LMFDB - Embedded newform 7623.2.a.cm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 2x + 1 $ x^4 - 2x^3 - 4x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.43091$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.301143 q^{2} -1.90931 q^{4} -1.30114 q^{5} -1.00000 q^{7} -1.17726 q^{8} +O(q^{10})$	$q+0.301143 q^{2} -1.90931 q^{4} -1.30114 q^{5} -1.00000 q^{7} -1.17726 q^{8} -0.391830 q^{10} -3.38772 q^{13} -0.301143 q^{14} +3.46410 q^{16} -0.478405 q^{17} +1.60817 q^{19} +2.48429 q^{20} -3.47841 q^{23} -3.30703 q^{25} -1.02019 q^{26} +1.90931 q^{28} -0.942507 q^{29} -1.17726 q^{31} +3.39771 q^{32} -0.144068 q^{34} +1.30114 q^{35} -3.55479 q^{37} +0.484289 q^{38} +1.53179 q^{40} -3.94839 q^{41} -2.87024 q^{43} -1.04750 q^{46} -12.1113 q^{47} +1.00000 q^{49} -0.995888 q^{50} +6.46821 q^{52} -5.28273 q^{53} +1.17726 q^{56} -0.283829 q^{58} +9.69345 q^{59} +3.00588 q^{61} -0.354524 q^{62} -5.90501 q^{64} +4.40791 q^{65} +7.74683 q^{67} +0.913425 q^{68} +0.391830 q^{70} +8.46568 q^{71} -13.3709 q^{73} -1.07050 q^{74} -3.07050 q^{76} -8.86612 q^{79} -4.50729 q^{80} -1.18903 q^{82} +11.4700 q^{83} +0.622473 q^{85} -0.864351 q^{86} -7.54068 q^{89} +3.38772 q^{91} +6.64136 q^{92} -3.64725 q^{94} -2.09246 q^{95} -1.71364 q^{97} +0.301143 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 4 * q^7	$ 4 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 4 q^{7} - 14 q^{10} + 2 q^{13} - 2 q^{14} + 2 q^{17} - 6 q^{19} - 8 q^{20} - 10 q^{23} - 4 q^{28} + 14 q^{29} + 12 q^{32} - 2 q^{34} + 6 q^{35} - 12 q^{37} - 16 q^{38} - 8 q^{40} + 16 q^{41} - 20 q^{43} - 8 q^{46} - 2 q^{47} + 4 q^{49} + 24 q^{50} + 40 q^{52} + 16 q^{53} - 8 q^{58} - 2 q^{59} - 2 q^{61} + 8 q^{62} - 16 q^{64} - 2 q^{65} - 20 q^{67} + 20 q^{68} + 14 q^{70} + 10 q^{71} + 10 q^{73} - 20 q^{74} - 28 q^{76} - 16 q^{79} - 12 q^{80} + 28 q^{82} + 18 q^{83} - 26 q^{86} + 14 q^{89} - 2 q^{91} + 8 q^{92} + 18 q^{94} + 22 q^{95} + 38 q^{97} + 2 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 4 * q^7 - 14 * q^10 + 2 * q^13 - 2 * q^14 + 2 * q^17 - 6 * q^19 - 8 * q^20 - 10 * q^23 - 4 * q^28 + 14 * q^29 + 12 * q^32 - 2 * q^34 + 6 * q^35 - 12 * q^37 - 16 * q^38 - 8 * q^40 + 16 * q^41 - 20 * q^43 - 8 * q^46 - 2 * q^47 + 4 * q^49 + 24 * q^50 + 40 * q^52 + 16 * q^53 - 8 * q^58 - 2 * q^59 - 2 * q^61 + 8 * q^62 - 16 * q^64 - 2 * q^65 - 20 * q^67 + 20 * q^68 + 14 * q^70 + 10 * q^71 + 10 * q^73 - 20 * q^74 - 28 * q^76 - 16 * q^79 - 12 * q^80 + 28 * q^82 + 18 * q^83 - 26 * q^86 + 14 * q^89 - 2 * q^91 + 8 * q^92 + 18 * q^94 + 22 * q^95 + 38 * q^97 + 2 * 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 $

$2$	0.301143	0.212940	0.106470	−	0.994316i	$-0.466045\pi$
$2$	0.301143	0.212940	0.106470	+	0.994316i	$0.466045\pi$
$3$	0	0
$3$	0	0
$4$	−1.90931	−0.954656
$5$	−1.30114	−0.581889	−0.290944	−	0.956740i	$-0.593969\pi$
$5$	−1.30114	−0.581889	−0.290944	+	0.956740i	$0.593969\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.17726	−0.416225
$9$	0	0
$10$	−0.391830	−0.123908
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.38772	−0.939584	−0.469792	−	0.882777i	$-0.655671\pi$
$13$	−3.38772	−0.939584	−0.469792	+	0.882777i	$0.655671\pi$
$14$	−0.301143	−0.0804838
$15$	0	0
$16$	3.46410	0.866025
$17$	−0.478405	−0.116030	−0.0580151	−	0.998316i	$-0.518477\pi$
$17$	−0.478405	−0.116030	−0.0580151	+	0.998316i	$0.518477\pi$
$18$	0	0
$19$	1.60817	0.368939	0.184470	−	0.982838i	$-0.440943\pi$
$19$	1.60817	0.368939	0.184470	+	0.982838i	$0.440943\pi$
$20$	2.48429	0.555504
$21$	0	0
$22$	0	0
$23$	−3.47841	−0.725298	−0.362649	−	0.931926i	$-0.618128\pi$
$23$	−3.47841	−0.725298	−0.362649	+	0.931926i	$0.618128\pi$
$24$	0	0
$25$	−3.30703	−0.661405
$26$	−1.02019	−0.200075
$27$	0	0
$28$	1.90931	0.360826
$29$	−0.942507	−0.175019	−0.0875095	−	0.996164i	$-0.527891\pi$
$29$	−0.942507	−0.175019	−0.0875095	+	0.996164i	$0.527891\pi$
$30$	0	0
$31$	−1.17726	−0.211443	−0.105721	−	0.994396i	$-0.533715\pi$
$31$	−1.17726	−0.211443	−0.105721	+	0.994396i	$0.533715\pi$
$32$	3.39771	0.600637
$33$	0	0
$34$	−0.144068	−0.0247075
$35$	1.30114	0.219933
$36$	0	0
$37$	−3.55479	−0.584404	−0.292202	−	0.956357i	$-0.594388\pi$
$37$	−3.55479	−0.584404	−0.292202	+	0.956357i	$0.594388\pi$
$38$	0.484289	0.0785621
$39$	0	0
$40$	1.53179	0.242197
$41$	−3.94839	−0.616635	−0.308318	−	0.951284i	$-0.599766\pi$
$41$	−3.94839	−0.616635	−0.308318	+	0.951284i	$0.599766\pi$
$42$	0	0
$43$	−2.87024	−0.437707	−0.218853	−	0.975758i	$-0.570232\pi$
$43$	−2.87024	−0.437707	−0.218853	+	0.975758i	$0.570232\pi$
$44$	0	0
$45$	0	0
$46$	−1.04750	−0.154445
$47$	−12.1113	−1.76662	−0.883311	−	0.468788i	$-0.844691\pi$
$47$	−12.1113	−1.76662	−0.883311	+	0.468788i	$0.844691\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.995888	−0.140840
$51$	0	0
$52$	6.46821	0.896980
$53$	−5.28273	−0.725638	−0.362819	−	0.931860i	$-0.618186\pi$
$53$	−5.28273	−0.725638	−0.362819	+	0.931860i	$0.618186\pi$
$54$	0	0
$55$	0	0
$56$	1.17726	0.157318
$57$	0	0
$58$	−0.283829	−0.0372686
$59$	9.69345	1.26198	0.630990	−	0.775791i	$-0.282649\pi$
