LMFDB - Embedded newform 5625.2.a.z.1.8

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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45 $ x^8 - 15x^6 + 70x^4 - 105*x^2 + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.66202$ 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)

$f(q)$	$=$	$q+2.66202 q^{2} +5.08634 q^{4} -3.46831 q^{7} +8.21589 q^{8} +O(q^{10})$	$q+2.66202 q^{2} +5.08634 q^{4} -3.46831 q^{7} +8.21589 q^{8} -3.43248 q^{11} -5.70437 q^{13} -9.23269 q^{14} +11.6982 q^{16} -1.89155 q^{17} +2.46831 q^{19} -9.13733 q^{22} -8.84431 q^{23} -15.1851 q^{26} -17.6410 q^{28} +8.98636 q^{29} -1.90746 q^{31} +14.7090 q^{32} -5.03535 q^{34} -5.09254 q^{37} +6.57068 q^{38} +6.35103 q^{41} +3.43296 q^{43} -17.4588 q^{44} -23.5437 q^{46} -8.61447 q^{47} +5.02915 q^{49} -29.0144 q^{52} -2.03360 q^{53} -28.4952 q^{56} +23.9219 q^{58} -8.47242 q^{59} -7.90746 q^{61} -5.07770 q^{62} +15.7592 q^{64} -6.95846 q^{67} -9.62108 q^{68} +2.91855 q^{71} +11.9280 q^{73} -13.5564 q^{74} +12.5546 q^{76} +11.9049 q^{77} -6.60564 q^{79} +16.9066 q^{82} -12.9993 q^{83} +9.13860 q^{86} -28.2009 q^{88} -11.5873 q^{89} +19.7845 q^{91} -44.9852 q^{92} -22.9319 q^{94} +2.17888 q^{97} +13.3877 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{4} - 10 q^{7}+O(q^{10}) $ 8 * q + 14 * q^4 - 10 * q^7	$ 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} + 22 q^{16} + 2 q^{19} - 10 q^{22} - 70 q^{28} - 6 q^{31} - 50 q^{34} - 50 q^{37} + 14 q^{49} - 80 q^{52} + 30 q^{58} - 54 q^{61} + 36 q^{64} - 10 q^{67} - 30 q^{73} + 56 q^{76} + 28 q^{79} - 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100}) $ 8 * q + 14 * q^4 - 10 * q^7 - 10 * q^13 + 22 * q^16 + 2 * q^19 - 10 * q^22 - 70 * q^28 - 6 * q^31 - 50 * q^34 - 50 * q^37 + 14 * q^49 - 80 * q^52 + 30 * q^58 - 54 * q^61 + 36 * q^64 - 10 * q^67 - 30 * q^73 + 56 * q^76 + 28 * q^79 - 20 * q^88 + 60 * q^91 - 40 * q^94

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 $

$2$	2.66202	1.88233	0.941166	−	0.337946i	$-0.109732\pi$
$2$	2.66202	1.88233	0.941166	+	0.337946i	$0.109732\pi$
$3$	0	0
$3$	0	0
$4$	5.08634	2.54317
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.46831	−1.31090	−0.655448	−	0.755240i	$-0.727520\pi$
$7$	−3.46831	−1.31090	−0.655448	+	0.755240i	$0.727520\pi$
$8$	8.21589	2.90476
$9$	0	0
$10$	0	0
$11$	−3.43248	−1.03493	−0.517466	−	0.855703i	$-0.673125\pi$
$11$	−3.43248	−1.03493	−0.517466	+	0.855703i	$0.673125\pi$
$12$	0	0
$13$	−5.70437	−1.58211	−0.791054	−	0.611746i	$-0.790468\pi$
$13$	−5.70437	−1.58211	−0.791054	+	0.611746i	$0.790468\pi$
$14$	−9.23269	−2.46754
$15$	0	0
$16$	11.6982	2.92454
$17$	−1.89155	−0.458769	−0.229384	−	0.973336i	$-0.573671\pi$
$17$	−1.89155	−0.458769	−0.229384	+	0.973336i	$0.573671\pi$
$18$	0	0
$19$	2.46831	0.566268	0.283134	−	0.959080i	$-0.408626\pi$
$19$	2.46831	0.566268	0.283134	+	0.959080i	$0.408626\pi$
$20$	0	0
$21$	0	0
$22$	−9.13733	−1.94809
$23$	−8.84431	−1.84417	−0.922083	−	0.386992i	$-0.873514\pi$
$23$	−8.84431	−1.84417	−0.922083	+	0.386992i	$0.873514\pi$
$24$	0	0
$25$	0	0
$26$	−15.1851	−2.97805
$27$	0	0
$28$	−17.6410	−3.33383
$29$	8.98636	1.66873	0.834363	−	0.551216i	$-0.185836\pi$
$29$	8.98636	1.66873	0.834363	+	0.551216i	$0.185836\pi$
$30$	0	0
$31$	−1.90746	−0.342591	−0.171295	−	0.985220i	$-0.554795\pi$
$31$	−1.90746	−0.342591	−0.171295	+	0.985220i	$0.554795\pi$
$32$	14.7090	2.60020
$33$	0	0
$34$	−5.03535	−0.863555
$35$	0	0
$36$	0	0
$37$	−5.09254	−0.837208	−0.418604	−	0.908169i	$-0.637480\pi$
$37$	−5.09254	−0.837208	−0.418604	+	0.908169i	$0.637480\pi$
$38$	6.57068	1.06590
$39$	0	0
$40$	0	0
$41$	6.35103	0.991864	0.495932	−	0.868361i	$-0.334826\pi$
$41$	6.35103	0.991864	0.495932	+	0.868361i	$0.334826\pi$
$42$	0	0
$43$	3.43296	0.523522	0.261761	−	0.965133i	$-0.415697\pi$
$43$	3.43296	0.523522	0.261761	+	0.965133i	$0.415697\pi$
$44$	−17.4588	−2.63201
$45$	0	0
$46$	−23.5437	−3.47133
$47$	−8.61447	−1.25655	−0.628275	−	0.777991i	$-0.716239\pi$
$47$	−8.61447	−1.25655	−0.628275	+	0.777991i	$0.716239\pi$
$48$	0	0
$49$	5.02915	0.718450
$50$	0	0
$51$	0	0
$52$	−29.0144	−4.02357
$53$	−2.03360	−0.279337	−0.139668	−	0.990198i	$-0.544604\pi$
$53$	−2.03360	−0.279337	−0.139668	+	0.990198i	$0.544604\pi$
$54$	0	0
$55$	0	0
$56$	−28.4952	−3.80784
$57$	0	0
$58$	23.9219	3.14109
$59$	−8.47242	−1.10302	−0.551508	−	0.834170i	$-0.685947\pi$
$59$	−8.47242	−1.10302	−0.551508	+	0.834170i	$0.685947\pi$
$60$	0	0
$61$	−7.90746	−1.01245	−0.506223	−	0.862402i	$-0.668959\pi$
$61$	−7.90746	−1.01245	−0.506223	+	0.862402i	$0.668959\pi$
$62$	−5.07770	−0.644869
$63$	0	0
$64$	15.7592	1.96990
$65$	0	0
$66$	0	0
$67$	−6.95846	−0.850111	−0.425055	−	0.905167i	$-0.639745\pi$
