LMFDB - Embedded newform 2576.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2576,2,Mod(1,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2576.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:	$20.5694635607$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	2576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.48119 q^{3} +2.48119 q^{5} +1.00000 q^{7} +3.15633 q^{9} +O(q^{10})$	$q-2.48119 q^{3} +2.48119 q^{5} +1.00000 q^{7} +3.15633 q^{9} -1.19394 q^{11} -2.96239 q^{13} -6.15633 q^{15} -0.481194 q^{17} +4.31265 q^{19} -2.48119 q^{21} -1.00000 q^{23} +1.15633 q^{25} -0.387873 q^{27} -4.15633 q^{29} -10.2193 q^{31} +2.96239 q^{33} +2.48119 q^{35} -5.35026 q^{37} +7.35026 q^{39} -11.9248 q^{41} +5.73813 q^{43} +7.83146 q^{45} -0.168544 q^{47} +1.00000 q^{49} +1.19394 q^{51} +8.70052 q^{53} -2.96239 q^{55} -10.7005 q^{57} +11.5696 q^{59} +9.05571 q^{61} +3.15633 q^{63} -7.35026 q^{65} -11.8945 q^{67} +2.48119 q^{69} -0.775746 q^{71} +8.88717 q^{73} -2.86907 q^{75} -1.19394 q^{77} -12.7308 q^{79} -8.50659 q^{81} -16.4387 q^{83} -1.19394 q^{85} +10.3127 q^{87} +13.7562 q^{89} -2.96239 q^{91} +25.3561 q^{93} +10.7005 q^{95} -0.0933212 q^{97} -3.76845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 2 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + 2 * q^5 + 3 * q^7 - q^9	$ 3 q - 2 q^{3} + 2 q^{5} + 3 q^{7} - q^{9} - 4 q^{11} + 2 q^{13} - 8 q^{15} + 4 q^{17} - 8 q^{19} - 2 q^{21} - 3 q^{23} - 7 q^{25} - 2 q^{27} - 2 q^{29} - 16 q^{31} - 2 q^{33} + 2 q^{35} - 6 q^{37} + 12 q^{39} - 14 q^{41} + 8 q^{43} + 8 q^{45} - 16 q^{47} + 3 q^{49} + 4 q^{51} + 6 q^{53} + 2 q^{55} - 12 q^{57} + 10 q^{59} + 10 q^{61} - q^{63} - 12 q^{65} - 16 q^{67} + 2 q^{69} - 4 q^{71} - 6 q^{73} - 4 q^{75} - 4 q^{77} - 16 q^{79} - 5 q^{81} - 20 q^{83} - 4 q^{85} + 10 q^{87} + 4 q^{89} + 2 q^{91} + 12 q^{93} + 12 q^{95} + 6 q^{97}+O(q^{100}) $ 3 * q - 2 * q^3 + 2 * q^5 + 3 * q^7 - q^9 - 4 * q^11 + 2 * q^13 - 8 * q^15 + 4 * q^17 - 8 * q^19 - 2 * q^21 - 3 * q^23 - 7 * q^25 - 2 * q^27 - 2 * q^29 - 16 * q^31 - 2 * q^33 + 2 * q^35 - 6 * q^37 + 12 * q^39 - 14 * q^41 + 8 * q^43 + 8 * q^45 - 16 * q^47 + 3 * q^49 + 4 * q^51 + 6 * q^53 + 2 * q^55 - 12 * q^57 + 10 * q^59 + 10 * q^61 - q^63 - 12 * q^65 - 16 * q^67 + 2 * q^69 - 4 * q^71 - 6 * q^73 - 4 * q^75 - 4 * q^77 - 16 * q^79 - 5 * q^81 - 20 * q^83 - 4 * q^85 + 10 * q^87 + 4 * q^89 + 2 * q^91 + 12 * q^93 + 12 * q^95 + 6 * q^97

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	0
$2$	0	0
$3$	−2.48119	−1.43252	−0.716259	−	0.697834i	$-0.754147\pi$
$3$	−2.48119	−1.43252	−0.716259	+	0.697834i	$0.754147\pi$
$4$	0	0
$5$	2.48119	1.10962	0.554812	−	0.831976i	$-0.312790\pi$
$5$	2.48119	1.10962	0.554812	+	0.831976i	$0.312790\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.15633	1.05211
$10$	0	0
$11$	−1.19394	−0.359985	−0.179993	−	0.983668i	$-0.557607\pi$
$11$	−1.19394	−0.359985	−0.179993	+	0.983668i	$0.557607\pi$
$12$	0	0
$13$	−2.96239	−0.821619	−0.410809	−	0.911721i	$-0.634754\pi$
$13$	−2.96239	−0.821619	−0.410809	+	0.911721i	$0.634754\pi$
$14$	0	0
$15$	−6.15633	−1.58956
$16$	0	0
$17$	−0.481194	−0.116707	−0.0583534	−	0.998296i	$-0.518585\pi$
$17$	−0.481194	−0.116707	−0.0583534	+	0.998296i	$0.518585\pi$
$18$	0	0
$19$	4.31265	0.989390	0.494695	−	0.869067i	$-0.335280\pi$
$19$	4.31265	0.989390	0.494695	+	0.869067i	$0.335280\pi$
$20$	0	0
$21$	−2.48119	−0.541441
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.15633	0.231265
$26$	0	0
$27$	−0.387873	−0.0746462
$28$	0	0
$29$	−4.15633	−0.771810	−0.385905	−	0.922538i	$-0.626111\pi$
$29$	−4.15633	−0.771810	−0.385905	+	0.922538i	$0.626111\pi$
$30$	0	0
$31$	−10.2193	−1.83545	−0.917723	−	0.397221i	$-0.869975\pi$
$31$	−10.2193	−1.83545	−0.917723	+	0.397221i	$0.869975\pi$
$32$	0	0
$33$	2.96239	0.515686
$34$	0	0
$35$	2.48119	0.419398
$36$	0	0
$37$	−5.35026	−0.879578	−0.439789	−	0.898101i	$-0.644947\pi$
$37$	−5.35026	−0.879578	−0.439789	+	0.898101i	$0.644947\pi$
$38$	0	0
$39$	7.35026	1.17698
$40$	0	0
$41$	−11.9248	−1.86234	−0.931169	−	0.364589i	$-0.881210\pi$
$41$	−11.9248	−1.86234	−0.931169	+	0.364589i	$0.881210\pi$
$42$	0	0
$43$	5.73813	0.875057	0.437529	−	0.899204i	$-0.355854\pi$
$43$	5.73813	0.875057	0.437529	+	0.899204i	$0.355854\pi$
$44$	0	0
$45$	7.83146	1.16744
$46$	0	0
$47$	−0.168544	−0.0245847	−0.0122923	−	0.999924i	$-0.503913\pi$
$47$	−0.168544	−0.0245847	−0.0122923	+	0.999924i	$0.503913\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.19394	0.167185
$52$	0	0
$53$	8.70052	1.19511	0.597554	−	0.801828i	$-0.296139\pi$
$53$	8.70052	1.19511	0.597554	+	0.801828i	$0.296139\pi$
$54$	0	0
$55$	−2.96239	−0.399448
$56$	0	0
$57$	−10.7005	−1.41732
$58$	0	0
$59$	11.5696	1.50623	0.753116	−	0.657887i	$-0.228550\pi$
$59$	11.5696	1.50623	0.753116	+	0.657887i	$0.228550\pi$
$60$	0	0