$59$	9.69345	1.26198	0.630990	+	0.775791i	$0.282649\pi$
$60$	0	0
$61$	3.00588	0.384864	0.192432	−	0.981310i	$-0.438363\pi$
$61$	3.00588	0.384864	0.192432	+	0.981310i	$0.438363\pi$
$62$	−0.354524	−0.0450246
$63$	0	0
$64$	−5.90501	−0.738126
$65$	4.40791	0.546733
$66$	0	0
$67$	7.74683	0.946426	0.473213	−	0.880948i	$-0.343094\pi$
$67$	7.74683	0.946426	0.473213	+	0.880948i	$0.343094\pi$
$68$	0.913425	0.110769
$69$	0	0
$70$	0.391830	0.0468326
$71$	8.46568	1.00469	0.502346	−	0.864667i	$-0.332471\pi$
$71$	8.46568	1.00469	0.502346	+	0.864667i	$0.332471\pi$
$72$	0	0
$73$	−13.3709	−1.56494	−0.782472	−	0.622686i	$-0.786041\pi$
$73$	−13.3709	−1.56494	−0.782472	+	0.622686i	$0.786041\pi$
$74$	−1.07050	−0.124443
$75$	0	0
$76$	−3.07050	−0.352210
$77$	0	0
$78$	0	0
$79$	−8.86612	−0.997517	−0.498758	−	0.866741i	$-0.666210\pi$
$79$	−8.86612	−0.997517	−0.498758	+	0.866741i	$0.666210\pi$
$80$	−4.50729	−0.503931
$81$	0	0
$82$	−1.18903	−0.131306
$83$	11.4700	1.25899	0.629497	−	0.777003i	$-0.283261\pi$
$83$	11.4700	1.25899	0.629497	+	0.777003i	$0.283261\pi$
$84$	0	0
$85$	0.622473	0.0675167
$86$	−0.864351	−0.0932054
$87$	0	0
$88$	0	0
$89$	−7.54068	−0.799311	−0.399655	−	0.916666i	$-0.630870\pi$
$89$	−7.54068	−0.799311	−0.399655	+	0.916666i	$0.630870\pi$
$90$	0	0
$91$	3.38772	0.355129
$92$	6.64136	0.692410
$93$	0	0
$94$	−3.64725	−0.376185
$95$	−2.09246	−0.214682
$96$	0	0
$97$	−1.71364	−0.173993	−0.0869967	−	0.996209i	$-0.527727\pi$
$97$	−1.71364	−0.173993	−0.0869967	+	0.996209i	$0.527727\pi$
$98$	0.301143	0.0304200
$99$	0	0
$100$	6.31415	0.631415
$101$	−9.53637	−0.948905	−0.474452	−	0.880281i	$-0.657354\pi$
$101$	−9.53637	−0.948905	−0.474452	+	0.880281i	$0.657354\pi$
$102$	0	0
$103$	−1.22634	−0.120834	−0.0604172	−	0.998173i	$-0.519243\pi$
$103$	−1.22634	−0.120834	−0.0604172	+	0.998173i	$0.519243\pi$
$104$	3.98823	0.391078
$105$	0	0
$106$	−1.59086	−0.154518
$107$	9.16296	0.885817	0.442908	−	0.896567i	$-0.353947\pi$
$107$	9.16296	0.885817	0.442908	+	0.896567i	$0.353947\pi$
$108$	0	0
$109$	−16.2268	−1.55425	−0.777123	−	0.629348i	$-0.783322\pi$
$109$	−16.2268	−1.55425	−0.777123	+	0.629348i	$0.783322\pi$
$110$	0	0
$111$	0	0
$112$	−3.46410	−0.327327
$113$	−17.4473	−1.64130	−0.820650	−	0.571431i	$-0.806389\pi$
$113$	−17.4473	−1.64130	−0.820650	+	0.571431i	$0.806389\pi$
$114$	0	0
$115$	4.52590	0.422043
$116$	1.79954	0.167083
$117$	0	0
$118$	2.91911	0.268726
$119$	0.478405	0.0438553
$120$	0	0
$121$	0	0
$122$	0.905201	0.0819530
$123$	0	0
$124$	2.24776	0.201855
$125$	10.8086	0.966753
$126$	0	0
$127$	−2.60248	−0.230933	−0.115466	−	0.993311i	$-0.536836\pi$
$127$	−2.60248	−0.230933	−0.115466	+	0.993311i	$0.536836\pi$
$128$	−8.57368	−0.757813
$129$	0	0
$130$	1.32741	0.116422
$131$	10.2853	0.898628	0.449314	−	0.893374i	$-0.351669\pi$
$131$	10.2853	0.898628	0.449314	+	0.893374i	$0.351669\pi$
$132$	0	0
$133$	−1.60817	−0.139446
$134$	2.33290	0.201532
$135$	0	0
$136$	0.563208	0.0482947
$137$	19.9098	1.70101	0.850504	−	0.525969i	$-0.176297\pi$
$137$	19.9098	1.70101	0.850504	+	0.525969i	$0.176297\pi$
$138$	0	0
$139$	−4.64089	−0.393635	−0.196818	−	0.980440i	$-0.563061\pi$
$139$	−4.64089	−0.393635	−0.196818	+	0.980440i	$0.563061\pi$
$140$	−2.48429	−0.209961
$141$	0	0
$142$	2.54938	0.213939
$143$	0	0
$144$	0	0
$145$	1.22634	0.101842
$146$	−4.02655	−0.333239
$147$	0	0
$148$	6.78720	0.557905
$149$	23.4807	1.92361	0.961805	−	0.273736i	$-0.0882593\pi$
$149$	23.4807	1.92361	0.961805	+	0.273736i	$0.0882593\pi$
$150$	0	0
$151$	−13.3893	−1.08961	−0.544803	−	0.838564i	$-0.683395\pi$
$151$	−13.3893	−1.08961	−0.544803	+	0.838564i	$0.683395\pi$
$152$	−1.89324	−0.153562
$153$	0	0
$154$	0	0
$155$	1.53179	0.123036
$156$	0	0
$157$	12.6414	1.00889	0.504445	−	0.863444i	$-0.331697\pi$
$157$	12.6414	1.00889	0.504445	+	0.863444i	$0.331697\pi$
$158$	−2.66997	−0.212411
$159$	0	0
$160$	−4.42091	−0.349504
$161$	3.47841	0.274137
$162$	0	0
$163$	−0.854355	−0.0669183	−0.0334591	−	0.999440i	$-0.510652\pi$
$163$	−0.854355	−0.0669183	−0.0334591	+	0.999440i	$0.510652\pi$
$164$	7.53871	0.588675
$165$	0	0
$166$	3.45411	0.268091
$167$	8.16727	0.632002	0.316001	−	0.948759i	$-0.397660\pi$
$167$	8.16727	0.632002	0.316001	+	0.948759i	$0.397660\pi$
$168$	0	0
$169$	−1.52337	−0.117182
$170$	0.187453	0.0143770
$171$	0	0
$172$	5.48018	0.417860
$173$	20.9111	1.58984	0.794920	−	0.606714i	$-0.207513\pi$
$173$	20.9111	1.58984	0.794920	+	0.606714i	$0.207513\pi$
$174$	0	0
$175$	3.30703	0.249988
$176$	0	0
$177$	0	0
$178$	−2.27082	−0.170205
$179$	−8.41230	−0.628765	−0.314382	−	0.949296i	$-0.601797\pi$
$179$	−8.41230	−0.628765	−0.314382	+	0.949296i	$0.601797\pi$
$180$	0	0
$181$	−20.6995	−1.53858	−0.769292	−	0.638898i	$-0.779391\pi$
$181$	−20.6995	−1.53858	−0.769292	+	0.638898i	$0.779391\pi$
$182$	1.02019	0.0756213
$183$	0	0
$184$	4.09499	0.301887
$185$	4.62529	0.340058
$186$	0	0
$187$	0	0
$188$	23.1244	1.68652
$189$	0	0
$190$	−0.630129	−0.0457144
$191$	14.6830	1.06242	0.531211	−	0.847239i	$-0.321737\pi$
$191$	14.6830	1.06242	0.531211	+	0.847239i	$0.321737\pi$
$192$	0	0
