LMFDB - Embedded newform 1232.4.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1232.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:	$72.6903531271$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 144x^{5} + 354x^{4} + 5172x^{3} - 6504x^{2} - 34432x + 18816 $ x^7 - 3x^6 - 144x^5 + 354x^4 + 5172x^3 - 6504x^2 - 34432x + 18816
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.14310$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.14310 q^{3} +18.9749 q^{5} -7.00000 q^{7} +39.3101 q^{9} +O(q^{10})$	$q-8.14310 q^{3} +18.9749 q^{5} -7.00000 q^{7} +39.3101 q^{9} -11.0000 q^{11} -53.8809 q^{13} -154.514 q^{15} +36.8135 q^{17} -1.35913 q^{19} +57.0017 q^{21} -26.7631 q^{23} +235.046 q^{25} -100.242 q^{27} +116.072 q^{29} +158.980 q^{31} +89.5741 q^{33} -132.824 q^{35} +162.513 q^{37} +438.758 q^{39} -454.155 q^{41} -384.147 q^{43} +745.903 q^{45} +264.345 q^{47} +49.0000 q^{49} -299.776 q^{51} +113.839 q^{53} -208.724 q^{55} +11.0675 q^{57} +602.586 q^{59} -701.481 q^{61} -275.170 q^{63} -1022.38 q^{65} -422.818 q^{67} +217.934 q^{69} -825.854 q^{71} -121.572 q^{73} -1914.00 q^{75} +77.0000 q^{77} +554.288 q^{79} -245.091 q^{81} +880.351 q^{83} +698.530 q^{85} -945.182 q^{87} -133.898 q^{89} +377.167 q^{91} -1294.59 q^{93} -25.7893 q^{95} +1379.84 q^{97} -432.411 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} + 11 q^{5} - 49 q^{7} + 108 q^{9}+O(q^{10}) $ 7 * q - 3 * q^3 + 11 * q^5 - 49 * q^7 + 108 * q^9	$ 7 q - 3 q^{3} + 11 q^{5} - 49 q^{7} + 108 q^{9} - 77 q^{11} + 26 q^{13} - 67 q^{15} + 44 q^{17} - 34 q^{19} + 21 q^{21} + 29 q^{23} + 182 q^{25} - 99 q^{27} + 94 q^{29} + 173 q^{31} + 33 q^{33} - 77 q^{35} + 255 q^{37} + 60 q^{39} + 508 q^{41} - 656 q^{43} + 466 q^{45} + 18 q^{47} + 343 q^{49} - 850 q^{51} + 1806 q^{53} - 121 q^{55} + 1154 q^{57} - 665 q^{59} + 608 q^{61} - 756 q^{63} + 588 q^{65} - 669 q^{67} + 2067 q^{69} - 1169 q^{71} + 380 q^{73} - 1254 q^{75} + 539 q^{77} + 110 q^{79} + 2847 q^{81} + 496 q^{83} + 3446 q^{85} + 582 q^{87} + 1321 q^{89} - 182 q^{91} + 1493 q^{93} + 50 q^{95} + 3927 q^{97} - 1188 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 + 11 * q^5 - 49 * q^7 + 108 * q^9 - 77 * q^11 + 26 * q^13 - 67 * q^15 + 44 * q^17 - 34 * q^19 + 21 * q^21 + 29 * q^23 + 182 * q^25 - 99 * q^27 + 94 * q^29 + 173 * q^31 + 33 * q^33 - 77 * q^35 + 255 * q^37 + 60 * q^39 + 508 * q^41 - 656 * q^43 + 466 * q^45 + 18 * q^47 + 343 * q^49 - 850 * q^51 + 1806 * q^53 - 121 * q^55 + 1154 * q^57 - 665 * q^59 + 608 * q^61 - 756 * q^63 + 588 * q^65 - 669 * q^67 + 2067 * q^69 - 1169 * q^71 + 380 * q^73 - 1254 * q^75 + 539 * q^77 + 110 * q^79 + 2847 * q^81 + 496 * q^83 + 3446 * q^85 + 582 * q^87 + 1321 * q^89 - 182 * q^91 + 1493 * q^93 + 50 * q^95 + 3927 * q^97 - 1188 * q^99

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