$67$	−6.95846	−0.850111	−0.425055	+	0.905167i	$0.639745\pi$
$68$	−9.62108	−1.16673
$69$	0	0
$70$	0	0
$71$	2.91855	0.346368	0.173184	−	0.984890i	$-0.444595\pi$
$71$	2.91855	0.346368	0.173184	+	0.984890i	$0.444595\pi$
$72$	0	0
$73$	11.9280	1.39607	0.698036	−	0.716062i	$-0.254057\pi$
$73$	11.9280	1.39607	0.698036	+	0.716062i	$0.254057\pi$
$74$	−13.5564	−1.57590
$75$	0	0
$76$	12.5546	1.44012
$77$	11.9049	1.35669
$78$	0	0
$79$	−6.60564	−0.743193	−0.371596	−	0.928394i	$-0.621189\pi$
$79$	−6.60564	−0.743193	−0.371596	+	0.928394i	$0.621189\pi$
$80$	0	0
$81$	0	0
$82$	16.9066	1.86702
$83$	−12.9993	−1.42686	−0.713429	−	0.700727i	$-0.752859\pi$
$83$	−12.9993	−1.42686	−0.713429	+	0.700727i	$0.752859\pi$
$84$	0	0
$85$	0	0
$86$	9.13860	0.985441
$87$	0	0
$88$	−28.2009	−3.00623
$89$	−11.5873	−1.22825	−0.614124	−	0.789209i	$-0.710491\pi$
$89$	−11.5873	−1.22825	−0.614124	+	0.789209i	$0.710491\pi$
$90$	0	0
$91$	19.7845	2.07398
$92$	−44.9852	−4.69003
$93$	0	0
$94$	−22.9319	−2.36524
$95$	0	0
$96$	0	0
$97$	2.17888	0.221231	0.110616	−	0.993863i	$-0.464718\pi$
$97$	2.17888	0.221231	0.110616	+	0.993863i	$0.464718\pi$
$98$	13.3877	1.35236
$99$	0	0
$100$	0	0
$101$	9.30399	0.925782	0.462891	−	0.886415i	$-0.346812\pi$
$101$	9.30399	0.925782	0.462891	+	0.886415i	$0.346812\pi$
$102$	0	0
$103$	0.176510	0.0173921	0.00869604	−	0.999962i	$-0.497232\pi$
$103$	0.176510	0.0173921	0.00869604	+	0.999962i	$0.497232\pi$
$104$	−46.8665	−4.59564
$105$	0	0
$106$	−5.41348	−0.525804
$107$	1.37761	0.133179	0.0665895	−	0.997780i	$-0.478788\pi$
$107$	1.37761	0.133179	0.0665895	+	0.997780i	$0.478788\pi$
$108$	0	0
$109$	1.89744	0.181742	0.0908708	−	0.995863i	$-0.471035\pi$
$109$	1.89744	0.181742	0.0908708	+	0.995863i	$0.471035\pi$
$110$	0	0
$111$	0	0
$112$	−40.5729	−3.83378
$113$	−2.35123	−0.221185	−0.110593	−	0.993866i	$-0.535275\pi$
$113$	−2.35123	−0.221185	−0.110593	+	0.993866i	$0.535275\pi$
$114$	0	0
$115$	0	0
$116$	45.7077	4.24385
$117$	0	0
$118$	−22.5537	−2.07624
$119$	6.56048	0.601398
$120$	0	0
$121$	0.781947	0.0710861
$122$	−21.0498	−1.90576
$123$	0	0
$124$	−9.70201	−0.871266
$125$	0	0
$126$	0	0
$127$	3.78069	0.335482	0.167741	−	0.985831i	$-0.446353\pi$
$127$	3.78069	0.335482	0.167741	+	0.985831i	$0.446353\pi$
$128$	12.5333	1.10780
$129$	0	0
$130$	0	0
$131$	−0.797155	−0.0696477	−0.0348239	−	0.999393i	$-0.511087\pi$
$131$	−0.797155	−0.0696477	−0.0348239	+	0.999393i	$0.511087\pi$
$132$	0	0
$133$	−8.56084	−0.742319
$134$	−18.5235	−1.60019
$135$	0	0
$136$	−15.5408	−1.33261
$137$	12.4188	1.06101	0.530507	−	0.847681i	$-0.322002\pi$
$137$	12.4188	1.06101	0.530507	+	0.847681i	$0.322002\pi$
$138$	0	0
$139$	12.2399	1.03817	0.519087	−	0.854721i	$-0.326272\pi$
$139$	12.2399	1.03817	0.519087	+	0.854721i	$0.326272\pi$
$140$	0	0
$141$	0	0
$142$	7.76922	0.651979
$143$	19.5802	1.63738
$144$	0	0
$145$	0	0
$146$	31.7527	2.62787
$147$	0	0
$148$	−25.9024	−2.12916
$149$	−1.31109	−0.107409	−0.0537044	−	0.998557i	$-0.517103\pi$
$149$	−1.31109	−0.107409	−0.0537044	+	0.998557i	$0.517103\pi$
$150$	0	0
$151$	2.57266	0.209360	0.104680	−	0.994506i	$-0.466618\pi$
$151$	2.57266	0.209360	0.104680	+	0.994506i	$0.466618\pi$
$152$	20.2793	1.64487
$153$	0	0
$154$	31.6911	2.55374
$155$	0	0
$156$	0	0
$157$	−15.7038	−1.25330	−0.626650	−	0.779301i	$-0.715574\pi$
$157$	−15.7038	−1.25330	−0.626650	+	0.779301i	$0.715574\pi$
$158$	−17.5843	−1.39893
$159$	0	0
$160$	0	0
$161$	30.6748	2.41751
$162$	0	0
$163$	7.42913	0.581894	0.290947	−	0.956739i	$-0.406030\pi$
$163$	7.42913	0.581894	0.290947	+	0.956739i	$0.406030\pi$
$164$	32.3035	2.52248
$165$	0	0
$166$	−34.6044	−2.68582
$167$	4.12200	0.318970	0.159485	−	0.987200i	$-0.449017\pi$
$167$	4.12200	0.318970	0.159485	+	0.987200i	$0.449017\pi$
$168$	0	0
$169$	19.5399	1.50307
$170$	0	0
$171$	0	0
$172$	17.4612	1.33140
$173$	19.7765	1.50358	0.751789	−	0.659404i	$-0.229191\pi$
$173$	19.7765	1.50358	0.751789	+	0.659404i	$0.229191\pi$
$174$	0	0
$175$	0	0
$176$	−40.1538	−3.02671
$177$	0	0
$178$	−30.8455	−2.31197
$179$	−7.18260	−0.536853	−0.268426	−	0.963300i	$-0.586504\pi$
$179$	−7.18260	−0.536853	−0.268426	+	0.963300i	$0.586504\pi$
$180$	0	0
$181$	15.9100	1.18258	0.591292	−	0.806458i	$-0.298618\pi$
$181$	15.9100	1.18258	0.591292	+	0.806458i	$0.298618\pi$
$182$	52.6667	3.90392
$183$	0	0
$184$	−72.6639	−5.35686
$185$	0	0
$186$	0	0
$187$	6.49272	0.474795
$188$	−43.8161	−3.19562
$189$	0	0
$190$	0	0
$191$	14.5190	1.05056	0.525278	−	0.850931i	$-0.323961\pi$
$191$	14.5190	1.05056	0.525278	+	0.850931i	$0.323961\pi$
$192$	0	0
$193$	13.0763	0.941254	0.470627	−	0.882332i	$-0.344028\pi$
$193$	13.0763	0.941254	0.470627	+	0.882332i	$0.344028\pi$
$194$	5.80021	0.416431
$195$	0	0
$196$	25.5800	1.82714