$193$	25.2657	1.81866	0.909332	−	0.416071i	$-0.136593\pi$
$193$	25.2657	1.81866	0.909332	+	0.416071i	$0.136593\pi$
$194$	−0.516049	−0.0370502
$195$	0	0
$196$	−1.90931	−0.136379
$197$	−10.3936	−0.740513	−0.370257	−	0.928929i	$-0.620730\pi$
$197$	−10.3936	−0.740513	−0.370257	+	0.928929i	$0.620730\pi$
$198$	0	0
$199$	11.1704	0.791850	0.395925	−	0.918283i	$-0.370424\pi$
$199$	11.1704	0.791850	0.395925	+	0.918283i	$0.370424\pi$
$200$	3.89324	0.275293
$201$	0	0
$202$	−2.87181	−0.202060
$203$	0.942507	0.0661510
$204$	0	0
$205$	5.13742	0.358813
$206$	−0.369302	−0.0257305
$207$	0	0
$208$	−11.7354	−0.813704
$209$	0	0
$210$	0	0
$211$	−11.1918	−0.770478	−0.385239	−	0.922817i	$-0.625881\pi$
$211$	−11.1918	−0.770478	−0.385239	+	0.922817i	$0.625881\pi$
$212$	10.0864	0.692735
$213$	0	0
$214$	2.75936	0.188626
$215$	3.73459	0.254697
$216$	0	0
$217$	1.17726	0.0799178
$218$	−4.88659	−0.330962
$219$	0	0
$220$	0	0
$221$	1.62070	0.109020
$222$	0	0
$223$	−20.4514	−1.36953	−0.684763	−	0.728766i	$-0.740094\pi$
$223$	−20.4514	−1.36953	−0.684763	+	0.728766i	$0.740094\pi$
$224$	−3.39771	−0.227019
$225$	0	0
$226$	−5.25412	−0.349499
$227$	28.0554	1.86210	0.931052	−	0.364886i	$-0.118892\pi$
$227$	28.0554	1.86210	0.931052	+	0.364886i	$0.118892\pi$
$228$	0	0
$229$	9.82178	0.649041	0.324521	−	0.945879i	$-0.394797\pi$
$229$	9.82178	0.649041	0.324521	+	0.945879i	$0.394797\pi$
$230$	1.36294	0.0898698
$231$	0	0
$232$	1.10958	0.0728473
$233$	−3.37294	−0.220969	−0.110484	−	0.993878i	$-0.535240\pi$
$233$	−3.37294	−0.220969	−0.110484	+	0.993878i	$0.535240\pi$
$234$	0	0
$235$	15.7586	1.02798
$236$	−18.5078	−1.20476
$237$	0	0
$238$	0.144068	0.00933856
$239$	25.1604	1.62749	0.813746	−	0.581220i	$-0.197425\pi$
$239$	25.1604	1.62749	0.813746	+	0.581220i	$0.197425\pi$
$240$	0	0
$241$	12.9077	0.831460	0.415730	−	0.909488i	$-0.363526\pi$
$241$	12.9077	0.831460	0.415730	+	0.909488i	$0.363526\pi$
$242$	0	0
$243$	0	0
$244$	−5.73917	−0.367413
$245$	−1.30114	−0.0831270
$246$	0	0
$247$	−5.44803	−0.346650
$248$	1.38595	0.0880077
$249$	0	0
$250$	3.25494	0.205861
$251$	10.8704	0.686135	0.343068	−	0.939311i	$-0.388534\pi$
$251$	10.8704	0.686135	0.343068	+	0.939311i	$0.388534\pi$
$252$	0	0
$253$	0	0
$254$	−0.783719	−0.0491749
$255$	0	0
$256$	9.22811	0.576757
$257$	−10.1400	−0.632513	−0.316257	−	0.948674i	$-0.602426\pi$
$257$	−10.1400	−0.632513	−0.316257	+	0.948674i	$0.602426\pi$
$258$	0	0
$259$	3.55479	0.220884
$260$	−8.41607	−0.521943
$261$	0	0
$262$	3.09733	0.191354
$263$	20.7929	1.28214	0.641072	−	0.767480i	$-0.278490\pi$
$263$	20.7929	1.28214	0.641072	+	0.767480i	$0.278490\pi$
$264$	0	0
$265$	6.87358	0.422241
$266$	−0.484289	−0.0296937
$267$	0	0
$268$	−14.7911	−0.903512
$269$	1.24235	0.0757476	0.0378738	−	0.999283i	$-0.487942\pi$
$269$	1.24235	0.0757476	0.0378738	+	0.999283i	$0.487942\pi$
$270$	0	0
$271$	−22.3996	−1.36068	−0.680340	−	0.732897i	$-0.738168\pi$
$271$	−22.3996	−1.36068	−0.680340	+	0.732897i	$0.738168\pi$
$272$	−1.65724	−0.100485
$273$	0	0
$274$	5.99569	0.362213
$275$	0	0
$276$	0	0
$277$	12.7509	0.766130	0.383065	−	0.923721i	$-0.374869\pi$
$277$	12.7509	0.766130	0.383065	+	0.923721i	$0.374869\pi$
$278$	−1.39757	−0.0838207
$279$	0	0
$280$	−1.53179	−0.0915417
$281$	−1.56957	−0.0936325	−0.0468163	−	0.998904i	$-0.514908\pi$
$281$	−1.56957	−0.0936325	−0.0468163	+	0.998904i	$0.514908\pi$
$282$	0	0
$283$	14.3841	0.855045	0.427522	−	0.904005i	$-0.359387\pi$
$283$	14.3841	0.855045	0.427522	+	0.904005i	$0.359387\pi$
$284$	−16.1636	−0.959135
$285$	0	0
$286$	0	0
$287$	3.94839	0.233066
$288$	0	0
$289$	−16.7711	−0.986537
$290$	0.369302	0.0216862
$291$	0	0
$292$	25.5292	1.49398
$293$	17.3984	1.01642	0.508212	−	0.861232i	$-0.330307\pi$
$293$	17.3984	1.01642	0.508212	+	0.861232i	$0.330307\pi$
$294$	0	0
$295$	−12.6126	−0.734332
$296$	4.18492	0.243243
$297$	0	0
$298$	7.07103	0.409614
$299$	11.7839	0.681478
$300$	0	0
$301$	2.87024	0.165438
$302$	−4.03209	−0.232021
$303$	0	0
$304$	5.57086	0.319511
$305$	−3.91108	−0.223948
$306$	0	0
$307$	11.3404	0.647232	0.323616	−	0.946189i	$-0.395101\pi$
$307$	11.3404	0.647232	0.323616	+	0.946189i	$0.395101\pi$
$308$	0	0
$309$	0	0
$310$	0.461287	0.0261993
$311$	−13.7568	−0.780078	−0.390039	−	0.920798i	$-0.627538\pi$
$311$	−13.7568	−0.780078	−0.390039	+	0.920798i	$0.627538\pi$
$312$	0	0
$313$	6.41359	0.362518	0.181259	−	0.983435i	$-0.441983\pi$
$313$	6.41359	0.362518	0.181259	+	0.983435i	$0.441983\pi$
$314$	3.80686	0.214833
$315$	0	0
$316$	16.9282	0.952286
$317$	−10.6661	−0.599070	−0.299535	−	0.954085i	$-0.596832\pi$
$317$	−10.6661	−0.599070	−0.299535	+	0.954085i	$0.596832\pi$
$318$	0	0
$319$	0	0
$320$	7.68326	0.429507
$321$	0	0
$322$	1.04750	0.0583747
$323$	−0.769357	−0.0428081
$324$	0	0
$325$	11.2033	0.621446
$326$	−0.257283	−0.0142496
$327$	0	0
$328$	4.64829	0.256659
$329$	12.1113	0.667720
$330$	0	0
$331$	9.96427	0.547686	0.273843	−	0.961774i	$-0.411705\pi$
$331$	9.96427	0.547686	0.273843	+	0.961774i	$0.411705\pi$
$332$	−21.8998	−1.20191
$333$	0	0
$334$	2.45951	0.134579
$335$	−10.0797	−0.550715
$336$	0	0