$61$	9.05571	1.15946	0.579732	−	0.814807i	$-0.303157\pi$
$61$	9.05571	1.15946	0.579732	+	0.814807i	$0.303157\pi$
$62$	0	0
$63$	3.15633	0.397660
$64$	0	0
$65$	−7.35026	−0.911688
$66$	0	0
$67$	−11.8945	−1.45314	−0.726570	−	0.687093i	$-0.758887\pi$
$67$	−11.8945	−1.45314	−0.726570	+	0.687093i	$0.758887\pi$
$68$	0	0
$69$	2.48119	0.298701
$70$	0	0
$71$	−0.775746	−0.0920641	−0.0460321	−	0.998940i	$-0.514658\pi$
$71$	−0.775746	−0.0920641	−0.0460321	+	0.998940i	$0.514658\pi$
$72$	0	0
$73$	8.88717	1.04016	0.520082	−	0.854116i	$-0.325901\pi$
$73$	8.88717	1.04016	0.520082	+	0.854116i	$0.325901\pi$
$74$	0	0
$75$	−2.86907	−0.331291
$76$	0	0
$77$	−1.19394	−0.136062
$78$	0	0
$79$	−12.7308	−1.43233	−0.716166	−	0.697930i	$-0.754105\pi$
$79$	−12.7308	−1.43233	−0.716166	+	0.697930i	$0.754105\pi$
$80$	0	0
$81$	−8.50659	−0.945176
$82$	0	0
$83$	−16.4387	−1.80438	−0.902189	−	0.431342i	$-0.858040\pi$
$83$	−16.4387	−1.80438	−0.902189	+	0.431342i	$0.858040\pi$
$84$	0	0
$85$	−1.19394	−0.129501
$86$	0	0
$87$	10.3127	1.10563
$88$	0	0
$89$	13.7562	1.45816	0.729079	−	0.684430i	$-0.239949\pi$
$89$	13.7562	1.45816	0.729079	+	0.684430i	$0.239949\pi$
$90$	0	0
$91$	−2.96239	−0.310543
$92$	0	0
$93$	25.3561	2.62931
$94$	0	0
$95$	10.7005	1.09785
$96$	0	0
$97$	−0.0933212	−0.00947533	−0.00473766	−	0.999989i	$-0.501508\pi$
$97$	−0.0933212	−0.00947533	−0.00473766	+	0.999989i	$0.501508\pi$
$98$	0	0
$99$	−3.76845	−0.378744
$100$	0	0
$101$	−2.64974	−0.263659	−0.131829	−	0.991272i	$-0.542085\pi$
$101$	−2.64974	−0.263659	−0.131829	+	0.991272i	$0.542085\pi$
$102$	0	0
$103$	−2.07522	−0.204478	−0.102239	−	0.994760i	$-0.532601\pi$
$103$	−2.07522	−0.204478	−0.102239	+	0.994760i	$0.532601\pi$
$104$	0	0
$105$	−6.15633	−0.600796
$106$	0	0
$107$	2.51388	0.243026	0.121513	−	0.992590i	$-0.461225\pi$
$107$	2.51388	0.243026	0.121513	+	0.992590i	$0.461225\pi$
$108$	0	0
$109$	6.18664	0.592573	0.296286	−	0.955099i	$-0.404252\pi$
$109$	6.18664	0.592573	0.296286	+	0.955099i	$0.404252\pi$
$110$	0	0
$111$	13.2750	1.26001
$112$	0	0
$113$	−11.1490	−1.04881	−0.524406	−	0.851468i	$-0.675713\pi$
$113$	−11.1490	−1.04881	−0.524406	+	0.851468i	$0.675713\pi$
$114$	0	0
$115$	−2.48119	−0.231373
$116$	0	0
$117$	−9.35026	−0.864432
$118$	0	0
$119$	−0.481194	−0.0441110
$120$	0	0
$121$	−9.57452	−0.870410
$122$	0	0
$123$	29.5877	2.66783
$124$	0	0
$125$	−9.53690	−0.853007
$126$	0	0
$127$	−0.649738	−0.0576549	−0.0288275	−	0.999584i	$-0.509177\pi$
$127$	−0.649738	−0.0576549	−0.0288275	+	0.999584i	$0.509177\pi$
$128$	0	0
$129$	−14.2374	−1.25354
$130$	0	0
$131$	−11.4944	−1.00427	−0.502134	−	0.864790i	$-0.667452\pi$
$131$	−11.4944	−1.00427	−0.502134	+	0.864790i	$0.667452\pi$
$132$	0	0
$133$	4.31265	0.373954
$134$	0	0
$135$	−0.962389	−0.0828292
$136$	0	0
$137$	−4.57452	−0.390827	−0.195414	−	0.980721i	$-0.562605\pi$
$137$	−4.57452	−0.390827	−0.195414	+	0.980721i	$0.562605\pi$
$138$	0	0
$139$	−13.0557	−1.10737	−0.553685	−	0.832726i	$-0.686779\pi$
$139$	−13.0557	−1.10737	−0.553685	+	0.832726i	$0.686779\pi$
$140$	0	0
$141$	0.418190	0.0352180
$142$	0	0
$143$	3.53690	0.295771
$144$	0	0
$145$	−10.3127	−0.856419
$146$	0	0
$147$	−2.48119	−0.204645
$148$	0	0
$149$	−9.22425	−0.755680	−0.377840	−	0.925871i	$-0.623333\pi$
$149$	−9.22425	−0.755680	−0.377840	+	0.925871i	$0.623333\pi$
$150$	0	0
$151$	−8.12601	−0.661285	−0.330643	−	0.943756i	$-0.607265\pi$
$151$	−8.12601	−0.661285	−0.330643	+	0.943756i	$0.607265\pi$
$152$	0	0
$153$	−1.51881	−0.122788
$154$	0	0
$155$	−25.3561	−2.03665
$156$	0	0
$157$	−7.51881	−0.600066	−0.300033	−	0.953929i	$-0.596998\pi$
$157$	−7.51881	−0.600066	−0.300033	+	0.953929i	$0.596998\pi$
$158$	0	0
$159$	−21.5877	−1.71202
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	24.1622	1.89253	0.946265	−	0.323392i	$-0.104823\pi$
$163$	24.1622	1.89253	0.946265	+	0.323392i	$0.104823\pi$
$164$	0	0
$165$	7.35026	0.572217
$166$	0	0
$167$	7.70545	0.596265	0.298133	−	0.954524i	$-0.403636\pi$
$167$	7.70545	0.596265	0.298133	+	0.954524i	$0.403636\pi$
$168$	0	0
$169$	−4.22425	−0.324943
$170$	0	0
$171$	13.6121	1.04095
$172$	0	0
$173$	−5.53690	−0.420963	−0.210482	−	0.977598i	$-0.567503\pi$
$173$	−5.53690	−0.420963	−0.210482	+	0.977598i	$0.567503\pi$
$174$	0	0
$175$	1.15633	0.0874100
$176$	0	0
$177$	−28.7064	−2.15771
$178$	0	0
$179$	−17.9003	−1.33793	−0.668967	−	0.743292i	$-0.733263\pi$
$179$	−17.9003	−1.33793	−0.668967	+	0.743292i	$0.733263\pi$
$180$	0	0
$181$	−0.793845	−0.0590060	−0.0295030	−	0.999565i	$-0.509392\pi$
$181$	−0.793845	−0.0590060	−0.0295030	+	0.999565i	$0.509392\pi$
$182$	0	0
$183$	−22.4690	−1.66095
$184$	0	0
$185$	−13.2750	−0.976000
$186$	0	0
$187$	0.574515	0.0420127
$188$	0	0
$189$	−0.387873	−0.0282136
$190$	0	0
$191$	−21.5877	−1.56203	−0.781015	−	0.624512i	$-0.785298\pi$
$191$	−21.5877	−1.56203	−0.781015	+	0.624512i	$0.785298\pi$