$197$	17.8849	1.27425	0.637124	−	0.770761i	$-0.280124\pi$
$197$	17.8849	1.27425	0.637124	+	0.770761i	$0.280124\pi$
$198$	0	0
$199$	−21.1565	−1.49974	−0.749871	−	0.661584i	$-0.769884\pi$
$199$	−21.1565	−1.49974	−0.749871	+	0.661584i	$0.769884\pi$
$200$	0	0
$201$	0	0
$202$	24.7674	1.74263
$203$	−31.1675	−2.18753
$204$	0	0
$205$	0	0
$206$	0.469874	0.0327377
$207$	0	0
$208$	−66.7308	−4.62695
$209$	−8.47242	−0.586050
$210$	0	0
$211$	13.4002	0.922507	0.461253	−	0.887268i	$-0.347400\pi$
$211$	13.4002	0.922507	0.461253	+	0.887268i	$0.347400\pi$
$212$	−10.3436	−0.710400
$213$	0	0
$214$	3.66724	0.250687
$215$	0	0
$216$	0	0
$217$	6.61567	0.449101
$218$	5.05101	0.342098
$219$	0	0
$220$	0	0
$221$	10.7901	0.725822
$222$	0	0
$223$	−25.2555	−1.69124	−0.845618	−	0.533788i	$-0.820768\pi$
$223$	−25.2555	−1.69124	−0.845618	+	0.533788i	$0.820768\pi$
$224$	−51.0152	−3.40860
$225$	0	0
$226$	−6.25902	−0.416344
$227$	1.89155	0.125547	0.0627734	−	0.998028i	$-0.480005\pi$
$227$	1.89155	0.125547	0.0627734	+	0.998028i	$0.480005\pi$
$228$	0	0
$229$	−4.78069	−0.315917	−0.157958	−	0.987446i	$-0.550491\pi$
$229$	−4.78069	−0.315917	−0.157958	+	0.987446i	$0.550491\pi$
$230$	0	0
$231$	0	0
$232$	73.8310	4.84724
$233$	3.44563	0.225731	0.112865	−	0.993610i	$-0.463997\pi$
$233$	3.44563	0.225731	0.112865	+	0.993610i	$0.463997\pi$
$234$	0	0
$235$	0	0
$236$	−43.0936	−2.80516
$237$	0	0
$238$	17.4641	1.13203
$239$	−24.8164	−1.60524	−0.802620	−	0.596490i	$-0.796561\pi$
$239$	−24.8164	−1.60524	−0.802620	+	0.596490i	$0.796561\pi$
$240$	0	0
$241$	−20.5045	−1.32081	−0.660407	−	0.750908i	$-0.729616\pi$
$241$	−20.5045	−1.32081	−0.660407	+	0.750908i	$0.729616\pi$
$242$	2.08156	0.133808
$243$	0	0
$244$	−40.2201	−2.57482
$245$	0	0
$246$	0	0
$247$	−14.0801	−0.895898
$248$	−15.6715	−0.995142
$249$	0	0
$250$	0	0
$251$	2.10825	0.133071	0.0665357	−	0.997784i	$-0.478805\pi$
$251$	2.10825	0.133071	0.0665357	+	0.997784i	$0.478805\pi$
$252$	0	0
$253$	30.3580	1.90859
$254$	10.0643	0.631488
$255$	0	0
$256$	1.84554	0.115346
$257$	1.45314	0.0906445	0.0453222	−	0.998972i	$-0.485569\pi$
$257$	1.45314	0.0906445	0.0453222	+	0.998972i	$0.485569\pi$
$258$	0	0
$259$	17.6625	1.09749
$260$	0	0
$261$	0	0
$262$	−2.12204	−0.131100
$263$	25.7688	1.58897	0.794485	−	0.607284i	$-0.207741\pi$
$263$	25.7688	1.58897	0.794485	+	0.607284i	$0.207741\pi$
$264$	0	0
$265$	0	0
$266$	−22.7891	−1.39729
$267$	0	0
$268$	−35.3931	−2.16198
$269$	−31.9778	−1.94972	−0.974859	−	0.222823i	$-0.928473\pi$
$269$	−31.9778	−1.94972	−0.974859	+	0.222823i	$0.928473\pi$
$270$	0	0
$271$	25.1821	1.52971	0.764853	−	0.644205i	$-0.222812\pi$
$271$	25.1821	1.52971	0.764853	+	0.644205i	$0.222812\pi$
$272$	−22.1277	−1.34169
$273$	0	0
$274$	33.0592	1.99718
$275$	0	0
$276$	0	0
$277$	−16.8847	−1.01450	−0.507252	−	0.861798i	$-0.669339\pi$
$277$	−16.8847	−1.01450	−0.507252	+	0.861798i	$0.669339\pi$
$278$	32.5828	1.95419
$279$	0	0
$280$	0	0
$281$	7.35764	0.438920	0.219460	−	0.975622i	$-0.429570\pi$
$281$	7.35764	0.438920	0.219460	+	0.975622i	$0.429570\pi$
$282$	0	0
$283$	−12.1220	−0.720581	−0.360290	−	0.932840i	$-0.617322\pi$
$283$	−12.1220	−0.720581	−0.360290	+	0.932840i	$0.617322\pi$
$284$	14.8447	0.880872
$285$	0	0
$286$	52.1228	3.08208
$287$	−22.0273	−1.30023
$288$	0	0
$289$	−13.4220	−0.789531
$290$	0	0
$291$	0	0
$292$	60.6701	3.55045
$293$	24.6612	1.44072	0.720362	−	0.693598i	$-0.243976\pi$
$293$	24.6612	1.44072	0.720362	+	0.693598i	$0.243976\pi$
$294$	0	0
$295$	0	0
$296$	−41.8397	−2.43189
$297$	0	0
$298$	−3.49015	−0.202179
$299$	50.4513	2.91767
$300$	0	0
$301$	−11.9066	−0.686283
$302$	6.84847	0.394085
$303$	0	0
$304$	28.8747	1.65608
$305$	0	0
$306$	0	0
$307$	19.9964	1.14125	0.570627	−	0.821210i	$-0.306700\pi$
$307$	19.9964	1.14125	0.570627	+	0.821210i	$0.306700\pi$
$308$	60.5524	3.45029
$309$	0	0
$310$	0	0
$311$	−3.11485	−0.176627	−0.0883136	−	0.996093i	$-0.528148\pi$
$311$	−3.11485	−0.176627	−0.0883136	+	0.996093i	$0.528148\pi$
$312$	0	0
$313$	10.1492	0.573664	0.286832	−	0.957981i	$-0.407398\pi$
$313$	10.1492	0.573664	0.286832	+	0.957981i	$0.407398\pi$
$314$	−41.8038	−2.35913
$315$	0	0
$316$	−33.5985	−1.89007
$317$	−8.18921	−0.459952	−0.229976	−	0.973196i	$-0.573865\pi$
$317$	−8.18921	−0.459952	−0.229976	+	0.973196i	$0.573865\pi$
$318$	0	0
$319$	−30.8455	−1.72702
$320$	0	0
$321$	0	0
$322$	81.6568	4.55056
$323$	−4.66893	−0.259786
$324$	0	0
$325$	0	0
$326$	19.7765	1.09532
$327$	0	0
$328$	52.1794	2.88113
$329$	29.8776	1.64721
$330$	0	0
$331$	19.3339	1.06269	0.531344	−	0.847156i	$-0.321687\pi$
$331$	19.3339	1.06269	0.531344	+	0.847156i	$0.321687\pi$
$332$	−66.1189	−3.62875
$333$	0	0
$334$	10.9728	0.600408
$335$	0	0
$336$	0	0
$337$	0.902375	0.0491555	0.0245778	−	0.999698i	$-0.492176\pi$