$337$	19.0653	1.03855	0.519276	−	0.854607i	$-0.326202\pi$
$337$	19.0653	1.03855	0.519276	+	0.854607i	$0.326202\pi$
$338$	−0.458751	−0.0249528
$339$	0	0
$340$	−1.18850	−0.0644553
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	3.37902	0.182184
$345$	0	0
$346$	6.29723	0.338541
$347$	0.262065	0.0140684	0.00703420	−	0.999975i	$-0.497761\pi$
$347$	0.262065	0.0140684	0.00703420	+	0.999975i	$0.497761\pi$
$348$	0	0
$349$	4.96728	0.265892	0.132946	−	0.991123i	$-0.457556\pi$
$349$	4.96728	0.265892	0.132946	+	0.991123i	$0.457556\pi$
$350$	0.995888	0.0532324
$351$	0	0
$352$	0	0
$353$	19.8104	1.05440	0.527201	−	0.849741i	$-0.323242\pi$
$353$	19.8104	1.05440	0.527201	+	0.849741i	$0.323242\pi$
$354$	0	0
$355$	−11.0151	−0.584619
$356$	14.3975	0.763067
$357$	0	0
$358$	−2.53330	−0.133889
$359$	−19.8539	−1.04785	−0.523924	−	0.851765i	$-0.675532\pi$
$359$	−19.8539	−1.04785	−0.523924	+	0.851765i	$0.675532\pi$
$360$	0	0
$361$	−16.4138	−0.863884
$362$	−6.23352	−0.327626
$363$	0	0
$364$	−6.46821	−0.339027
$365$	17.3974	0.910623
$366$	0	0
$367$	30.9087	1.61342	0.806710	−	0.590947i	$-0.201246\pi$
$367$	30.9087	1.61342	0.806710	+	0.590947i	$0.201246\pi$
$368$	−12.0495	−0.628126
$369$	0	0
$370$	1.39287	0.0724120
$371$	5.28273	0.274266
$372$	0	0
$373$	37.1879	1.92552	0.962759	−	0.270361i	$-0.0871431\pi$
$373$	37.1879	1.92552	0.962759	+	0.270361i	$0.0871431\pi$
$374$	0	0
$375$	0	0
$376$	14.2582	0.735312
$377$	3.19295	0.164445
$378$	0	0
$379$	−19.9729	−1.02594	−0.512969	−	0.858407i	$-0.671454\pi$
$379$	−19.9729	−1.02594	−0.512969	+	0.858407i	$0.671454\pi$
$380$	3.99516	0.204947
$381$	0	0
$382$	4.42168	0.226233
$383$	−20.3331	−1.03897	−0.519486	−	0.854479i	$-0.673877\pi$
$383$	−20.3331	−1.03897	−0.519486	+	0.854479i	$0.673877\pi$
$384$	0	0
$385$	0	0
$386$	7.60859	0.387267
$387$	0	0
$388$	3.27187	0.166104
$389$	−10.7754	−0.546336	−0.273168	−	0.961966i	$-0.588072\pi$
$389$	−10.7754	−0.546336	−0.273168	+	0.961966i	$0.588072\pi$
$390$	0	0
$391$	1.66409	0.0841565
$392$	−1.17726	−0.0594607
$393$	0	0
$394$	−3.12996	−0.157685
$395$	11.5361	0.580444
$396$	0	0
$397$	−0.247762	−0.0124348	−0.00621740	−	0.999981i	$-0.501979\pi$
$397$	−0.247762	−0.0124348	−0.00621740	+	0.999981i	$0.501979\pi$
$398$	3.36389	0.168617
$399$	0	0
$400$	−11.4559	−0.572794
$401$	24.8441	1.24065	0.620327	−	0.784343i	$-0.287000\pi$
$401$	24.8441	1.24065	0.620327	+	0.784343i	$0.287000\pi$
$402$	0	0
$403$	3.98823	0.198668
$404$	18.2079	0.905878
$405$	0	0
$406$	0.283829	0.0140862
$407$	0	0
$408$	0	0
$409$	18.5332	0.916409	0.458204	−	0.888847i	$-0.348493\pi$
$409$	18.5332	0.916409	0.458204	+	0.888847i	$0.348493\pi$
$410$	1.54710	0.0764057
$411$	0	0
$412$	2.34146	0.115355
$413$	−9.69345	−0.476983
$414$	0	0
$415$	−14.9241	−0.732595
$416$	−11.5105	−0.564349
$417$	0	0
$418$	0	0
$419$	19.5238	0.953802	0.476901	−	0.878957i	$-0.341760\pi$
$419$	19.5238	0.953802	0.476901	+	0.878957i	$0.341760\pi$
$420$	0	0
$421$	−8.78687	−0.428246	−0.214123	−	0.976807i	$-0.568689\pi$
$421$	−8.78687	−0.428246	−0.214123	+	0.976807i	$0.568689\pi$
$422$	−3.37035	−0.164066
$423$	0	0
$424$	6.21915	0.302029
$425$	1.58210	0.0767430
$426$	0	0
$427$	−3.00588	−0.145465
$428$	−17.4950	−0.845651
$429$	0	0
$430$	1.12464	0.0542352
$431$	−18.1532	−0.874407	−0.437203	−	0.899363i	$-0.644031\pi$
$431$	−18.1532	−0.874407	−0.437203	+	0.899363i	$0.644031\pi$
$432$	0	0
$433$	21.3423	1.02564	0.512822	−	0.858495i	$-0.328600\pi$
$433$	21.3423	1.02564	0.512822	+	0.858495i	$0.328600\pi$
$434$	0.354524	0.0170177
$435$	0	0
$436$	30.9821	1.48377
$437$	−5.59387	−0.267591
$438$	0	0
$439$	−7.59086	−0.362292	−0.181146	−	0.983456i	$-0.557981\pi$
$439$	−7.59086	−0.362292	−0.181146	+	0.983456i	$0.557981\pi$
$440$	0	0
$441$	0	0
$442$	0.488063	0.0232148
$443$	37.8911	1.80026	0.900130	−	0.435621i	$-0.143471\pi$
$443$	37.8911	1.80026	0.900130	+	0.435621i	$0.143471\pi$
$444$	0	0
$445$	9.81150	0.465110
$446$	−6.15879	−0.291627
$447$	0	0
$448$	5.90501	0.278985
$449$	−15.2640	−0.720354	−0.360177	−	0.932884i	$-0.617284\pi$
$449$	−15.2640	−0.720354	−0.360177	+	0.932884i	$0.617284\pi$
$450$	0	0
$451$	0	0
$452$	33.3123	1.56688
$453$	0	0
$454$	8.44870	0.396517
$455$	−4.40791	−0.206646
$456$	0	0
$457$	−16.7762	−0.784757	−0.392379	−	0.919804i	$-0.628348\pi$
$457$	−16.7762	−0.784757	−0.392379	+	0.919804i	$0.628348\pi$
$458$	2.95776	0.138207
$459$	0	0
$460$	−8.64136	−0.402906
$461$	−36.0385	−1.67848	−0.839239	−	0.543763i	$-0.816999\pi$
$461$	−36.0385	−1.67848	−0.839239	+	0.543763i	$0.816999\pi$
$462$	0	0
$463$	−25.7197	−1.19530	−0.597648	−	0.801758i	$-0.703898\pi$
$463$	−25.7197	−1.19530	−0.597648	+	0.801758i	$0.703898\pi$
$464$	−3.26494	−0.151571
$465$	0	0
$466$	−1.01574	−0.0470531
$467$	−27.6780	−1.28079	−0.640394	−	0.768047i	$-0.721229\pi$
$467$	−27.6780	−1.28079	−0.640394	+	0.768047i	$0.721229\pi$
$468$	0	0
$469$	−7.74683	−0.357715
$470$	4.74559	0.218898
$471$	0	0
$472$	−11.4117	−0.525267
$473$	0	0
$474$	0	0
$475$	−5.31826	−0.244019
$476$	−0.913425	−0.0418668
$477$	0	0
$478$	7.57688	0.346559
$479$	13.8413	0.632427	0.316214	−	0.948688i	$-0.397588\pi$