$192$	0	0
$193$	−24.0508	−1.73121	−0.865607	−	0.500725i	$-0.833067\pi$
$193$	−24.0508	−1.73121	−0.865607	+	0.500725i	$0.833067\pi$
$194$	0	0
$195$	18.2374	1.30601
$196$	0	0
$197$	6.62530	0.472033	0.236017	−	0.971749i	$-0.424158\pi$
$197$	6.62530	0.472033	0.236017	+	0.971749i	$0.424158\pi$
$198$	0	0
$199$	−11.2243	−0.795666	−0.397833	−	0.917458i	$-0.630238\pi$
$199$	−11.2243	−0.795666	−0.397833	+	0.917458i	$0.630238\pi$
$200$	0	0
$201$	29.5125	2.08165
$202$	0	0
$203$	−4.15633	−0.291717
$204$	0	0
$205$	−29.5877	−2.06649
$206$	0	0
$207$	−3.15633	−0.219380
$208$	0	0
$209$	−5.14903	−0.356166
$210$	0	0
$211$	2.05079	0.141182	0.0705909	−	0.997505i	$-0.477512\pi$
$211$	2.05079	0.141182	0.0705909	+	0.997505i	$0.477512\pi$
$212$	0	0
$213$	1.92478	0.131884
$214$	0	0
$215$	14.2374	0.970985
$216$	0	0
$217$	−10.2193	−0.693733
$218$	0	0
$219$	−22.0508	−1.49005
$220$	0	0
$221$	1.42548	0.0958885
$222$	0	0
$223$	−21.5550	−1.44343	−0.721715	−	0.692190i	$-0.756646\pi$
$223$	−21.5550	−1.44343	−0.721715	+	0.692190i	$0.756646\pi$
$224$	0	0
$225$	3.64974	0.243316
$226$	0	0
$227$	1.29948	0.0862493	0.0431246	−	0.999070i	$-0.486269\pi$
$227$	1.29948	0.0862493	0.0431246	+	0.999070i	$0.486269\pi$
$228$	0	0
$229$	−9.88224	−0.653037	−0.326518	−	0.945191i	$-0.605876\pi$
$229$	−9.88224	−0.653037	−0.326518	+	0.945191i	$0.605876\pi$
$230$	0	0
$231$	2.96239	0.194911
$232$	0	0
$233$	−22.8568	−1.49740	−0.748701	−	0.662908i	$-0.769322\pi$
$233$	−22.8568	−1.49740	−0.748701	+	0.662908i	$0.769322\pi$
$234$	0	0
$235$	−0.418190	−0.0272797
$236$	0	0
$237$	31.5877	2.05184
$238$	0	0
$239$	2.05079	0.132654	0.0663271	−	0.997798i	$-0.478872\pi$
$239$	2.05079	0.132654	0.0663271	+	0.997798i	$0.478872\pi$
$240$	0	0
$241$	24.3961	1.57149	0.785746	−	0.618549i	$-0.212279\pi$
$241$	24.3961	1.57149	0.785746	+	0.618549i	$0.212279\pi$
$242$	0	0
$243$	22.2701	1.42863
$244$	0	0
$245$	2.48119	0.158518
$246$	0	0
$247$	−12.7757	−0.812901
$248$	0	0
$249$	40.7875	2.58480
$250$	0	0
$251$	11.1636	0.704641	0.352321	−	0.935879i	$-0.385393\pi$
$251$	11.1636	0.704641	0.352321	+	0.935879i	$0.385393\pi$
$252$	0	0
$253$	1.19394	0.0750621
$254$	0	0
$255$	2.96239	0.185512
$256$	0	0
$257$	3.29948	0.205816	0.102908	−	0.994691i	$-0.467185\pi$
$257$	3.29948	0.205816	0.102908	+	0.994691i	$0.467185\pi$
$258$	0	0
$259$	−5.35026	−0.332449
$260$	0	0
$261$	−13.1187	−0.812028
$262$	0	0
$263$	25.3561	1.56353	0.781763	−	0.623575i	$-0.214321\pi$
$263$	25.3561	1.56353	0.781763	+	0.623575i	$0.214321\pi$
$264$	0	0
$265$	21.5877	1.32612
$266$	0	0
$267$	−34.1319	−2.08884
$268$	0	0
$269$	−3.08840	−0.188303	−0.0941514	−	0.995558i	$-0.530014\pi$
$269$	−3.08840	−0.188303	−0.0941514	+	0.995558i	$0.530014\pi$
$270$	0	0
$271$	8.65799	0.525935	0.262968	−	0.964805i	$-0.415299\pi$
$271$	8.65799	0.525935	0.262968	+	0.964805i	$0.415299\pi$
$272$	0	0
$273$	7.35026	0.444858
$274$	0	0
$275$	−1.38058	−0.0832520
$276$	0	0
$277$	6.08110	0.365378	0.182689	−	0.983171i	$-0.441520\pi$
$277$	6.08110	0.365378	0.182689	+	0.983171i	$0.441520\pi$
$278$	0	0
$279$	−32.2555	−1.93109
$280$	0	0
$281$	−23.0640	−1.37588	−0.687940	−	0.725767i	$-0.741485\pi$
$281$	−23.0640	−1.37588	−0.687940	+	0.725767i	$0.741485\pi$
$282$	0	0
$283$	6.44851	0.383324	0.191662	−	0.981461i	$-0.438612\pi$
$283$	6.44851	0.383324	0.191662	+	0.981461i	$0.438612\pi$
$284$	0	0
$285$	−26.5501	−1.57269
$286$	0	0
$287$	−11.9248	−0.703897
$288$	0	0
$289$	−16.7685	−0.986380
$290$	0	0
$291$	0.231548	0.0135736
$292$	0	0
$293$	10.0181	0.585264	0.292632	−	0.956225i	$-0.405469\pi$
$293$	10.0181	0.585264	0.292632	+	0.956225i	$0.405469\pi$
$294$	0	0
$295$	28.7064	1.67135
$296$	0	0
$297$	0.463096	0.0268716
$298$	0	0
$299$	2.96239	0.171319
$300$	0	0
$301$	5.73813	0.330741
$302$	0	0
$303$	6.57452	0.377696
$304$	0	0
$305$	22.4690	1.28657
$306$	0	0
$307$	6.55642	0.374194	0.187097	−	0.982341i	$-0.440092\pi$
$307$	6.55642	0.374194	0.187097	+	0.982341i	$0.440092\pi$
$308$	0	0
$309$	5.14903	0.292918
$310$	0	0
$311$	28.0689	1.59164	0.795820	−	0.605533i	$-0.207040\pi$
$311$	28.0689	1.59164	0.795820	+	0.605533i	$0.207040\pi$
$312$	0	0
$313$	31.9330	1.80496	0.902481	−	0.430730i	$-0.141744\pi$
$313$	31.9330	1.80496	0.902481	+	0.430730i	$0.141744\pi$
$314$	0	0
$315$	7.83146	0.441253
$316$	0	0
$317$	9.30536	0.522641	0.261320	−	0.965252i	$-0.415842\pi$
$317$	9.30536	0.522641	0.261320	+	0.965252i	$0.415842\pi$
$318$	0	0
$319$	4.96239	0.277840
$320$	0	0
$321$	−6.23743	−0.348139
$322$	0	0
$323$	−2.07522	−0.115468
$324$	0	0
$325$	−3.42548	−0.190012
$326$	0	0
$327$	−15.3503	−0.848871
$328$	0	0
$329$	−0.168544	−0.00929213
$330$	0	0
$331$	−24.2882	−1.33500	−0.667500	−	0.744609i	$-0.732636\pi$
$331$	−24.2882	−1.33500	−0.667500	+	0.744609i	$0.732636\pi$
$332$	0	0
$333$	−16.8872	−0.925411
$334$	0	0
$335$	−29.5125	−1.61244
$336$	0	0