$337$	0.902375	0.0491555	0.0245778	+	0.999698i	$0.492176\pi$
$338$	52.0155	2.82927
$339$	0	0
$340$	0	0
$341$	6.54734	0.354558
$342$	0	0
$343$	6.83551	0.369083
$344$	28.2048	1.52070
$345$	0	0
$346$	52.6454	2.83023
$347$	−0.676375	−0.0363097	−0.0181548	−	0.999835i	$-0.505779\pi$
$347$	−0.676375	−0.0363097	−0.0181548	+	0.999835i	$0.505779\pi$
$348$	0	0
$349$	10.0604	0.538523	0.269262	−	0.963067i	$-0.413220\pi$
$349$	10.0604	0.538523	0.269262	+	0.963067i	$0.413220\pi$
$350$	0	0
$351$	0	0
$352$	−50.4883	−2.69104
$353$	−7.29023	−0.388020	−0.194010	−	0.981000i	$-0.562149\pi$
$353$	−7.29023	−0.388020	−0.194010	+	0.981000i	$0.562149\pi$
$354$	0	0
$355$	0	0
$356$	−58.9368	−3.12365
$357$	0	0
$358$	−19.1202	−1.01053
$359$	15.8301	0.835479	0.417739	−	0.908567i	$-0.362823\pi$
$359$	15.8301	0.835479	0.417739	+	0.908567i	$0.362823\pi$
$360$	0	0
$361$	−12.9075	−0.679340
$362$	42.3528	2.22601
$363$	0	0
$364$	100.631	5.27449
$365$	0	0
$366$	0	0
$367$	−23.5116	−1.22730	−0.613649	−	0.789579i	$-0.710299\pi$
$367$	−23.5116	−1.22730	−0.613649	+	0.789579i	$0.710299\pi$
$368$	−103.462	−5.39335
$369$	0	0
$370$	0	0
$371$	7.05315	0.366181
$372$	0	0
$373$	8.27467	0.428446	0.214223	−	0.976785i	$-0.431278\pi$
$373$	8.27467	0.428446	0.214223	+	0.976785i	$0.431278\pi$
$374$	17.2837	0.893721
$375$	0	0
$376$	−70.7756	−3.64997
$377$	−51.2616	−2.64010
$378$	0	0
$379$	−27.3648	−1.40564	−0.702819	−	0.711369i	$-0.748075\pi$
$379$	−27.3648	−1.40564	−0.702819	+	0.711369i	$0.748075\pi$
$380$	0	0
$381$	0	0
$382$	38.6498	1.97749
$383$	−15.6880	−0.801620	−0.400810	−	0.916161i	$-0.631271\pi$
$383$	−15.6880	−0.801620	−0.400810	+	0.916161i	$0.631271\pi$
$384$	0	0
$385$	0	0
$386$	34.8094	1.77175
$387$	0	0
$388$	11.0825	0.562629
$389$	−20.6950	−1.04928	−0.524638	−	0.851325i	$-0.675799\pi$
$389$	−20.6950	−1.04928	−0.524638	+	0.851325i	$0.675799\pi$
$390$	0	0
$391$	16.7295	0.846046
$392$	41.3190	2.08692
$393$	0	0
$394$	47.6100	2.39856
$395$	0	0
$396$	0	0
$397$	−10.3616	−0.520033	−0.260016	−	0.965604i	$-0.583728\pi$
$397$	−10.3616	−0.520033	−0.260016	+	0.965604i	$0.583728\pi$
$398$	−56.3189	−2.82301
$399$	0	0
$400$	0	0
$401$	0.600848	0.0300049	0.0150025	−	0.999887i	$-0.495224\pi$
$401$	0.600848	0.0300049	0.0150025	+	0.999887i	$0.495224\pi$
$402$	0	0
$403$	10.8809	0.542015
$404$	47.3233	2.35442
$405$	0	0
$406$	−82.9683	−4.11765
$407$	17.4801	0.866454
$408$	0	0
$409$	−20.0064	−0.989253	−0.494626	−	0.869106i	$-0.664695\pi$
$409$	−20.0064	−0.989253	−0.494626	+	0.869106i	$0.664695\pi$
$410$	0	0
$411$	0	0
$412$	0.897792	0.0442310
$413$	29.3850	1.44594
$414$	0	0
$415$	0	0
$416$	−83.9055	−4.11381
$417$	0	0
$418$	−22.5537	−1.10314
$419$	−10.8245	−0.528813	−0.264407	−	0.964411i	$-0.585176\pi$
$419$	−10.8245	−0.528813	−0.264407	+	0.964411i	$0.585176\pi$
$420$	0	0
$421$	−31.3321	−1.52703	−0.763516	−	0.645789i	$-0.776528\pi$
$421$	−31.3321	−1.52703	−0.763516	+	0.645789i	$0.776528\pi$
$422$	35.6715	1.73646
$423$	0	0
$424$	−16.7078	−0.811405
$425$	0	0
$426$	0	0
$427$	27.4255	1.32721
$428$	7.00702	0.338697
$429$	0	0
$430$	0	0
$431$	31.6470	1.52438	0.762191	−	0.647353i	$-0.224124\pi$
$431$	31.6470	1.52438	0.762191	+	0.647353i	$0.224124\pi$
$432$	0	0
$433$	−19.9075	−0.956692	−0.478346	−	0.878172i	$-0.658763\pi$
$433$	−19.9075	−0.956692	−0.478346	+	0.878172i	$0.658763\pi$
$434$	17.6110	0.845356
$435$	0	0
$436$	9.65101	0.462200
$437$	−21.8305	−1.04429
$438$	0	0
$439$	−23.7707	−1.13451	−0.567256	−	0.823542i	$-0.691995\pi$
$439$	−23.7707	−1.13451	−0.567256	+	0.823542i	$0.691995\pi$
$440$	0	0
$441$	0	0
$442$	28.7235	1.36624
$443$	8.36336	0.397355	0.198678	−	0.980065i	$-0.436335\pi$
$443$	8.36336	0.397355	0.198678	+	0.980065i	$0.436335\pi$
$444$	0	0
$445$	0	0
$446$	−67.2307	−3.18347
$447$	0	0
$448$	−54.6577	−2.58234
$449$	35.2889	1.66539	0.832693	−	0.553734i	$-0.186798\pi$
$449$	35.2889	1.66539	0.832693	+	0.553734i	$0.186798\pi$
$450$	0	0
$451$	−21.7998	−1.02651
$452$	−11.9592	−0.562512
$453$	0	0
$454$	5.03535	0.236320
$455$	0	0
$456$	0	0
$457$	−12.9534	−0.605933	−0.302967	−	0.953001i	$-0.597977\pi$
$457$	−12.9534	−0.605933	−0.302967	+	0.953001i	$0.597977\pi$
$458$	−12.7263	−0.594660
$459$	0	0
$460$	0	0
$461$	−29.8908	−1.39215	−0.696076	−	0.717968i	$-0.745072\pi$
$461$	−29.8908	−1.39215	−0.696076	+	0.717968i	$0.745072\pi$
$462$	0	0
$463$	14.1155	0.656002	0.328001	−	0.944677i	$-0.393625\pi$
$463$	14.1155	0.656002	0.328001	+	0.944677i	$0.393625\pi$
$464$	105.124	4.88026
$465$	0	0
$466$	9.17233	0.424900
$467$	−4.00068	−0.185129	−0.0925647	−	0.995707i	$-0.529506\pi$
$467$	−4.00068	−0.185129	−0.0925647	+	0.995707i	$0.529506\pi$
$468$	0	0
$469$	24.1341	1.11441
$470$	0	0
$471$	0	0
$472$	−69.6085	−3.20399
$473$	−11.7836	−0.541810