$479$	13.8413	0.632427	0.316214	+	0.948688i	$0.397588\pi$
$480$	0	0
$481$	12.0426	0.549096
$482$	3.88707	0.177051
$483$	0	0
$484$	0	0
$485$	2.22968	0.101245
$486$	0	0
$487$	−10.9228	−0.494959	−0.247480	−	0.968893i	$-0.579602\pi$
$487$	−10.9228	−0.494959	−0.247480	+	0.968893i	$0.579602\pi$
$488$	−3.53871	−0.160190
$489$	0	0
$490$	−0.391830	−0.0177011
$491$	−26.6839	−1.20423	−0.602114	−	0.798410i	$-0.705675\pi$
$491$	−26.6839	−1.20423	−0.602114	+	0.798410i	$0.705675\pi$
$492$	0	0
$493$	0.450900	0.0203075
$494$	−1.64063	−0.0738156
$495$	0	0
$496$	−4.07816	−0.183115
$497$	−8.46568	−0.379738
$498$	0	0
$499$	−36.1374	−1.61773	−0.808867	−	0.587992i	$-0.799919\pi$
$499$	−36.1374	−1.61773	−0.808867	+	0.587992i	$0.799919\pi$
$500$	−20.6371	−0.922917
$501$	0	0
$502$	3.27355	0.146106
$503$	27.2718	1.21599	0.607994	−	0.793942i	$-0.291974\pi$
$503$	27.2718	1.21599	0.607994	+	0.793942i	$0.291974\pi$
$504$	0	0
$505$	12.4082	0.552157
$506$	0	0
$507$	0	0
$508$	4.96895	0.220462
$509$	−35.5186	−1.57434	−0.787168	−	0.616739i	$-0.788453\pi$
$509$	−35.5186	−1.57434	−0.787168	+	0.616739i	$0.788453\pi$
$510$	0	0
$511$	13.3709	0.591493
$512$	19.9263	0.880628
$513$	0	0
$514$	−3.05358	−0.134687
$515$	1.59564	0.0703122
$516$	0	0
$517$	0	0
$518$	1.07050	0.0470351
$519$	0	0
$520$	−5.18926	−0.227564
$521$	−19.1607	−0.839446	−0.419723	−	0.907652i	$-0.637873\pi$
$521$	−19.1607	−0.839446	−0.419723	+	0.907652i	$0.637873\pi$
$522$	0	0
$523$	−4.44268	−0.194265	−0.0971323	−	0.995271i	$-0.530967\pi$
$523$	−4.44268	−0.194265	−0.0971323	+	0.995271i	$0.530967\pi$
$524$	−19.6378	−0.857881
$525$	0	0
$526$	6.26163	0.273020
$527$	0.563208	0.0245337
$528$	0	0
$529$	−10.9007	−0.473943
$530$	2.06993	0.0899121
$531$	0	0
$532$	3.07050	0.133123
$533$	13.3760	0.579380
$534$	0	0
$535$	−11.9223	−0.515447
$536$	−9.12005	−0.393926
$537$	0	0
$538$	0.374126	0.0161297
$539$	0	0
$540$	0	0
$541$	23.7902	1.02282	0.511411	−	0.859336i	$-0.329123\pi$
$541$	23.7902	1.02282	0.511411	+	0.859336i	$0.329123\pi$
$542$	−6.74549	−0.289744
$543$	0	0
$544$	−1.62548	−0.0696920
$545$	21.1134	0.904399
$546$	0	0
$547$	22.0421	0.942452	0.471226	−	0.882013i	$-0.343812\pi$
$547$	22.0421	0.942452	0.471226	+	0.882013i	$0.343812\pi$
$548$	−38.0140	−1.62388
$549$	0	0
$550$	0	0
$551$	−1.51571	−0.0645715
$552$	0	0
$553$	8.86612	0.377026
$554$	3.83986	0.163140
$555$	0	0
$556$	8.86091	0.375786
$557$	29.7068	1.25872	0.629359	−	0.777115i	$-0.283318\pi$
$557$	29.7068	1.25872	0.629359	+	0.777115i	$0.283318\pi$
$558$	0	0
$559$	9.72355	0.411262
$560$	4.50729	0.190468
$561$	0	0
$562$	−0.472664	−0.0199381
$563$	7.06070	0.297573	0.148786	−	0.988869i	$-0.452463\pi$
$563$	7.06070	0.297573	0.148786	+	0.988869i	$0.452463\pi$
$564$	0	0
$565$	22.7014	0.955054
$566$	4.33166	0.182073
$567$	0	0
$568$	−9.96632	−0.418178
$569$	22.6262	0.948538	0.474269	−	0.880380i	$-0.342712\pi$
$569$	22.6262	0.948538	0.474269	+	0.880380i	$0.342712\pi$
$570$	0	0
$571$	18.4644	0.772713	0.386356	−	0.922350i	$-0.373733\pi$
$571$	18.4644	0.772713	0.386356	+	0.922350i	$0.373733\pi$
$572$	0	0
$573$	0	0
$574$	1.18903	0.0496292
$575$	11.5032	0.479716
$576$	0	0
$577$	2.19908	0.0915488	0.0457744	−	0.998952i	$-0.485424\pi$
$577$	2.19908	0.0915488	0.0457744	+	0.998952i	$0.485424\pi$
$578$	−5.05051	−0.210073
$579$	0	0
$580$	−2.34146	−0.0972238
$581$	−11.4700	−0.475855
$582$	0	0
$583$	0	0
$584$	15.7410	0.651368
$585$	0	0
$586$	5.23940	0.216438
$587$	19.1979	0.792383	0.396192	−	0.918168i	$-0.370332\pi$
$587$	19.1979	0.792383	0.396192	+	0.918168i	$0.370332\pi$
$588$	0	0
$589$	−1.89324	−0.0780095
$590$	−3.79818	−0.156369
$591$	0	0
$592$	−12.3141	−0.506108
$593$	−19.1328	−0.785691	−0.392846	−	0.919604i	$-0.628509\pi$
$593$	−19.1328	−0.785691	−0.392846	+	0.919604i	$0.628509\pi$
$594$	0	0
$595$	−0.622473	−0.0255189
$596$	−44.8319	−1.83639
$597$	0	0
$598$	3.54863	0.145114
$599$	4.42406	0.180762	0.0903812	−	0.995907i	$-0.471191\pi$
$599$	4.42406	0.180762	0.0903812	+	0.995907i	$0.471191\pi$
$600$	0	0
$601$	39.9277	1.62869	0.814343	−	0.580384i	$-0.197098\pi$
$601$	39.9277	1.62869	0.814343	+	0.580384i	$0.197098\pi$
$602$	0.864351	0.0352283
$603$	0	0
$604$	25.5644	1.04020
$605$	0	0
$606$	0	0
$607$	22.7804	0.924626	0.462313	−	0.886717i	$-0.347020\pi$
$607$	22.7804	0.924626	0.462313	+	0.886717i	$0.347020\pi$
$608$	5.46410	0.221599
$609$	0	0
$610$	−1.17780	−0.0476876
$611$	41.0298	1.65989
$612$	0	0
$613$	−40.4017	−1.63181	−0.815904	−	0.578187i	$-0.803760\pi$
$613$	−40.4017	−1.63181	−0.815904	+	0.578187i	$0.803760\pi$
$614$	3.41509	0.137822
$615$	0	0
$616$	0	0
$617$	−21.2137	−0.854033	−0.427017	−	0.904244i	$-0.640435\pi$
$617$	−21.2137	−0.854033	−0.427017	+	0.904244i	$0.640435\pi$
$618$	0	0
$619$	13.0445	0.524302	0.262151	−	0.965027i	$-0.415568\pi$
$619$	13.0445	0.524302	0.262151	+	0.965027i	$0.415568\pi$
$620$	−2.92466	−0.117457
$621$	0	0
$622$	−4.14277	−0.166110
$623$	7.54068	0.302111
$624$	0	0
$625$	2.47156	0.0988625
$626$	1.93141	0.0771946
$627$	0	0
$628$	−24.1363	−0.963144
$629$	1.70063	0.0678085
$630$	0	0