$337$	−7.11283	−0.387461	−0.193730	−	0.981055i	$-0.562059\pi$
$337$	−7.11283	−0.387461	−0.193730	+	0.981055i	$0.562059\pi$
$338$	0	0
$339$	27.6629	1.50244
$340$	0	0
$341$	12.2012	0.660734
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	6.15633	0.331445
$346$	0	0
$347$	30.5745	1.64133	0.820663	−	0.571413i	$-0.193604\pi$
$347$	30.5745	1.64133	0.820663	+	0.571413i	$0.193604\pi$
$348$	0	0
$349$	26.3733	1.41173	0.705865	−	0.708347i	$-0.250559\pi$
$349$	26.3733	1.41173	0.705865	+	0.708347i	$0.250559\pi$
$350$	0	0
$351$	1.14903	0.0613307
$352$	0	0
$353$	−3.67750	−0.195734	−0.0978668	−	0.995200i	$-0.531202\pi$
$353$	−3.67750	−0.195734	−0.0978668	+	0.995200i	$0.531202\pi$
$354$	0	0
$355$	−1.92478	−0.102157
$356$	0	0
$357$	1.19394	0.0631898
$358$	0	0
$359$	−32.2677	−1.70303	−0.851513	−	0.524333i	$-0.824315\pi$
$359$	−32.2677	−1.70303	−0.851513	+	0.524333i	$0.824315\pi$
$360$	0	0
$361$	−0.401047	−0.0211077
$362$	0	0
$363$	23.7562	1.24688
$364$	0	0
$365$	22.0508	1.15419
$366$	0	0
$367$	0.901754	0.0470712	0.0235356	−	0.999723i	$-0.492508\pi$
$367$	0.901754	0.0470712	0.0235356	+	0.999723i	$0.492508\pi$
$368$	0	0
$369$	−37.6385	−1.95938
$370$	0	0
$371$	8.70052	0.451709
$372$	0	0
$373$	−7.67276	−0.397281	−0.198640	−	0.980072i	$-0.563653\pi$
$373$	−7.67276	−0.397281	−0.198640	+	0.980072i	$0.563653\pi$
$374$	0	0
$375$	23.6629	1.22195
$376$	0	0
$377$	12.3127	0.634134
$378$	0	0
$379$	−18.9683	−0.974334	−0.487167	−	0.873309i	$-0.661970\pi$
$379$	−18.9683	−0.974334	−0.487167	+	0.873309i	$0.661970\pi$
$380$	0	0
$381$	1.61213	0.0825918
$382$	0	0
$383$	−22.0508	−1.12674	−0.563371	−	0.826204i	$-0.690496\pi$
$383$	−22.0508	−1.12674	−0.563371	+	0.826204i	$0.690496\pi$
$384$	0	0
$385$	−2.96239	−0.150977
$386$	0	0
$387$	18.1114	0.920655
$388$	0	0
$389$	−24.2130	−1.22765	−0.613824	−	0.789443i	$-0.710369\pi$
$389$	−24.2130	−1.22765	−0.613824	+	0.789443i	$0.710369\pi$
$390$	0	0
$391$	0.481194	0.0243350
$392$	0	0
$393$	28.5198	1.43863
$394$	0	0
$395$	−31.5877	−1.58935
$396$	0	0
$397$	22.4387	1.12616	0.563082	−	0.826401i	$-0.309616\pi$
$397$	22.4387	1.12616	0.563082	+	0.826401i	$0.309616\pi$
$398$	0	0
$399$	−10.7005	−0.535696
$400$	0	0
$401$	17.0376	0.850818	0.425409	−	0.905001i	$-0.360130\pi$
$401$	17.0376	0.850818	0.425409	+	0.905001i	$0.360130\pi$
$402$	0	0
$403$	30.2736	1.50804
$404$	0	0
$405$	−21.1065	−1.04879
$406$	0	0
$407$	6.38787	0.316635
$408$	0	0
$409$	−8.67609	−0.429005	−0.214502	−	0.976723i	$-0.568813\pi$
$409$	−8.67609	−0.429005	−0.214502	+	0.976723i	$0.568813\pi$
$410$	0	0
$411$	11.3503	0.559867
$412$	0	0
$413$	11.5696	0.569302
$414$	0	0
$415$	−40.7875	−2.00218
$416$	0	0
$417$	32.3938	1.58633
$418$	0	0
$419$	21.7137	1.06078	0.530392	−	0.847753i	$-0.322045\pi$
$419$	21.7137	1.06078	0.530392	+	0.847753i	$0.322045\pi$
$420$	0	0
$421$	−34.4749	−1.68020	−0.840101	−	0.542430i	$-0.817504\pi$
$421$	−34.4749	−1.68020	−0.840101	+	0.542430i	$0.817504\pi$
$422$	0	0
$423$	−0.531980	−0.0258657
$424$	0	0
$425$	−0.556417	−0.0269902
$426$	0	0
$427$	9.05571	0.438237
$428$	0	0
$429$	−8.77575	−0.423697
$430$	0	0
$431$	18.5501	0.893526	0.446763	−	0.894652i	$-0.352577\pi$
$431$	18.5501	0.893526	0.446763	+	0.894652i	$0.352577\pi$
$432$	0	0
$433$	0.117759	0.00565912	0.00282956	−	0.999996i	$-0.499099\pi$
$433$	0.117759	0.00565912	0.00282956	+	0.999996i	$0.499099\pi$
$434$	0	0
$435$	25.5877	1.22684
$436$	0	0
$437$	−4.31265	−0.206302
$438$	0	0
$439$	−22.2701	−1.06289	−0.531447	−	0.847091i	$-0.678352\pi$
$439$	−22.2701	−1.06289	−0.531447	+	0.847091i	$0.678352\pi$
$440$	0	0
$441$	3.15633	0.150301
$442$	0	0
$443$	15.1636	0.720445	0.360223	−	0.932866i	$-0.382701\pi$
$443$	15.1636	0.720445	0.360223	+	0.932866i	$0.382701\pi$
$444$	0	0
$445$	34.1319	1.61801
$446$	0	0
$447$	22.8872	1.08253
$448$	0	0
$449$	32.8832	1.55185	0.775927	−	0.630823i	$-0.217282\pi$
$449$	32.8832	1.55185	0.775927	+	0.630823i	$0.217282\pi$
$450$	0	0
$451$	14.2374	0.670414
$452$	0	0
$453$	20.1622	0.947303
$454$	0	0
$455$	−7.35026	−0.344586
$456$	0	0
$457$	−25.3865	−1.18753	−0.593764	−	0.804639i	$-0.702359\pi$
$457$	−25.3865	−1.18753	−0.593764	+	0.804639i	$0.702359\pi$
$458$	0	0
$459$	0.186642	0.00871172
$460$	0	0
$461$	11.6121	0.540831	0.270415	−	0.962744i	$-0.412839\pi$
$461$	11.6121	0.540831	0.270415	+	0.962744i	$0.412839\pi$
$462$	0	0
$463$	7.41090	0.344414	0.172207	−	0.985061i	$-0.444910\pi$
$463$	7.41090	0.344414	0.172207	+	0.985061i	$0.444910\pi$
$464$	0	0
$465$	62.9135	2.91754
$466$	0	0
$467$	27.3865	1.26729	0.633647	−	0.773622i	$-0.281557\pi$
$467$	27.3865	1.26729	0.633647	+	0.773622i	$0.281557\pi$
$468$	0	0
$469$	−11.8945	−0.549235
$470$	0	0
$471$	18.6556	0.859605
$472$	0	0
$473$	−6.85097	−0.315008
$474$	0	0
$475$	4.98683	0.228811
$476$	0	0
$477$	27.4617	1.25738
$478$	0	0
$479$	−33.5633	−1.53354	−0.766772	−	0.641919i	$-0.778138\pi$