$474$	0	0
$475$	0	0
$476$	33.3688	1.52946
$477$	0	0
$478$	−66.0618	−3.02159
$479$	19.2413	0.879156	0.439578	−	0.898204i	$-0.355128\pi$
$479$	19.2413	0.879156	0.439578	+	0.898204i	$0.355128\pi$
$480$	0	0
$481$	29.0497	1.32455
$482$	−54.5835	−2.48621
$483$	0	0
$484$	3.97725	0.180784
$485$	0	0
$486$	0	0
$487$	13.9219	0.630859	0.315430	−	0.948949i	$-0.397851\pi$
$487$	13.9219	0.630859	0.315430	+	0.948949i	$0.397851\pi$
$488$	−64.9669	−2.94091
$489$	0	0
$490$	0	0
$491$	−14.5058	−0.654639	−0.327319	−	0.944914i	$-0.606145\pi$
$491$	−14.5058	−0.654639	−0.327319	+	0.944914i	$0.606145\pi$
$492$	0	0
$493$	−16.9982	−0.765559
$494$	−37.4816	−1.68638
$495$	0	0
$496$	−22.3138	−1.00192
$497$	−10.1224	−0.454052
$498$	0	0
$499$	1.26905	0.0568103	0.0284052	−	0.999596i	$-0.490957\pi$
$499$	1.26905	0.0568103	0.0284052	+	0.999596i	$0.490957\pi$
$500$	0	0
$501$	0	0
$502$	5.61219	0.250484
$503$	−8.50684	−0.379301	−0.189651	−	0.981852i	$-0.560736\pi$
$503$	−8.50684	−0.379301	−0.189651	+	0.981852i	$0.560736\pi$
$504$	0	0
$505$	0	0
$506$	80.8134	3.59260
$507$	0	0
$508$	19.2299	0.853188
$509$	−36.4250	−1.61451	−0.807254	−	0.590204i	$-0.799047\pi$
$509$	−36.4250	−1.61451	−0.807254	+	0.590204i	$0.799047\pi$
$510$	0	0
$511$	−41.3701	−1.83011
$512$	−20.1538	−0.890680
$513$	0	0
$514$	3.86829	0.170623
$515$	0	0
$516$	0	0
$517$	29.5690	1.30044
$518$	47.0178	2.06585
$519$	0	0
$520$	0	0
$521$	31.9990	1.40190	0.700951	−	0.713209i	$-0.252759\pi$
$521$	31.9990	1.40190	0.700951	+	0.713209i	$0.252759\pi$
$522$	0	0
$523$	−29.0448	−1.27004	−0.635020	−	0.772496i	$-0.719008\pi$
$523$	−29.0448	−1.27004	−0.635020	+	0.772496i	$0.719008\pi$
$524$	−4.05460	−0.177126
$525$	0	0
$526$	68.5969	2.99097
$527$	3.60807	0.157170
$528$	0	0
$529$	55.2218	2.40095
$530$	0	0
$531$	0	0
$532$	−43.5434	−1.88784
$533$	−36.2287	−1.56924
$534$	0	0
$535$	0	0
$536$	−57.1700	−2.46937
$537$	0	0
$538$	−85.1254	−3.67001
$539$	−17.2625	−0.743547
$540$	0	0
$541$	−18.2546	−0.784827	−0.392414	−	0.919789i	$-0.628360\pi$
$541$	−18.2546	−0.784827	−0.392414	+	0.919789i	$0.628360\pi$
$542$	67.0353	2.87941
$543$	0	0
$544$	−27.8228	−1.19289
$545$	0	0
$546$	0	0
$547$	−6.92805	−0.296222	−0.148111	−	0.988971i	$-0.547319\pi$
$547$	−6.92805	−0.296222	−0.148111	+	0.988971i	$0.547319\pi$
$548$	63.1665	2.69834
$549$	0	0
$550$	0	0
$551$	22.1811	0.944946
$552$	0	0
$553$	22.9104	0.974249
$554$	−44.9474	−1.90963
$555$	0	0
$556$	62.2563	2.64025
$557$	−11.5873	−0.490969	−0.245484	−	0.969401i	$-0.578947\pi$
$557$	−11.5873	−0.490969	−0.245484	+	0.969401i	$0.578947\pi$
$558$	0	0
$559$	−19.5829	−0.828268
$560$	0	0
$561$	0	0
$562$	19.5862	0.826192
$563$	−4.92004	−0.207355	−0.103677	−	0.994611i	$-0.533061\pi$
$563$	−4.92004	−0.207355	−0.103677	+	0.994611i	$0.533061\pi$
$564$	0	0
$565$	0	0
$566$	−32.2691	−1.35637
$567$	0	0
$568$	23.9785	1.00611
$569$	37.5316	1.57341	0.786704	−	0.617331i	$-0.211786\pi$
$569$	37.5316	1.57341	0.786704	+	0.617331i	$0.211786\pi$
$570$	0	0
$571$	−31.5956	−1.32224	−0.661118	−	0.750282i	$-0.729918\pi$
$571$	−31.5956	−1.32224	−0.661118	+	0.750282i	$0.729918\pi$
$572$	99.5914	4.16413
$573$	0	0
$574$	−58.6371	−2.44747
$575$	0	0
$576$	0	0
$577$	−36.4055	−1.51558	−0.757790	−	0.652499i	$-0.773721\pi$
$577$	−36.4055	−1.51558	−0.757790	+	0.652499i	$0.773721\pi$
$578$	−35.7297	−1.48616
$579$	0	0
$580$	0	0
$581$	45.0856	1.87046
$582$	0	0
$583$	6.98030	0.289095
$584$	97.9996	4.05525
$585$	0	0
$586$	65.6486	2.71192
$587$	35.5061	1.46550	0.732748	−	0.680500i	$-0.238237\pi$
$587$	35.5061	1.46550	0.732748	+	0.680500i	$0.238237\pi$
$588$	0	0
$589$	−4.70820	−0.193998
$590$	0	0
$591$	0	0
$592$	−59.5734	−2.44845
$593$	45.7086	1.87703	0.938513	−	0.345244i	$-0.112204\pi$
$593$	45.7086	1.87703	0.938513	+	0.345244i	$0.112204\pi$
$594$	0	0
$595$	0	0
$596$	−6.66866	−0.273159
$597$	0	0
$598$	134.302	5.49202
$599$	3.58625	0.146530	0.0732652	−	0.997312i	$-0.476658\pi$
$599$	3.58625	0.146530	0.0732652	+	0.997312i	$0.476658\pi$
$600$	0	0
$601$	10.0366	0.409400	0.204700	−	0.978825i	$-0.434378\pi$
$601$	10.0366	0.409400	0.204700	+	0.978825i	$0.434378\pi$
$602$	−31.6955	−1.29181
$603$	0	0
$604$	13.0854	0.532439
$605$	0	0
$606$	0	0
$607$	−34.8691	−1.41529	−0.707646	−	0.706567i	$-0.750243\pi$
$607$	−34.8691	−1.41529	−0.707646	+	0.706567i	$0.750243\pi$
$608$	36.3063	1.47241
$609$	0	0
$610$	0	0
$611$	49.1402	1.98800
$612$	0	0
$613$	−23.8674	−0.963994	−0.481997	−	0.876173i	$-0.660088\pi$
$613$	−23.8674	−0.963994	−0.481997	+	0.876173i	$0.660088\pi$
$614$	53.2307	2.14822
$615$	0	0
$616$	97.8095	3.94086
$617$	−6.81159	−0.274224	−0.137112	−	0.990556i	$-0.543782\pi$
$617$	−6.81159	−0.274224	−0.137112	+	0.990556i	$0.543782\pi$
$618$	0	0