$631$	37.0202	1.47375	0.736875	−	0.676029i	$-0.236301\pi$
$631$	37.0202	1.47375	0.736875	+	0.676029i	$0.236301\pi$
$632$	10.4378	0.415191
$633$	0	0
$634$	−3.21203	−0.127566
$635$	3.38620	0.134377
$636$	0	0
$637$	−3.38772	−0.134226
$638$	0	0
$639$	0	0
$640$	11.1556	0.440963
$641$	−21.3679	−0.843980	−0.421990	−	0.906600i	$-0.638668\pi$
$641$	−21.3679	−0.843980	−0.421990	+	0.906600i	$0.638668\pi$
$642$	0	0
$643$	18.1845	0.717128	0.358564	−	0.933505i	$-0.383267\pi$
$643$	18.1845	0.717128	0.358564	+	0.933505i	$0.383267\pi$
$644$	−6.64136	−0.261706
$645$	0	0
$646$	−0.231686	−0.00911558
$647$	9.43913	0.371091	0.185545	−	0.982636i	$-0.440595\pi$
$647$	9.43913	0.371091	0.185545	+	0.982636i	$0.440595\pi$
$648$	0	0
$649$	0	0
$650$	3.37379	0.132331
$651$	0	0
$652$	1.63123	0.0638839
$653$	21.9975	0.860827	0.430414	−	0.902632i	$-0.358368\pi$
$653$	21.9975	0.860827	0.430414	+	0.902632i	$0.358368\pi$
$654$	0	0
$655$	−13.3826	−0.522901
$656$	−13.6776	−0.534022
$657$	0	0
$658$	3.64725	0.142185
$659$	46.9739	1.82984	0.914922	−	0.403631i	$-0.132252\pi$
$659$	46.9739	1.82984	0.914922	+	0.403631i	$0.132252\pi$
$660$	0	0
$661$	−11.4368	−0.444840	−0.222420	−	0.974951i	$-0.571396\pi$
$661$	−11.4368	−0.444840	−0.222420	+	0.974951i	$0.571396\pi$
$662$	3.00067	0.116624
$663$	0	0
$664$	−13.5032	−0.524025
$665$	2.09246	0.0811421
$666$	0	0
$667$	3.27842	0.126941
$668$	−15.5939	−0.603345
$669$	0	0
$670$	−3.03544	−0.117269
$671$	0	0
$672$	0	0
$673$	4.68715	0.180676	0.0903381	−	0.995911i	$-0.471205\pi$
$673$	4.68715	0.180676	0.0903381	+	0.995911i	$0.471205\pi$
$674$	5.74138	0.221150
$675$	0	0
$676$	2.90858	0.111869
$677$	−8.71493	−0.334942	−0.167471	−	0.985877i	$-0.553560\pi$
$677$	−8.71493	−0.334942	−0.167471	+	0.985877i	$0.553560\pi$
$678$	0	0
$679$	1.71364	0.0657633
$680$	−0.732814	−0.0281021
$681$	0	0
$682$	0	0
$683$	39.2999	1.50377	0.751884	−	0.659295i	$-0.229145\pi$
$683$	39.2999	1.50377	0.751884	+	0.659295i	$0.229145\pi$
$684$	0	0
$685$	−25.9055	−0.989798
$686$	−0.301143	−0.0114977
$687$	0	0
$688$	−9.94279	−0.379065
$689$	17.8964	0.681798
$690$	0	0
$691$	−21.4190	−0.814819	−0.407409	−	0.913246i	$-0.633568\pi$
$691$	−21.4190	−0.814819	−0.407409	+	0.913246i	$0.633568\pi$
$692$	−39.9258	−1.51775
$693$	0	0
$694$	0.0789191	0.00299573
$695$	6.03846	0.229052
$696$	0	0
$697$	1.88893	0.0715483
$698$	1.49586	0.0566192
$699$	0	0
$700$	−6.31415	−0.238652
$701$	24.6197	0.929875	0.464937	−	0.885344i	$-0.346077\pi$
$701$	24.6197	0.929875	0.464937	+	0.885344i	$0.346077\pi$
$702$	0	0
$703$	−5.71670	−0.215610
$704$	0	0
$705$	0	0
$706$	5.96576	0.224524
$707$	9.53637	0.358652
$708$	0	0
$709$	−10.9882	−0.412672	−0.206336	−	0.978481i	$-0.566154\pi$
$709$	−10.9882	−0.412672	−0.206336	+	0.978481i	$0.566154\pi$
$710$	−3.31711	−0.124489
$711$	0	0
$712$	8.87736	0.332693
$713$	4.09499	0.153359
$714$	0	0
$715$	0	0
$716$	16.0617	0.600254
$717$	0	0
$718$	−5.97885	−0.223129
$719$	−33.4441	−1.24725	−0.623627	−	0.781722i	$-0.714342\pi$
$719$	−33.4441	−1.24725	−0.623627	+	0.781722i	$0.714342\pi$
$720$	0	0
$721$	1.22634	0.0456711
$722$	−4.94290	−0.183956
$723$	0	0
$724$	39.5219	1.46882
$725$	3.11690	0.115759
$726$	0	0
$727$	44.9100	1.66562	0.832810	−	0.553560i	$-0.186731\pi$
$727$	44.9100	1.66562	0.832810	+	0.553560i	$0.186731\pi$
$728$	−3.98823	−0.147814
$729$	0	0
$730$	5.23911	0.193908
$731$	1.37314	0.0507872
$732$	0	0
$733$	−23.4850	−0.867437	−0.433718	−	0.901048i	$-0.642799\pi$
$733$	−23.4850	−0.867437	−0.433718	+	0.901048i	$0.642799\pi$
$734$	9.30793	0.343562
$735$	0	0
$736$	−11.8186	−0.435640
$737$	0	0
$738$	0	0
$739$	−17.2692	−0.635259	−0.317630	−	0.948215i	$-0.602887\pi$
$739$	−17.2692	−0.635259	−0.317630	+	0.948215i	$0.602887\pi$
$740$	−8.83112	−0.324639
$741$	0	0
$742$	1.59086	0.0584022
$743$	−7.15344	−0.262434	−0.131217	−	0.991354i	$-0.541888\pi$
$743$	−7.15344	−0.262434	−0.131217	+	0.991354i	$0.541888\pi$
$744$	0	0
$745$	−30.5517	−1.11933
$746$	11.1989	0.410020
$747$	0	0
$748$	0	0
$749$	−9.16296	−0.334807
$750$	0	0
$751$	−52.3091	−1.90879	−0.954394	−	0.298551i	$-0.903497\pi$
$751$	−52.3091	−1.90879	−0.954394	+	0.298551i	$0.903497\pi$
$752$	−41.9549	−1.52994
$753$	0	0
$754$	0.961534	0.0350170
$755$	17.4214	0.634029
$756$	0	0
$757$	−11.0149	−0.400344	−0.200172	−	0.979761i	$-0.564150\pi$
$757$	−11.0149	−0.400344	−0.200172	+	0.979761i	$0.564150\pi$
$758$	−6.01469	−0.218464
$759$	0	0
$760$	2.46337	0.0893559
$761$	−11.2716	−0.408596	−0.204298	−	0.978909i	$-0.565491\pi$
$761$	−11.2716	−0.408596	−0.204298	+	0.978909i	$0.565491\pi$
$762$	0	0
$763$	16.2268	0.587450
$764$	−28.0344	−1.01425
$765$	0	0
$766$	−6.12317	−0.221239
$767$	−32.8387	−1.18574
$768$	0	0
$769$	1.37733	0.0496678	0.0248339	−	0.999692i	$-0.492094\pi$
$769$	1.37733	0.0496678	0.0248339	+	0.999692i	$0.492094\pi$
$770$	0	0
$771$	0	0
$772$	−48.2401	−1.73620
$773$	−1.30100	−0.0467937	−0.0233969	−	0.999726i	$-0.507448\pi$
$773$	−1.30100	−0.0467937	−0.0233969	+	0.999726i	$0.507448\pi$
$774$	0	0
$775$	3.89324	0.139849
$776$	2.01740	0.0724204
$777$	0	0
$778$	−3.24495	−0.116337