$479$	−33.5633	−1.53354	−0.766772	+	0.641919i	$0.778138\pi$
$480$	0	0
$481$	15.8496	0.722677
$482$	0	0
$483$	2.48119	0.112898
$484$	0	0
$485$	−0.231548	−0.0105141
$486$	0	0
$487$	−23.8496	−1.08073	−0.540363	−	0.841432i	$-0.681713\pi$
$487$	−23.8496	−1.08073	−0.540363	+	0.841432i	$0.681713\pi$
$488$	0	0
$489$	−59.9511	−2.71108
$490$	0	0
$491$	−3.47627	−0.156882	−0.0784409	−	0.996919i	$-0.524994\pi$
$491$	−3.47627	−0.156882	−0.0784409	+	0.996919i	$0.524994\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	−9.35026	−0.420263
$496$	0	0
$497$	−0.775746	−0.0347970
$498$	0	0
$499$	14.3634	0.642996	0.321498	−	0.946910i	$-0.395814\pi$
$499$	14.3634	0.642996	0.321498	+	0.946910i	$0.395814\pi$
$500$	0	0
$501$	−19.1187	−0.854161
$502$	0	0
$503$	21.8641	0.974874	0.487437	−	0.873158i	$-0.337932\pi$
$503$	21.8641	0.974874	0.487437	+	0.873158i	$0.337932\pi$
$504$	0	0
$505$	−6.57452	−0.292562
$506$	0	0
$507$	10.4812	0.465486
$508$	0	0
$509$	−35.1490	−1.55795	−0.778977	−	0.627053i	$-0.784261\pi$
$509$	−35.1490	−1.55795	−0.778977	+	0.627053i	$0.784261\pi$
$510$	0	0
$511$	8.88717	0.393145
$512$	0	0
$513$	−1.67276	−0.0738542
$514$	0	0
$515$	−5.14903	−0.226893
$516$	0	0
$517$	0.201231	0.00885012
$518$	0	0
$519$	13.7381	0.603037
$520$	0	0
$521$	11.5042	0.504009	0.252004	−	0.967726i	$-0.418910\pi$
$521$	11.5042	0.504009	0.252004	+	0.967726i	$0.418910\pi$
$522$	0	0
$523$	27.0738	1.18385	0.591927	−	0.805991i	$-0.298367\pi$
$523$	27.0738	1.18385	0.591927	+	0.805991i	$0.298367\pi$
$524$	0	0
$525$	−2.86907	−0.125216
$526$	0	0
$527$	4.91748	0.214209
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	36.5174	1.58472
$532$	0	0
$533$	35.3258	1.53013
$534$	0	0
$535$	6.23743	0.269668
$536$	0	0
$537$	44.4142	1.91662
$538$	0	0
$539$	−1.19394	−0.0514265
$540$	0	0
$541$	26.8568	1.15467	0.577333	−	0.816509i	$-0.304093\pi$
$541$	26.8568	1.15467	0.577333	+	0.816509i	$0.304093\pi$
$542$	0	0
$543$	1.96968	0.0845272
$544$	0	0
$545$	15.3503	0.657533
$546$	0	0
$547$	9.79877	0.418965	0.209483	−	0.977812i	$-0.432822\pi$
$547$	9.79877	0.418965	0.209483	+	0.977812i	$0.432822\pi$
$548$	0	0
$549$	28.5828	1.21988
$550$	0	0
$551$	−17.9248	−0.763621
$552$	0	0
$553$	−12.7308	−0.541370
$554$	0	0
$555$	32.9380	1.39814
$556$	0	0
$557$	3.67276	0.155620	0.0778099	−	0.996968i	$-0.475207\pi$
$557$	3.67276	0.155620	0.0778099	+	0.996968i	$0.475207\pi$
$558$	0	0
$559$	−16.9986	−0.718964
$560$	0	0
$561$	−1.42548	−0.0601840
$562$	0	0
$563$	−21.2506	−0.895606	−0.447803	−	0.894132i	$-0.647793\pi$
$563$	−21.2506	−0.895606	−0.447803	+	0.894132i	$0.647793\pi$
$564$	0	0
$565$	−27.6629	−1.16379
$566$	0	0
$567$	−8.50659	−0.357243
$568$	0	0
$569$	16.4241	0.688533	0.344266	−	0.938872i	$-0.388128\pi$
$569$	16.4241	0.688533	0.344266	+	0.938872i	$0.388128\pi$
$570$	0	0
$571$	−15.4109	−0.644926	−0.322463	−	0.946582i	$-0.604511\pi$
$571$	−15.4109	−0.644926	−0.322463	+	0.946582i	$0.604511\pi$
$572$	0	0
$573$	53.5633	2.23764
$574$	0	0
$575$	−1.15633	−0.0482221
$576$	0	0
$577$	36.9478	1.53816	0.769079	−	0.639154i	$-0.220715\pi$
$577$	36.9478	1.53816	0.769079	+	0.639154i	$0.220715\pi$
$578$	0	0
$579$	59.6747	2.47999
$580$	0	0
$581$	−16.4387	−0.681990
$582$	0	0
$583$	−10.3879	−0.430222
$584$	0	0
$585$	−23.1998	−0.959194
$586$	0	0
$587$	12.8691	0.531163	0.265582	−	0.964088i	$-0.414436\pi$
$587$	12.8691	0.531163	0.265582	+	0.964088i	$0.414436\pi$
$588$	0	0
$589$	−44.0724	−1.81597
$590$	0	0
$591$	−16.4387	−0.676196
$592$	0	0
$593$	−5.16362	−0.212044	−0.106022	−	0.994364i	$-0.533811\pi$
$593$	−5.16362	−0.212044	−0.106022	+	0.994364i	$0.533811\pi$
$594$	0	0
$595$	−1.19394	−0.0489466
$596$	0	0
$597$	27.8496	1.13981
$598$	0	0
$599$	26.0508	1.06441	0.532203	−	0.846617i	$-0.321364\pi$
$599$	26.0508	1.06441	0.532203	+	0.846617i	$0.321364\pi$
$600$	0	0
$601$	11.0230	0.449638	0.224819	−	0.974400i	$-0.427821\pi$
$601$	11.0230	0.449638	0.224819	+	0.974400i	$0.427821\pi$
$602$	0	0
$603$	−37.5428	−1.52886
$604$	0	0
$605$	−23.7562	−0.965828
$606$	0	0
$607$	−4.64340	−0.188470	−0.0942349	−	0.995550i	$-0.530040\pi$
$607$	−4.64340	−0.188470	−0.0942349	+	0.995550i	$0.530040\pi$
$608$	0	0
$609$	10.3127	0.417890
$610$	0	0
$611$	0.499293	0.0201992
$612$	0	0
$613$	−27.3357	−1.10408	−0.552039	−	0.833818i	$-0.686150\pi$
$613$	−27.3357	−1.10408	−0.552039	+	0.833818i	$0.686150\pi$
$614$	0	0
$615$	73.4128	2.96029
$616$	0	0
$617$	−23.9003	−0.962191	−0.481096	−	0.876668i	$-0.659761\pi$
$617$	−23.9003	−0.962191	−0.481096	+	0.876668i	$0.659761\pi$
$618$	0	0
$619$	−15.4109	−0.619416	−0.309708	−	0.950832i	$-0.600231\pi$
$619$	−15.4109	−0.619416	−0.309708	+	0.950832i	$0.600231\pi$
$620$	0	0
$621$	0.387873	0.0155648
$622$	0	0
$623$	13.7562	0.551132
$624$	0	0
$625$	−29.4445	−1.17778
$626$	0	0
$627$	12.7757	0.510214
$628$	0	0
$629$	2.57452	0.102653
$630$	0	0