$619$	−24.7256	−0.993808	−0.496904	−	0.867806i	$-0.665530\pi$
$619$	−24.7256	−0.993808	−0.496904	+	0.867806i	$0.665530\pi$
$620$	0	0
$621$	0	0
$622$	−8.29180	−0.332471
$623$	40.1882	1.61011
$624$	0	0
$625$	0	0
$626$	27.0172	1.07983
$627$	0	0
$628$	−79.8749	−3.18735
$629$	9.63280	0.384085
$630$	0	0
$631$	−23.4178	−0.932250	−0.466125	−	0.884719i	$-0.654350\pi$
$631$	−23.4178	−0.932250	−0.466125	+	0.884719i	$0.654350\pi$
$632$	−54.2713	−2.15879
$633$	0	0
$634$	−21.7998	−0.865781
$635$	0	0
$636$	0	0
$637$	−28.6882	−1.13667
$638$	−82.1114	−3.25082
$639$	0	0
$640$	0	0
$641$	26.9591	1.06482	0.532410	−	0.846487i	$-0.321287\pi$
$641$	26.9591	1.06482	0.532410	+	0.846487i	$0.321287\pi$
$642$	0	0
$643$	9.56996	0.377403	0.188701	−	0.982035i	$-0.439572\pi$
$643$	9.56996	0.377403	0.188701	+	0.982035i	$0.439572\pi$
$644$	156.022	6.14814
$645$	0	0
$646$	−12.4288	−0.489004
$647$	−20.0941	−0.789981	−0.394991	−	0.918685i	$-0.629252\pi$
$647$	−20.0941	−0.789981	−0.394991	+	0.918685i	$0.629252\pi$
$648$	0	0
$649$	29.0815	1.14155
$650$	0	0
$651$	0	0
$652$	37.7871	1.47986
$653$	−40.7211	−1.59354	−0.796770	−	0.604282i	$-0.793460\pi$
$653$	−40.7211	−1.59354	−0.796770	+	0.604282i	$0.793460\pi$
$654$	0	0
$655$	0	0
$656$	74.2955	2.90075
$657$	0	0
$658$	79.5348	3.10059
$659$	11.4254	0.445070	0.222535	−	0.974925i	$-0.428567\pi$
$659$	11.4254	0.445070	0.222535	+	0.974925i	$0.428567\pi$
$660$	0	0
$661$	−33.5985	−1.30683	−0.653416	−	0.756999i	$-0.726665\pi$
$661$	−33.5985	−1.30683	−0.653416	+	0.756999i	$0.726665\pi$
$662$	51.4672	2.00033
$663$	0	0
$664$	−106.801	−4.14468
$665$	0	0
$666$	0	0
$667$	−79.4782	−3.07741
$668$	20.9659	0.811196
$669$	0	0
$670$	0	0
$671$	27.1422	1.04781
$672$	0	0
$673$	−27.7871	−1.07111	−0.535557	−	0.844499i	$-0.679898\pi$
$673$	−27.7871	−1.07111	−0.535557	+	0.844499i	$0.679898\pi$
$674$	2.40214	0.0925270
$675$	0	0
$676$	99.3865	3.82256
$677$	31.3520	1.20496	0.602478	−	0.798135i	$-0.294180\pi$
$677$	31.3520	1.20496	0.602478	+	0.798135i	$0.294180\pi$
$678$	0	0
$679$	−7.55701	−0.290012
$680$	0	0
$681$	0	0
$682$	17.4291	0.667396
$683$	0.655106	0.0250669	0.0125335	−	0.999921i	$-0.496010\pi$
$683$	0.655106	0.0250669	0.0125335	+	0.999921i	$0.496010\pi$
$684$	0	0
$685$	0	0
$686$	18.1963	0.694736
$687$	0	0
$688$	40.1594	1.53106
$689$	11.6004	0.441941
$690$	0	0
$691$	28.9381	1.10086	0.550428	−	0.834883i	$-0.314464\pi$
$691$	28.9381	1.10086	0.550428	+	0.834883i	$0.314464\pi$
$692$	100.590	3.82385
$693$	0	0
$694$	−1.80052	−0.0683469
$695$	0	0
$696$	0	0
$697$	−12.0133	−0.455036
$698$	26.7811	1.01368
$699$	0	0
$700$	0	0
$701$	31.0249	1.17179	0.585896	−	0.810386i	$-0.300743\pi$
$701$	31.0249	1.17179	0.585896	+	0.810386i	$0.300743\pi$
$702$	0	0
$703$	−12.5699	−0.474084
$704$	−54.0932	−2.03871
$705$	0	0
$706$	−19.4067	−0.730382
$707$	−32.2691	−1.21360
$708$	0	0
$709$	41.2676	1.54984	0.774918	−	0.632062i	$-0.217791\pi$
$709$	41.2676	1.54984	0.774918	+	0.632062i	$0.217791\pi$
$710$	0	0
$711$	0	0
$712$	−95.1998	−3.56776
$713$	16.8702	0.631794
$714$	0	0
$715$	0	0
$716$	−36.5331	−1.36531
$717$	0	0
$718$	42.1399	1.57265
$719$	2.15581	0.0803980	0.0401990	−	0.999192i	$-0.487201\pi$
$719$	2.15581	0.0803980	0.0401990	+	0.999192i	$0.487201\pi$
$720$	0	0
$721$	−0.612192	−0.0227992
$722$	−34.3599	−1.27874
$723$	0	0
$724$	80.9239	3.00751
$725$	0	0
$726$	0	0
$727$	−43.8091	−1.62479	−0.812396	−	0.583107i	$-0.801837\pi$
$727$	−43.8091	−1.62479	−0.812396	+	0.583107i	$0.801837\pi$
$728$	162.548	6.02441
$729$	0	0
$730$	0	0
$731$	−6.49362	−0.240175
$732$	0	0
$733$	38.6326	1.42693	0.713464	−	0.700692i	$-0.247125\pi$
$733$	38.6326	1.42693	0.713464	+	0.700692i	$0.247125\pi$
$734$	−62.5884	−2.31018
$735$	0	0
$736$	−130.091	−4.79521
$737$	23.8848	0.879808
$738$	0	0
$739$	20.4743	0.753159	0.376579	−	0.926384i	$-0.377100\pi$
$739$	20.4743	0.753159	0.376579	+	0.926384i	$0.377100\pi$
$740$	0	0
$741$	0	0
$742$	18.7756	0.689275
$743$	−4.86490	−0.178476	−0.0892379	−	0.996010i	$-0.528443\pi$
$743$	−4.86490	−0.178476	−0.0892379	+	0.996010i	$0.528443\pi$
$744$	0	0
$745$	0	0
$746$	22.0273	0.806478
$747$	0	0
$748$	33.0242	1.20748
$749$	−4.77799	−0.174584
$750$	0	0
$751$	−36.7353	−1.34049	−0.670245	−	0.742140i	$-0.733811\pi$
$751$	−36.7353	−1.34049	−0.670245	+	0.742140i	$0.733811\pi$
$752$	−100.774	−3.67484
$753$	0	0
$754$	−136.459	−4.96955
$755$	0	0
$756$	0	0
$757$	−13.1609	−0.478340	−0.239170	−	0.970978i	$-0.576875\pi$
$757$	−13.1609	−0.478340	−0.239170	+	0.970978i	$0.576875\pi$
$758$	−72.8457	−2.64587
$759$	0	0
$760$	0	0
$761$	17.9859	0.651987	0.325994	−	0.945372i	$-0.394301\pi$
$761$	17.9859	0.651987	0.325994	+	0.945372i	$0.394301\pi$
$762$	0	0
$763$	−6.58089	−0.238244
$764$	73.8484	2.67174
$765$	0	0
$766$	−41.7618	−1.50891