$779$	−6.34968	−0.227501
$780$	0	0
$781$	0	0
$782$	0.501128	0.0179203
$783$	0	0
$784$	3.46410	0.123718
$785$	−16.4482	−0.587062
$786$	0	0
$787$	50.2646	1.79174	0.895871	−	0.444315i	$-0.146553\pi$
$787$	50.2646	1.79174	0.895871	+	0.444315i	$0.146553\pi$
$788$	19.8446	0.706936
$789$	0	0
$790$	3.47401	0.123600
$791$	17.4473	0.620353
$792$	0	0
$793$	−10.1831	−0.361612
$794$	−0.0746117	−0.00264787
$795$	0	0
$796$	−21.3278	−0.755945
$797$	7.51222	0.266096	0.133048	−	0.991110i	$-0.457523\pi$
$797$	7.51222	0.266096	0.133048	+	0.991110i	$0.457523\pi$
$798$	0	0
$799$	5.79413	0.204982
$800$	−11.2363	−0.397264
$801$	0	0
$802$	7.48162	0.264185
$803$	0	0
$804$	0	0
$805$	−4.52590	−0.159517
$806$	1.20103	0.0423044
$807$	0	0
$808$	11.2268	0.394958
$809$	6.79104	0.238760	0.119380	−	0.992849i	$-0.461909\pi$
$809$	6.79104	0.238760	0.119380	+	0.992849i	$0.461909\pi$
$810$	0	0
$811$	−32.0227	−1.12447	−0.562235	−	0.826978i	$-0.690058\pi$
$811$	−32.0227	−1.12447	−0.562235	+	0.826978i	$0.690058\pi$
$812$	−1.79954	−0.0631515
$813$	0	0
$814$	0	0
$815$	1.11164	0.0389390
$816$	0	0
$817$	−4.61583	−0.161487
$818$	5.58115	0.195140
$819$	0	0
$820$	−9.80894	−0.342543
$821$	−35.3145	−1.23249	−0.616243	−	0.787556i	$-0.711346\pi$
$821$	−35.3145	−1.23249	−0.616243	+	0.787556i	$0.711346\pi$
$822$	0	0
$823$	−2.00171	−0.0697753	−0.0348877	−	0.999391i	$-0.511107\pi$
$823$	−2.00171	−0.0697753	−0.0348877	+	0.999391i	$0.511107\pi$
$824$	1.44372	0.0502943
$825$	0	0
$826$	−2.91911	−0.101569
$827$	8.42435	0.292943	0.146472	−	0.989215i	$-0.453208\pi$
$827$	8.42435	0.292943	0.146472	+	0.989215i	$0.453208\pi$
$828$	0	0
$829$	14.1931	0.492945	0.246473	−	0.969150i	$-0.420728\pi$
$829$	14.1931	0.492945	0.246473	+	0.969150i	$0.420728\pi$
$830$	−4.49429	−0.155999
$831$	0	0
$832$	20.0045	0.693531
$833$	−0.478405	−0.0165758
$834$	0	0
$835$	−10.6268	−0.367755
$836$	0	0
$837$	0	0
$838$	5.87947	0.203103
$839$	−8.44234	−0.291462	−0.145731	−	0.989324i	$-0.546553\pi$
$839$	−8.44234	−0.291462	−0.145731	+	0.989324i	$0.546553\pi$
$840$	0	0
$841$	−28.1117	−0.969368
$842$	−2.64610	−0.0911907
$843$	0	0
$844$	21.3687	0.735542
$845$	1.98212	0.0681869
$846$	0	0
$847$	0	0
$848$	−18.2999	−0.628421
$849$	0	0
$850$	0.476438	0.0163417
$851$	12.3650	0.423867
$852$	0	0
$853$	−57.3123	−1.96234	−0.981168	−	0.193158i	$-0.938127\pi$
$853$	−57.3123	−1.96234	−0.981168	+	0.193158i	$0.938127\pi$
$854$	−0.905201	−0.0309753
$855$	0	0
$856$	−10.7872	−0.368699
$857$	13.5681	0.463479	0.231739	−	0.972778i	$-0.425558\pi$
$857$	13.5681	0.463479	0.231739	+	0.972778i	$0.425558\pi$
$858$	0	0
$859$	−15.4832	−0.528280	−0.264140	−	0.964484i	$-0.585088\pi$
$859$	−15.4832	−0.528280	−0.264140	+	0.964484i	$0.585088\pi$
$860$	−7.13049	−0.243148
$861$	0	0
$862$	−5.46670	−0.186196
$863$	21.5553	0.733752	0.366876	−	0.930270i	$-0.380427\pi$
$863$	21.5553	0.733752	0.366876	+	0.930270i	$0.380427\pi$
$864$	0	0
$865$	−27.2083	−0.925110
$866$	6.42708	0.218401
$867$	0	0
$868$	−2.24776	−0.0762940
$869$	0	0
$870$	0	0
$871$	−26.2441	−0.889246
$872$	19.1032	0.646916
$873$	0	0
$874$	−1.68455	−0.0569809
$875$	−10.8086	−0.365398
$876$	0	0
$877$	18.6275	0.629007	0.314504	−	0.949256i	$-0.398162\pi$
$877$	18.6275	0.629007	0.314504	+	0.949256i	$0.398162\pi$
$878$	−2.28593	−0.0771465
$879$	0	0
$880$	0	0
$881$	−13.7010	−0.461600	−0.230800	−	0.973001i	$-0.574134\pi$
$881$	−13.7010	−0.461600	−0.230800	+	0.973001i	$0.574134\pi$
$882$	0	0
$883$	−29.4327	−0.990489	−0.495244	−	0.868754i	$-0.664921\pi$
$883$	−29.4327	−0.990489	−0.495244	+	0.868754i	$0.664921\pi$
$884$	−3.09443	−0.104077
$885$	0	0
$886$	11.4106	0.383348
$887$	14.4530	0.485283	0.242641	−	0.970116i	$-0.421986\pi$
$887$	14.4530	0.485283	0.242641	+	0.970116i	$0.421986\pi$
$888$	0	0
$889$	2.60248	0.0872844
$890$	2.95467	0.0990406
$891$	0	0
$892$	39.0481	1.30743
$893$	−19.4771	−0.651777
$894$	0	0
$895$	10.9456	0.365871
$896$	8.57368	0.286427
$897$	0	0
$898$	−4.59666	−0.153392
$899$	1.10958	0.0370065
$900$	0	0
$901$	2.52728	0.0841960
$902$	0	0
$903$	0	0
$904$	20.5400	0.683150
$905$	26.9330	0.895285
$906$	0	0
$907$	33.5011	1.11239	0.556193	−	0.831053i	$-0.312261\pi$
$907$	33.5011	1.11239	0.556193	+	0.831053i	$0.312261\pi$
$908$	−53.5666	−1.77767
$909$	0	0
$910$	−1.32741	−0.0440032
$911$	21.2100	0.702718	0.351359	−	0.936241i	$-0.385720\pi$
$911$	21.2100	0.702718	0.351359	+	0.936241i	$0.385720\pi$
$912$	0	0
$913$	0	0
$914$	−5.05203	−0.167106
$915$	0	0
$916$	−18.7528	−0.619612
$917$	−10.2853	−0.339649
$918$	0	0
$919$	−46.8324	−1.54486	−0.772430	−	0.635100i	$-0.780959\pi$
$919$	−46.8324	−1.54486	−0.772430	+	0.635100i	$0.780959\pi$
$920$	−5.32817	−0.175665
$921$	0	0
$922$	−10.8527	−0.357415
$923$	−28.6793	−0.943992
$924$	0	0
$925$	11.7558	0.386528
$926$	−7.74531	−0.254527
$927$	0	0
$928$	−3.20237	−0.105123
$929$	10.9210	0.358307	0.179154	−	0.983821i	$-0.442664\pi$
$929$	10.9210	0.358307	0.179154	+	0.983821i	$0.442664\pi$
$930$	0	0
$931$	1.60817	0.0527056
$932$	6.44000	0.210949
$933$	0	0
$934$	−8.33505	−0.272731
$935$	0	0
$936$	0	0
$937$	−7.85086	−0.256476	−0.128238	−	0.991743i	$-0.540932\pi$