$631$	−42.6556	−1.69809	−0.849047	−	0.528318i	$-0.822823\pi$
$631$	−42.6556	−1.69809	−0.849047	+	0.528318i	$0.822823\pi$
$632$	0	0
$633$	−5.08840	−0.202246
$634$	0	0
$635$	−1.61213	−0.0639753
$636$	0	0
$637$	−2.96239	−0.117374
$638$	0	0
$639$	−2.44851	−0.0968615
$640$	0	0
$641$	8.42407	0.332731	0.166365	−	0.986064i	$-0.446797\pi$
$641$	8.42407	0.332731	0.166365	+	0.986064i	$0.446797\pi$
$642$	0	0
$643$	3.89843	0.153739	0.0768695	−	0.997041i	$-0.475508\pi$
$643$	3.89843	0.153739	0.0768695	+	0.997041i	$0.475508\pi$
$644$	0	0
$645$	−35.3258	−1.39095
$646$	0	0
$647$	34.7186	1.36493	0.682465	−	0.730918i	$-0.260908\pi$
$647$	34.7186	1.36493	0.682465	+	0.730918i	$0.260908\pi$
$648$	0	0
$649$	−13.8134	−0.542222
$650$	0	0
$651$	25.3561	0.993786
$652$	0	0
$653$	−1.62672	−0.0636583	−0.0318291	−	0.999493i	$-0.510133\pi$
$653$	−1.62672	−0.0636583	−0.0318291	+	0.999493i	$0.510133\pi$
$654$	0	0
$655$	−28.5198	−1.11436
$656$	0	0
$657$	28.0508	1.09437
$658$	0	0
$659$	31.8496	1.24068	0.620341	−	0.784332i	$-0.286994\pi$
$659$	31.8496	1.24068	0.620341	+	0.784332i	$0.286994\pi$
$660$	0	0
$661$	1.77082	0.0688770	0.0344385	−	0.999407i	$-0.489036\pi$
$661$	1.77082	0.0688770	0.0344385	+	0.999407i	$0.489036\pi$
$662$	0	0
$663$	−3.53690	−0.137362
$664$	0	0
$665$	10.7005	0.414949
$666$	0	0
$667$	4.15633	0.160934
$668$	0	0
$669$	53.4821	2.06774
$670$	0	0
$671$	−10.8119	−0.417390
$672$	0	0
$673$	−7.68147	−0.296099	−0.148049	−	0.988980i	$-0.547299\pi$
$673$	−7.68147	−0.296099	−0.148049	+	0.988980i	$0.547299\pi$
$674$	0	0
$675$	−0.448507	−0.0172631
$676$	0	0
$677$	−21.0919	−0.810628	−0.405314	−	0.914178i	$-0.632838\pi$
$677$	−21.0919	−0.810628	−0.405314	+	0.914178i	$0.632838\pi$
$678$	0	0
$679$	−0.0933212	−0.00358134
$680$	0	0
$681$	−3.22425	−0.123554
$682$	0	0
$683$	11.1636	0.427164	0.213582	−	0.976925i	$-0.431487\pi$
$683$	11.1636	0.427164	0.213582	+	0.976925i	$0.431487\pi$
$684$	0	0
$685$	−11.3503	−0.433671
$686$	0	0
$687$	24.5198	0.935487
$688$	0	0
$689$	−25.7743	−0.981924
$690$	0	0
$691$	−18.0689	−0.687373	−0.343686	−	0.939084i	$-0.611676\pi$
$691$	−18.0689	−0.687373	−0.343686	+	0.939084i	$0.611676\pi$
$692$	0	0
$693$	−3.76845	−0.143152
$694$	0	0
$695$	−32.3938	−1.22877
$696$	0	0
$697$	5.73813	0.217347
$698$	0	0
$699$	56.7123	2.14506
$700$	0	0
$701$	−20.5256	−0.775243	−0.387621	−	0.921819i	$-0.626703\pi$
$701$	−20.5256	−0.775243	−0.387621	+	0.921819i	$0.626703\pi$
$702$	0	0
$703$	−23.0738	−0.870245
$704$	0	0
$705$	1.03761	0.0390787
$706$	0	0
$707$	−2.64974	−0.0996537
$708$	0	0
$709$	14.9624	0.561924	0.280962	−	0.959719i	$-0.409346\pi$
$709$	14.9624	0.561924	0.280962	+	0.959719i	$0.409346\pi$
$710$	0	0
$711$	−40.1827	−1.50697
$712$	0	0
$713$	10.2193	0.382717
$714$	0	0
$715$	8.77575	0.328194
$716$	0	0
$717$	−5.08840	−0.190030
$718$	0	0
$719$	1.84131	0.0686691	0.0343345	−	0.999410i	$-0.489069\pi$
$719$	1.84131	0.0686691	0.0343345	+	0.999410i	$0.489069\pi$
$720$	0	0
$721$	−2.07522	−0.0772853
$722$	0	0
$723$	−60.5315	−2.25119
$724$	0	0
$725$	−4.80606	−0.178493
$726$	0	0
$727$	−24.7757	−0.918882	−0.459441	−	0.888208i	$-0.651950\pi$
$727$	−24.7757	−0.918882	−0.459441	+	0.888208i	$0.651950\pi$
$728$	0	0
$729$	−29.7367	−1.10136
$730$	0	0
$731$	−2.76116	−0.102125
$732$	0	0
$733$	0.677686	0.0250309	0.0125154	−	0.999922i	$-0.496016\pi$
$733$	0.677686	0.0250309	0.0125154	+	0.999922i	$0.496016\pi$
$734$	0	0
$735$	−6.15633	−0.227079
$736$	0	0
$737$	14.2012	0.523109
$738$	0	0
$739$	11.1030	0.408430	0.204215	−	0.978926i	$-0.434536\pi$
$739$	11.1030	0.408430	0.204215	+	0.978926i	$0.434536\pi$
$740$	0	0
$741$	31.6991	1.16450
$742$	0	0
$743$	53.4168	1.95967	0.979836	−	0.199805i	$-0.0640308\pi$
$743$	53.4168	1.95967	0.979836	+	0.199805i	$0.0640308\pi$
$744$	0	0
$745$	−22.8872	−0.838521
$746$	0	0
$747$	−51.8858	−1.89840
$748$	0	0
$749$	2.51388	0.0918552
$750$	0	0
$751$	−11.7929	−0.430329	−0.215164	−	0.976578i	$-0.569029\pi$
$751$	−11.7929	−0.430329	−0.215164	+	0.976578i	$0.569029\pi$
$752$	0	0
$753$	−27.6991	−1.00941
$754$	0	0
$755$	−20.1622	−0.733778
$756$	0	0
$757$	12.4241	0.451561	0.225780	−	0.974178i	$-0.427507\pi$
$757$	12.4241	0.451561	0.225780	+	0.974178i	$0.427507\pi$
$758$	0	0
$759$	−2.96239	−0.107528
$760$	0	0
$761$	1.66291	0.0602805	0.0301403	−	0.999546i	$-0.490405\pi$
$761$	1.66291	0.0602805	0.0301403	+	0.999546i	$0.490405\pi$
$762$	0	0
$763$	6.18664	0.223971
$764$	0	0
$765$	−3.76845	−0.136249
$766$	0	0
$767$	−34.2736	−1.23755
$768$	0	0
$769$	−12.7938	−0.461358	−0.230679	−	0.973030i	$-0.574095\pi$
$769$	−12.7938	−0.461358	−0.230679	+	0.973030i	$0.574095\pi$
$770$	0	0
$771$	−8.18664	−0.294835
$772$	0	0
$773$	−10.5320	−0.378809	−0.189404	−	0.981899i	$-0.560656\pi$
$773$	−10.5320	−0.378809	−0.189404	+	0.981899i	$0.560656\pi$
$774$	0	0
$775$	−11.8169	−0.424474
$776$	0	0
$777$	13.2750	0.476239
$778$	0	0