$767$	48.3299	1.74509
$768$	0	0
$769$	3.28562	0.118483	0.0592413	−	0.998244i	$-0.481132\pi$
$769$	3.28562	0.118483	0.0592413	+	0.998244i	$0.481132\pi$
$770$	0	0
$771$	0	0
$772$	66.5106	2.39377
$773$	−10.0265	−0.360628	−0.180314	−	0.983609i	$-0.557711\pi$
$773$	−10.0265	−0.360628	−0.180314	+	0.983609i	$0.557711\pi$
$774$	0	0
$775$	0	0
$776$	17.9014	0.642624
$777$	0	0
$778$	−55.0904	−1.97509
$779$	15.6763	0.561661
$780$	0	0
$781$	−10.0179	−0.358467
$782$	44.5342	1.59254
$783$	0	0
$784$	58.8319	2.10114
$785$	0	0
$786$	0	0
$787$	−43.1508	−1.53816	−0.769080	−	0.639153i	$-0.779285\pi$
$787$	−43.1508	−1.53816	−0.769080	+	0.639153i	$0.779285\pi$
$788$	90.9688	3.24063
$789$	0	0
$790$	0	0
$791$	8.15479	0.289951
$792$	0	0
$793$	45.1071	1.60180
$794$	−27.5827	−0.978874
$795$	0	0
$796$	−107.609	−3.81410
$797$	23.2836	0.824748	0.412374	−	0.911015i	$-0.364700\pi$
$797$	23.2836	0.824748	0.412374	+	0.911015i	$0.364700\pi$
$798$	0	0
$799$	16.2947	0.576466
$800$	0	0
$801$	0	0
$802$	1.59947	0.0564792
$803$	−40.9428	−1.44484
$804$	0	0
$805$	0	0
$806$	28.9651	1.02025
$807$	0	0
$808$	76.4406	2.68917
$809$	−6.18914	−0.217598	−0.108799	−	0.994064i	$-0.534701\pi$
$809$	−6.18914	−0.217598	−0.108799	+	0.994064i	$0.534701\pi$
$810$	0	0
$811$	−2.53809	−0.0891245	−0.0445623	−	0.999007i	$-0.514189\pi$
$811$	−2.53809	−0.0891245	−0.0445623	+	0.999007i	$0.514189\pi$
$812$	−158.528	−5.56325
$813$	0	0
$814$	46.5322	1.63095
$815$	0	0
$816$	0	0
$817$	8.47360	0.296454
$818$	−53.2574	−1.86210
$819$	0	0
$820$	0	0
$821$	−44.4179	−1.55019	−0.775097	−	0.631842i	$-0.782299\pi$
$821$	−44.4179	−1.55019	−0.775097	+	0.631842i	$0.782299\pi$
$822$	0	0
$823$	−19.6434	−0.684724	−0.342362	−	0.939568i	$-0.611227\pi$
$823$	−19.6434	−0.684724	−0.342362	+	0.939568i	$0.611227\pi$
$824$	1.45019	0.0505198
$825$	0	0
$826$	78.2233	2.72174
$827$	31.1557	1.08339	0.541695	−	0.840575i	$-0.317783\pi$
$827$	31.1557	1.08339	0.541695	+	0.840575i	$0.317783\pi$
$828$	0	0
$829$	2.46831	0.0857278	0.0428639	−	0.999081i	$-0.486352\pi$
$829$	2.46831	0.0857278	0.0428639	+	0.999081i	$0.486352\pi$
$830$	0	0
$831$	0	0
$832$	−89.8964	−3.11660
$833$	−9.51290	−0.329602
$834$	0	0
$835$	0	0
$836$	−43.0936	−1.49042
$837$	0	0
$838$	−28.8151	−0.995401
$839$	−46.0334	−1.58925	−0.794625	−	0.607100i	$-0.792333\pi$
$839$	−46.0334	−1.58925	−0.794625	+	0.607100i	$0.792333\pi$
$840$	0	0
$841$	51.7547	1.78464
$842$	−83.4065	−2.87438
$843$	0	0
$844$	68.1579	2.34609
$845$	0	0
$846$	0	0
$847$	−2.71203	−0.0931866
$848$	−23.7894	−0.816932
$849$	0	0
$850$	0	0
$851$	45.0400	1.54395
$852$	0	0
$853$	9.24700	0.316611	0.158306	−	0.987390i	$-0.449397\pi$
$853$	9.24700	0.316611	0.158306	+	0.987390i	$0.449397\pi$
$854$	73.0072	2.49825
$855$	0	0
$856$	11.3183	0.386853
$857$	−32.3505	−1.10507	−0.552536	−	0.833489i	$-0.686340\pi$
$857$	−32.3505	−1.10507	−0.552536	+	0.833489i	$0.686340\pi$
$858$	0	0
$859$	−14.0883	−0.480688	−0.240344	−	0.970688i	$-0.577260\pi$
$859$	−14.0883	−0.480688	−0.240344	+	0.970688i	$0.577260\pi$
$860$	0	0
$861$	0	0
$862$	84.2448	2.86939
$863$	−52.5301	−1.78815	−0.894073	−	0.447921i	$-0.852164\pi$
$863$	−52.5301	−1.78815	−0.894073	+	0.447921i	$0.852164\pi$
$864$	0	0
$865$	0	0
$866$	−52.9940	−1.80081
$867$	0	0
$868$	33.6495	1.14214
$869$	22.6738	0.769155
$870$	0	0
$871$	39.6936	1.34497
$872$	15.5891	0.527915
$873$	0	0
$874$	−58.1131	−1.96571
$875$	0	0
$876$	0	0
$877$	6.58359	0.222312	0.111156	−	0.993803i	$-0.464545\pi$
$877$	6.58359	0.222312	0.111156	+	0.993803i	$0.464545\pi$
$878$	−63.2779	−2.13553
$879$	0	0
$880$	0	0
$881$	−36.4594	−1.22835	−0.614174	−	0.789171i	$-0.710511\pi$
$881$	−36.4594	−1.22835	−0.614174	+	0.789171i	$0.710511\pi$
$882$	0	0
$883$	−17.5190	−0.589560	−0.294780	−	0.955565i	$-0.595246\pi$
$883$	−17.5190	−0.589560	−0.294780	+	0.955565i	$0.595246\pi$
$884$	54.8822	1.84589
$885$	0	0
$886$	22.2634	0.747954
$887$	−24.6277	−0.826917	−0.413459	−	0.910523i	$-0.635679\pi$
$887$	−24.6277	−0.826917	−0.413459	+	0.910523i	$0.635679\pi$
$888$	0	0
$889$	−13.1126	−0.439782
$890$	0	0
$891$	0	0
$892$	−128.458	−4.30110
$893$	−21.2632	−0.711544
$894$	0	0
$895$	0	0
$896$	−43.4694	−1.45221
$897$	0	0
$898$	93.9397	3.13481
$899$	−17.1412	−0.571689
$900$	0	0
$901$	3.84666	0.128151
$902$	−58.0315	−1.93224
$903$	0	0
$904$	−19.3175	−0.642490
$905$	0	0
$906$	0	0
$907$	4.47955	0.148741	0.0743705	−	0.997231i	$-0.476305\pi$
$907$	4.47955	0.148741	0.0743705	+	0.997231i	$0.476305\pi$
$908$	9.62108	0.319287
$909$	0	0
$910$	0	0
$911$	41.9576	1.39012	0.695058	−	0.718953i	$-0.255379\pi$
$911$	41.9576	1.39012	0.695058	+	0.718953i	$0.255379\pi$
$912$	0	0
$913$	44.6199	1.47670
$914$	−34.4821	−1.14057
$915$	0	0
$916$	−24.3162	−0.803430
$917$	2.76478	0.0913010