$937$	−7.85086	−0.256476	−0.128238	+	0.991743i	$0.540932\pi$
$938$	−2.33290	−0.0761720
$939$	0	0
$940$	−30.0881	−0.981365
$941$	−27.6122	−0.900132	−0.450066	−	0.892995i	$-0.648599\pi$
$941$	−27.6122	−0.900132	−0.450066	+	0.892995i	$0.648599\pi$
$942$	0	0
$943$	13.7341	0.447244
$944$	33.5791	1.09291
$945$	0	0
$946$	0	0
$947$	−56.9267	−1.84987	−0.924934	−	0.380127i	$-0.875880\pi$
$947$	−56.9267	−1.84987	−0.924934	+	0.380127i	$0.875880\pi$
$948$	0	0
$949$	45.2968	1.47040
$950$	−1.60156	−0.0519614
$951$	0	0
$952$	−0.563208	−0.0182537
$953$	−25.2881	−0.819163	−0.409581	−	0.912274i	$-0.634325\pi$
$953$	−25.2881	−0.819163	−0.409581	+	0.912274i	$0.634325\pi$
$954$	0	0
$955$	−19.1047	−0.618212
$956$	−48.0391	−1.55370
$957$	0	0
$958$	4.16822	0.134669
$959$	−19.9098	−0.642921
$960$	0	0
$961$	−29.6141	−0.955292
$962$	3.62655	0.116925
$963$	0	0
$964$	−24.6449	−0.793759
$965$	−32.8743	−1.05826
$966$	0	0
$967$	19.2122	0.617824	0.308912	−	0.951091i	$-0.400035\pi$
$967$	19.2122	0.617824	0.308912	+	0.951091i	$0.400035\pi$
$968$	0	0
$969$	0	0
$970$	0.671454	0.0215591
$971$	−50.9532	−1.63517	−0.817583	−	0.575811i	$-0.804686\pi$
$971$	−50.9532	−1.63517	−0.817583	+	0.575811i	$0.804686\pi$
$972$	0	0
$973$	4.64089	0.148780
$974$	−3.28932	−0.105397
$975$	0	0
$976$	10.4127	0.333302
$977$	54.7278	1.75090	0.875449	−	0.483310i	$-0.160565\pi$
$977$	54.7278	1.75090	0.875449	+	0.483310i	$0.160565\pi$
$978$	0	0
$979$	0	0
$980$	2.48429	0.0793577
$981$	0	0
$982$	−8.03568	−0.256429
$983$	−6.20739	−0.197985	−0.0989924	−	0.995088i	$-0.531562\pi$
$983$	−6.20739	−0.197985	−0.0989924	+	0.995088i	$0.531562\pi$
$984$	0	0
$985$	13.5236	0.430897
$986$	0.135785	0.00432429
$987$	0	0
$988$	10.4020	0.330931
$989$	9.98384	0.317468
$990$	0	0
$991$	−4.64753	−0.147634	−0.0738168	−	0.997272i	$-0.523518\pi$
$991$	−4.64753	−0.147634	−0.0738168	+	0.997272i	$0.523518\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	−2.54938	−0.0808614
$995$	−14.5343	−0.460769
$996$	0	0
$997$	34.0235	1.07754	0.538768	−	0.842454i	$-0.318890\pi$
$997$	34.0235	1.07754	0.538768	+	0.842454i	$0.318890\pi$
$998$	−10.8825	−0.344480
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cm.1.2		4
3.2	odd	2		2541.2.a.bk.1.3	✓	4
11.10	odd	2		7623.2.a.cf.1.3		4
33.32	even	2		2541.2.a.bo.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.bk.1.3	✓	4	3.2	odd	2
2541.2.a.bo.1.2	yes	4	33.32	even	2
7623.2.a.cf.1.3		4	11.10	odd	2
7623.2.a.cm.1.2		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+0.301143 q^{2} -1.90931 q^{4} -1.30114 q^{5} -1.00000 q^{7} -1.17726 q^{8} +O(q^{10})\)	\(q+0.301143 q^{2} -1.90931 q^{4} -1.30114 q^{5} -1.00000 q^{7} -1.17726 q^{8} -0.391830 q^{10} -3.38772 q^{13} -0.301143 q^{14} +3.46410 q^{16} -0.478405 q^{17} +1.60817 q^{19} +2.48429 q^{20} -3.47841 q^{23} -3.30703 q^{25} -1.02019 q^{26} +1.90931 q^{28} -0.942507 q^{29} -1.17726 q^{31} +3.39771 q^{32} -0.144068 q^{34} +1.30114 q^{35} -3.55479 q^{37} +0.484289 q^{38} +1.53179 q^{40} -3.94839 q^{41} -2.87024 q^{43} -1.04750 q^{46} -12.1113 q^{47} +1.00000 q^{49} -0.995888 q^{50} +6.46821 q^{52} -5.28273 q^{53} +1.17726 q^{56} -0.283829 q^{58} +9.69345 q^{59} +3.00588 q^{61} -0.354524 q^{62} -5.90501 q^{64} +4.40791 q^{65} +7.74683 q^{67} +0.913425 q^{68} +0.391830 q^{70} +8.46568 q^{71} -13.3709 q^{73} -1.07050 q^{74} -3.07050 q^{76} -8.86612 q^{79} -4.50729 q^{80} -1.18903 q^{82} +11.4700 q^{83} +0.622473 q^{85} -0.864351 q^{86} -7.54068 q^{89} +3.38772 q^{91} +6.64136 q^{92} -3.64725 q^{94} -2.09246 q^{95} -1.71364 q^{97} +0.301143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 4 * q^7	\( 4 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 4 q^{7} - 14 q^{10} + 2 q^{13} - 2 q^{14} + 2 q^{17} - 6 q^{19} - 8 q^{20} - 10 q^{23} - 4 q^{28} + 14 q^{29} + 12 q^{32} - 2 q^{34} + 6 q^{35} - 12 q^{37} - 16 q^{38} - 8 q^{40} + 16 q^{41} - 20 q^{43} - 8 q^{46} - 2 q^{47} + 4 q^{49} + 24 q^{50} + 40 q^{52} + 16 q^{53} - 8 q^{58} - 2 q^{59} - 2 q^{61} + 8 q^{62} - 16 q^{64} - 2 q^{65} - 20 q^{67} + 20 q^{68} + 14 q^{70} + 10 q^{71} + 10 q^{73} - 20 q^{74} - 28 q^{76} - 16 q^{79} - 12 q^{80} + 28 q^{82} + 18 q^{83} - 26 q^{86} + 14 q^{89} - 2 q^{91} + 8 q^{92} + 18 q^{94} + 22 q^{95} + 38 q^{97} + 2 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 4 * q^7 - 14 * q^10 + 2 * q^13 - 2 * q^14 + 2 * q^17 - 6 * q^19 - 8 * q^20 - 10 * q^23 - 4 * q^28 + 14 * q^29 + 12 * q^32 - 2 * q^34 + 6 * q^35 - 12 * q^37 - 16 * q^38 - 8 * q^40 + 16 * q^41 - 20 * q^43 - 8 * q^46 - 2 * q^47 + 4 * q^49 + 24 * q^50 + 40 * q^52 + 16 * q^53 - 8 * q^58 - 2 * q^59 - 2 * q^61 + 8 * q^62 - 16 * q^64 - 2 * q^65 - 20 * q^67 + 20 * q^68 + 14 * q^70 + 10 * q^71 + 10 * q^73 - 20 * q^74 - 28 * q^76 - 16 * q^79 - 12 * q^80 + 28 * q^82 + 18 * q^83 - 26 * q^86 + 14 * q^89 - 2 * q^91 + 8 * q^92 + 18 * q^94 + 22 * q^95 + 38 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

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Database

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