$779$	−51.4274	−1.84258
$780$	0	0
$781$	0.926192	0.0331417
$782$	0	0
$783$	1.61213	0.0576127
$784$	0	0
$785$	−18.6556	−0.665848
$786$	0	0
$787$	−47.4518	−1.69148	−0.845738	−	0.533599i	$-0.820839\pi$
$787$	−47.4518	−1.69148	−0.845738	+	0.533599i	$0.820839\pi$
$788$	0	0
$789$	−62.9135	−2.23978
$790$	0	0
$791$	−11.1490	−0.396414
$792$	0	0
$793$	−26.8265	−0.952638
$794$	0	0
$795$	−53.5633	−1.89969
$796$	0	0
$797$	−34.4177	−1.21914	−0.609569	−	0.792733i	$-0.708658\pi$
$797$	−34.4177	−1.21914	−0.609569	+	0.792733i	$0.708658\pi$
$798$	0	0
$799$	0.0811024	0.00286920
$800$	0	0
$801$	43.4191	1.53414
$802$	0	0
$803$	−10.6107	−0.374444
$804$	0	0
$805$	−2.48119	−0.0874506
$806$	0	0
$807$	7.66291	0.269747
$808$	0	0
$809$	17.5007	0.615292	0.307646	−	0.951501i	$-0.400459\pi$
$809$	17.5007	0.615292	0.307646	+	0.951501i	$0.400459\pi$
$810$	0	0
$811$	−23.3439	−0.819716	−0.409858	−	0.912149i	$-0.634422\pi$
$811$	−23.3439	−0.819716	−0.409858	+	0.912149i	$0.634422\pi$
$812$	0	0
$813$	−21.4821	−0.753412
$814$	0	0
$815$	59.9511	2.10000
$816$	0	0
$817$	24.7466	0.865773
$818$	0	0
$819$	−9.35026	−0.326725
$820$	0	0
$821$	37.7196	1.31642	0.658211	−	0.752833i	$-0.271313\pi$
$821$	37.7196	1.31642	0.658211	+	0.752833i	$0.271313\pi$
$822$	0	0
$823$	14.5091	0.505757	0.252878	−	0.967498i	$-0.418623\pi$
$823$	14.5091	0.505757	0.252878	+	0.967498i	$0.418623\pi$
$824$	0	0
$825$	3.42548	0.119260
$826$	0	0
$827$	−5.67276	−0.197261	−0.0986306	−	0.995124i	$-0.531446\pi$
$827$	−5.67276	−0.197261	−0.0986306	+	0.995124i	$0.531446\pi$
$828$	0	0
$829$	−15.5223	−0.539112	−0.269556	−	0.962985i	$-0.586877\pi$
$829$	−15.5223	−0.539112	−0.269556	+	0.962985i	$0.586877\pi$
$830$	0	0
$831$	−15.0884	−0.523411
$832$	0	0
$833$	−0.481194	−0.0166724
$834$	0	0
$835$	19.1187	0.661630
$836$	0	0
$837$	3.96380	0.137009
$838$	0	0
$839$	16.7513	0.578319	0.289160	−	0.957281i	$-0.406624\pi$
$839$	16.7513	0.578319	0.289160	+	0.957281i	$0.406624\pi$
$840$	0	0
$841$	−11.7250	−0.404309
$842$	0	0
$843$	57.2262	1.97097
$844$	0	0
$845$	−10.4812	−0.360564
$846$	0	0
$847$	−9.57452	−0.328984
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	5.35026	0.183405
$852$	0	0
$853$	11.4861	0.393277	0.196639	−	0.980476i	$-0.436997\pi$
$853$	11.4861	0.393277	0.196639	+	0.980476i	$0.436997\pi$
$854$	0	0
$855$	33.7743	1.15506
$856$	0	0
$857$	49.4274	1.68841	0.844204	−	0.536022i	$-0.180074\pi$
$857$	49.4274	1.68841	0.844204	+	0.536022i	$0.180074\pi$
$858$	0	0
$859$	36.1051	1.23189	0.615945	−	0.787789i	$-0.288774\pi$
$859$	36.1051	1.23189	0.615945	+	0.787789i	$0.288774\pi$
$860$	0	0
$861$	29.5877	1.00835
$862$	0	0
$863$	42.5012	1.44676	0.723379	−	0.690451i	$-0.242588\pi$
$863$	42.5012	1.44676	0.723379	+	0.690451i	$0.242588\pi$
$864$	0	0
$865$	−13.7381	−0.467111
$866$	0	0
$867$	41.6058	1.41301
$868$	0	0
$869$	15.1998	0.515618
$870$	0	0
$871$	35.2360	1.19393
$872$	0	0
$873$	−0.294552	−0.00996907
$874$	0	0
$875$	−9.53690	−0.322406
$876$	0	0
$877$	13.0943	0.442162	0.221081	−	0.975255i	$-0.429041\pi$
$877$	13.0943	0.442162	0.221081	+	0.975255i	$0.429041\pi$
$878$	0	0
$879$	−24.8568	−0.838401
$880$	0	0
$881$	27.0919	0.912750	0.456375	−	0.889788i	$-0.349148\pi$
$881$	27.0919	0.912750	0.456375	+	0.889788i	$0.349148\pi$
$882$	0	0
$883$	−0.126008	−0.00424051	−0.00212025	−	0.999998i	$-0.500675\pi$
$883$	−0.126008	−0.00424051	−0.00212025	+	0.999998i	$0.500675\pi$
$884$	0	0
$885$	−71.2262	−2.39424
$886$	0	0
$887$	−25.2424	−0.847555	−0.423778	−	0.905766i	$-0.639296\pi$
$887$	−25.2424	−0.847555	−0.423778	+	0.905766i	$0.639296\pi$
$888$	0	0
$889$	−0.649738	−0.0217915
$890$	0	0
$891$	10.1563	0.340250
$892$	0	0
$893$	−0.726871	−0.0243238
$894$	0	0
$895$	−44.4142	−1.48460
$896$	0	0
$897$	−7.35026	−0.245418
$898$	0	0
$899$	42.4749	1.41662
$900$	0	0
$901$	−4.18664	−0.139477
$902$	0	0
$903$	−14.2374	−0.473792
$904$	0	0
$905$	−1.96968	−0.0654745
$906$	0	0
$907$	8.77575	0.291394	0.145697	−	0.989329i	$-0.453458\pi$
$907$	8.77575	0.291394	0.145697	+	0.989329i	$0.453458\pi$
$908$	0	0
$909$	−8.36344	−0.277398
$910$	0	0
$911$	−6.95651	−0.230479	−0.115240	−	0.993338i	$-0.536764\pi$
$911$	−6.95651	−0.230479	−0.115240	+	0.993338i	$0.536764\pi$
$912$	0	0
$913$	19.6267	0.649549
$914$	0	0
$915$	−55.7499	−1.84303
$916$	0	0
$917$	−11.4944	−0.379577
$918$	0	0
$919$	−29.9697	−0.988609	−0.494304	−	0.869289i	$-0.664577\pi$
$919$	−29.9697	−0.988609	−0.494304	+	0.869289i	$0.664577\pi$
$920$	0	0
$921$	−16.2677	−0.536040
$922$	0	0
$923$	2.29806	0.0756416
$924$	0	0
$925$	−6.18664	−0.203416
$926$	0	0
$927$	−6.55008	−0.215133
$928$	0	0
$929$	34.3536	1.12710	0.563552	−	0.826080i	$-0.309434\pi$
$929$	34.3536	1.12710	0.563552	+	0.826080i	$0.309434\pi$
$930$	0	0
$931$	4.31265	0.141341
$932$	0	0
$933$	−69.6444	−2.28005
$934$	0	0
$935$	1.42548	0.0466183
$936$	0	0
