 - 19 * q^77 + 35 * q^79 - 6 * q^82 - 16 * q^83 - 3 * q^86 - 30 * q^88 + 35 * q^89 - 12 * q^91 + 23 * q^92 - 9 * q^94 + 12 * q^97 + q^98

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