$918$	0	0
$919$	−37.6692	−1.24259	−0.621297	−	0.783575i	$-0.713394\pi$
$919$	−37.6692	−1.24259	−0.621297	+	0.783575i	$0.713394\pi$
$920$	0	0
$921$	0	0
$922$	−79.5698	−2.62049
$923$	−16.6485	−0.547992
$924$	0	0
$925$	0	0
$926$	37.5757	1.23481
$927$	0	0
$928$	132.180	4.33903
$929$	17.1493	0.562649	0.281325	−	0.959613i	$-0.409226\pi$
$929$	17.1493	0.562649	0.281325	+	0.959613i	$0.409226\pi$
$930$	0	0
$931$	12.4135	0.406835
$932$	17.5256	0.574071
$933$	0	0
$934$	−10.6499	−0.348475
$935$	0	0
$936$	0	0
$937$	−37.1944	−1.21509	−0.607544	−	0.794286i	$-0.707845\pi$
$937$	−37.1944	−1.21509	−0.607544	+	0.794286i	$0.707845\pi$
$938$	64.2453	2.09768
$939$	0	0
$940$	0	0
$941$	−29.5387	−0.962935	−0.481468	−	0.876464i	$-0.659896\pi$
$941$	−29.5387	−0.962935	−0.481468	+	0.876464i	$0.659896\pi$
$942$	0	0
$943$	−56.1705	−1.82916
$944$	−99.1119	−3.22582
$945$	0	0
$946$	−31.3681	−1.01987
$947$	34.0869	1.10767	0.553837	−	0.832625i	$-0.313163\pi$
$947$	34.0869	1.10767	0.553837	+	0.832625i	$0.313163\pi$
$948$	0	0
$949$	−68.0421	−2.20874
$950$	0	0
$951$	0	0
$952$	53.9002	1.74692
$953$	−9.67588	−0.313432	−0.156716	−	0.987644i	$-0.550091\pi$
$953$	−9.67588	−0.313432	−0.156716	+	0.987644i	$0.550091\pi$
$954$	0	0
$955$	0	0
$956$	−126.225	−4.08240
$957$	0	0
$958$	51.2206	1.65486
$959$	−43.0724	−1.39088
$960$	0	0
$961$	−27.3616	−0.882632
$962$	77.3309	2.49325
$963$	0	0
$964$	−104.293	−3.35905
$965$	0	0
$966$	0	0
$967$	17.2537	0.554842	0.277421	−	0.960748i	$-0.410520\pi$
$967$	17.2537	0.554842	0.277421	+	0.960748i	$0.410520\pi$
$968$	6.42440	0.206488
$969$	0	0
$970$	0	0
$971$	1.92509	0.0617789	0.0308895	−	0.999523i	$-0.490166\pi$
$971$	1.92509	0.0617789	0.0308895	+	0.999523i	$0.490166\pi$
$972$	0	0
$973$	−42.4517	−1.36094
$974$	37.0602	1.18749
$975$	0	0
$976$	−92.5029	−2.96095
$977$	26.1185	0.835605	0.417803	−	0.908538i	$-0.362800\pi$
$977$	26.1185	0.835605	0.417803	+	0.908538i	$0.362800\pi$
$978$	0	0
$979$	39.7731	1.27116
$980$	0	0
$981$	0	0
$982$	−38.6148	−1.23225
$983$	9.28360	0.296101	0.148050	−	0.988980i	$-0.452700\pi$
$983$	9.28360	0.296101	0.148050	+	0.988980i	$0.452700\pi$
$984$	0	0
$985$	0	0
$986$	−45.2494	−1.44104
$987$	0	0
$988$	−71.6164	−2.27842
$989$	−30.3622	−0.965461
$990$	0	0
$991$	41.1337	1.30666	0.653328	−	0.757075i	$-0.273372\pi$
$991$	41.1337	1.30666	0.653328	+	0.757075i	$0.273372\pi$
$992$	−28.0568	−0.890805
$993$	0	0
$994$	−26.9461	−0.854677
$995$	0	0
$996$	0	0
$997$	−4.60496	−0.145840	−0.0729202	−	0.997338i	$-0.523232\pi$
$997$	−4.60496	−0.145840	−0.0729202	+	0.997338i	$0.523232\pi$
$998$	3.37823	0.106936
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.z.1.8	yes	8
3.2	odd	2	inner	5625.2.a.z.1.1	✓	8
5.4	even	2		5625.2.a.bb.1.1	yes	8
15.14	odd	2		5625.2.a.bb.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5625.2.a.z.1.1	✓	8	3.2	odd	2	inner
5625.2.a.z.1.8	yes	8	1.1	even	1	trivial
5625.2.a.bb.1.1	yes	8	5.4	even	2
5625.2.a.bb.1.8	yes	8	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.66202 q^{2} +5.08634 q^{4} -3.46831 q^{7} +8.21589 q^{8} +O(q^{10})\)	\(q+2.66202 q^{2} +5.08634 q^{4} -3.46831 q^{7} +8.21589 q^{8} -3.43248 q^{11} -5.70437 q^{13} -9.23269 q^{14} +11.6982 q^{16} -1.89155 q^{17} +2.46831 q^{19} -9.13733 q^{22} -8.84431 q^{23} -15.1851 q^{26} -17.6410 q^{28} +8.98636 q^{29} -1.90746 q^{31} +14.7090 q^{32} -5.03535 q^{34} -5.09254 q^{37} +6.57068 q^{38} +6.35103 q^{41} +3.43296 q^{43} -17.4588 q^{44} -23.5437 q^{46} -8.61447 q^{47} +5.02915 q^{49} -29.0144 q^{52} -2.03360 q^{53} -28.4952 q^{56} +23.9219 q^{58} -8.47242 q^{59} -7.90746 q^{61} -5.07770 q^{62} +15.7592 q^{64} -6.95846 q^{67} -9.62108 q^{68} +2.91855 q^{71} +11.9280 q^{73} -13.5564 q^{74} +12.5546 q^{76} +11.9049 q^{77} -6.60564 q^{79} +16.9066 q^{82} -12.9993 q^{83} +9.13860 q^{86} -28.2009 q^{88} -11.5873 q^{89} +19.7845 q^{91} -44.9852 q^{92} -22.9319 q^{94} +2.17888 q^{97} +13.3877 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{4} - 10 q^{7}+O(q^{10}) \) 8 * q + 14 * q^4 - 10 * q^7	\( 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} + 22 q^{16} + 2 q^{19} - 10 q^{22} - 70 q^{28} - 6 q^{31} - 50 q^{34} - 50 q^{37} + 14 q^{49} - 80 q^{52} + 30 q^{58} - 54 q^{61} + 36 q^{64} - 10 q^{67} - 30 q^{73} + 56 q^{76} + 28 q^{79} - 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100}) \) 8 * q + 14 * q^4 - 10 * q^7 - 10 * q^13 + 22 * q^16 + 2 * q^19 - 10 * q^22 - 70 * q^28 - 6 * q^31 - 50 * q^34 - 50 * q^37 + 14 * q^49 - 80 * q^52 + 30 * q^58 - 54 * q^61 + 36 * q^64 - 10 * q^67 - 30 * q^73 + 56 * q^76 + 28 * q^79 - 20 * q^88 + 60 * q^91 - 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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