$937$	−9.96731	−0.325618	−0.162809	−	0.986658i	$-0.552055\pi$
$937$	−9.96731	−0.325618	−0.162809	+	0.986658i	$0.552055\pi$
$938$	0	0
$939$	−79.2320	−2.58564
$940$	0	0
$941$	7.62038	0.248417	0.124209	−	0.992256i	$-0.460361\pi$
$941$	7.62038	0.248417	0.124209	+	0.992256i	$0.460361\pi$
$942$	0	0
$943$	11.9248	0.388324
$944$	0	0
$945$	−0.962389	−0.0313065
$946$	0	0
$947$	16.1866	0.525995	0.262998	−	0.964796i	$-0.415289\pi$
$947$	16.1866	0.525995	0.262998	+	0.964796i	$0.415289\pi$
$948$	0	0
$949$	−26.3272	−0.854618
$950$	0	0
$951$	−23.0884	−0.748693
$952$	0	0
$953$	40.7123	1.31880	0.659400	−	0.751792i	$-0.270810\pi$
$953$	40.7123	1.31880	0.659400	+	0.751792i	$0.270810\pi$
$954$	0	0
$955$	−53.5633	−1.73327
$956$	0	0
$957$	−12.3127	−0.398011
$958$	0	0
$959$	−4.57452	−0.147719
$960$	0	0
$961$	73.4347	2.36886
$962$	0	0
$963$	7.93463	0.255690
$964$	0	0
$965$	−59.6747	−1.92100
$966$	0	0
$967$	−3.83780	−0.123415	−0.0617076	−	0.998094i	$-0.519655\pi$
$967$	−3.83780	−0.123415	−0.0617076	+	0.998094i	$0.519655\pi$
$968$	0	0
$969$	5.14903	0.165411
$970$	0	0
$971$	0.348847	0.0111950	0.00559752	−	0.999984i	$-0.498218\pi$
$971$	0.348847	0.0111950	0.00559752	+	0.999984i	$0.498218\pi$
$972$	0	0
$973$	−13.0557	−0.418547
$974$	0	0
$975$	8.49929	0.272195
$976$	0	0
$977$	−3.43724	−0.109967	−0.0549836	−	0.998487i	$-0.517511\pi$
$977$	−3.43724	−0.109967	−0.0549836	+	0.998487i	$0.517511\pi$
$978$	0	0
$979$	−16.4241	−0.524916
$980$	0	0
$981$	19.5271	0.623451
$982$	0	0
$983$	9.94921	0.317331	0.158665	−	0.987332i	$-0.449281\pi$
$983$	9.94921	0.317331	0.158665	+	0.987332i	$0.449281\pi$
$984$	0	0
$985$	16.4387	0.523779
$986$	0	0
$987$	0.418190	0.0133111
$988$	0	0
$989$	−5.73813	−0.182462
$990$	0	0
$991$	−10.2981	−0.327129	−0.163564	−	0.986533i	$-0.552299\pi$
$991$	−10.2981	−0.327129	−0.163564	+	0.986533i	$0.552299\pi$
$992$	0	0
$993$	60.2638	1.91241
$994$	0	0
$995$	−27.8496	−0.882890
$996$	0	0
$997$	−1.68735	−0.0534389	−0.0267194	−	0.999643i	$-0.508506\pi$
$997$	−1.68735	−0.0534389	−0.0267194	+	0.999643i	$0.508506\pi$
$998$	0	0
$999$	2.07522	0.0656571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.v.1.1		3
4.3	odd	2		161.2.a.c.1.2	✓	3
12.11	even	2		1449.2.a.m.1.2		3
20.19	odd	2		4025.2.a.j.1.2		3
28.27	even	2		1127.2.a.f.1.2		3
92.91	even	2		3703.2.a.c.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.c.1.2	✓	3	4.3	odd	2
1127.2.a.f.1.2		3	28.27	even	2
1449.2.a.m.1.2		3	12.11	even	2
2576.2.a.v.1.1		3	1.1	even	1	trivial
3703.2.a.c.1.2		3	92.91	even	2
4025.2.a.j.1.2		3	20.19	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:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.a (trivial)

\(f(q)\)	\(=\)	\(q-2.48119 q^{3} +2.48119 q^{5} +1.00000 q^{7} +3.15633 q^{9} +O(q^{10})\)	\(q-2.48119 q^{3} +2.48119 q^{5} +1.00000 q^{7} +3.15633 q^{9} -1.19394 q^{11} -2.96239 q^{13} -6.15633 q^{15} -0.481194 q^{17} +4.31265 q^{19} -2.48119 q^{21} -1.00000 q^{23} +1.15633 q^{25} -0.387873 q^{27} -4.15633 q^{29} -10.2193 q^{31} +2.96239 q^{33} +2.48119 q^{35} -5.35026 q^{37} +7.35026 q^{39} -11.9248 q^{41} +5.73813 q^{43} +7.83146 q^{45} -0.168544 q^{47} +1.00000 q^{49} +1.19394 q^{51} +8.70052 q^{53} -2.96239 q^{55} -10.7005 q^{57} +11.5696 q^{59} +9.05571 q^{61} +3.15633 q^{63} -7.35026 q^{65} -11.8945 q^{67} +2.48119 q^{69} -0.775746 q^{71} +8.88717 q^{73} -2.86907 q^{75} -1.19394 q^{77} -12.7308 q^{79} -8.50659 q^{81} -16.4387 q^{83} -1.19394 q^{85} +10.3127 q^{87} +13.7562 q^{89} -2.96239 q^{91} +25.3561 q^{93} +10.7005 q^{95} -0.0933212 q^{97} -3.76845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 2 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + 2 * q^5 + 3 * q^7 - q^9	\( 3 q - 2 q^{3} + 2 q^{5} + 3 q^{7} - q^{9} - 4 q^{11} + 2 q^{13} - 8 q^{15} + 4 q^{17} - 8 q^{19} - 2 q^{21} - 3 q^{23} - 7 q^{25} - 2 q^{27} - 2 q^{29} - 16 q^{31} - 2 q^{33} + 2 q^{35} - 6 q^{37} + 12 q^{39} - 14 q^{41} + 8 q^{43} + 8 q^{45} - 16 q^{47} + 3 q^{49} + 4 q^{51} + 6 q^{53} + 2 q^{55} - 12 q^{57} + 10 q^{59} + 10 q^{61} - q^{63} - 12 q^{65} - 16 q^{67} + 2 q^{69} - 4 q^{71} - 6 q^{73} - 4 q^{75} - 4 q^{77} - 16 q^{79} - 5 q^{81} - 20 q^{83} - 4 q^{85} + 10 q^{87} + 4 q^{89} + 2 q^{91} + 12 q^{93} + 12 q^{95} + 6 q^{97}+O(q^{100}) \) 3 * q - 2 * q^3 + 2 * q^5 + 3 * q^7 - q^9 - 4 * q^11 + 2 * q^13 - 8 * q^15 + 4 * q^17 - 8 * q^19 - 2 * q^21 - 3 * q^23 - 7 * q^25 - 2 * q^27 - 2 * q^29 - 16 * q^31 - 2 * q^33 + 2 * q^35 - 6 * q^37 + 12 * q^39 - 14 * q^41 + 8 * q^43 + 8 * q^45 - 16 * q^47 + 3 * q^49 + 4 * q^51 + 6 * q^53 + 2 * q^55 - 12 * q^57 + 10 * q^59 + 10 * q^61 - q^63 - 12 * q^65 - 16 * q^67 + 2 * q^69 - 4 * q^71 - 6 * q^73 - 4 * q^75 - 4 * q^77 - 16 * q^79 - 5 * q^81 - 20 * q^83 - 4 * q^85 + 10 * q^87 + 4 * q^89 + 2 * q^91 + 12 * q^93 + 12 * q^95 + 6 * q^97

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