$3$	−8.14310	−1.56714	−0.783570	−	0.621303i	$-0.786603\pi$
$3$	−8.14310	−1.56714	−0.783570	+	0.621303i	$0.786603\pi$
$4$	0	0
$5$	18.9749	1.69716	0.848582	−	0.529064i	$-0.177457\pi$
$5$	18.9749	1.69716	0.848582	+	0.529064i	$0.177457\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	39.3101	1.45593
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−53.8809	−1.14953	−0.574765	−	0.818319i	$-0.694906\pi$
$13$	−53.8809	−1.14953	−0.574765	+	0.818319i	$0.694906\pi$
$14$	0	0
$15$	−154.514	−2.65969
$16$	0	0
$17$	36.8135	0.525210	0.262605	−	0.964903i	$-0.415418\pi$
$17$	36.8135	0.525210	0.262605	+	0.964903i	$0.415418\pi$
$18$	0	0
$19$	−1.35913	−0.0164108	−0.00820541	−	0.999966i	$-0.502612\pi$
$19$	−1.35913	−0.0164108	−0.00820541	+	0.999966i	$0.502612\pi$
$20$	0	0
$21$	57.0017	0.592323
$22$	0	0
$23$	−26.7631	−0.242630	−0.121315	−	0.992614i	$-0.538711\pi$
$23$	−26.7631	−0.242630	−0.121315	+	0.992614i	$0.538711\pi$
$24$	0	0
$25$	235.046	1.88036
$26$	0	0
$27$	−100.242	−0.714503
$28$	0	0
$29$	116.072	0.743240	0.371620	−	0.928385i	$-0.378803\pi$
$29$	116.072	0.743240	0.371620	+	0.928385i	$0.378803\pi$
$30$	0	0
$31$	158.980	0.921087	0.460543	−	0.887637i	$-0.347655\pi$
$31$	158.980	0.921087	0.460543	+	0.887637i	$0.347655\pi$
$32$	0	0
$33$	89.5741	0.472511
$34$	0	0
$35$	−132.824	−0.641468
$36$	0	0
$37$	162.513	0.722081	0.361041	−	0.932550i	$-0.382422\pi$
$37$	162.513	0.722081	0.361041	+	0.932550i	$0.382422\pi$
$38$	0	0
$39$	438.758	1.80147
$40$	0	0
$41$	−454.155	−1.72993	−0.864964	−	0.501833i	$-0.832659\pi$
$41$	−454.155	−1.72993	−0.864964	+	0.501833i	$0.832659\pi$
$42$	0	0
$43$	−384.147	−1.36237	−0.681185	−	0.732111i	$-0.738535\pi$
$43$	−384.147	−1.36237	−0.681185	+	0.732111i	$0.738535\pi$
$44$	0	0
$45$	745.903	2.47095
$46$	0	0
$47$	264.345	0.820396	0.410198	−	0.911997i	$-0.365460\pi$
$47$	264.345	0.820396	0.410198	+	0.911997i	$0.365460\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−299.776	−0.823078
$52$	0	0
$53$	113.839	0.295037	0.147518	−	0.989059i	$-0.452871\pi$
$53$	113.839	0.295037	0.147518	+	0.989059i	$0.452871\pi$
$54$	0	0
$55$	−208.724	−0.511714
$56$	0	0
$57$	11.0675	0.0257181
$58$	0	0
$59$	602.586	1.32966	0.664831	−	0.746994i	$-0.268504\pi$
$59$	602.586	1.32966	0.664831	+	0.746994i	$0.268504\pi$
$60$	0	0
$61$	−701.481	−1.47238	−0.736192	−	0.676773i	$-0.763378\pi$
$61$	−701.481	−1.47238	−0.736192	+	0.676773i	$0.763378\pi$
$62$	0	0
$63$	−275.170	−0.550289
$64$	0	0
$65$	−1022.38	−1.95094
$66$	0	0
$67$	−422.818	−0.770976	−0.385488	−	0.922713i	$-0.625967\pi$
$67$	−422.818	−0.770976	−0.385488	+	0.922713i	$0.625967\pi$
$68$	0	0
$69$	217.934	0.380235
$70$	0	0
$71$	−825.854	−1.38043	−0.690217	−	0.723602i	$-0.742485\pi$
$71$	−825.854	−1.38043	−0.690217	+	0.723602i	$0.742485\pi$
$72$	0	0
$73$	−121.572	−0.194917	−0.0974583	−	0.995240i	$-0.531071\pi$
$73$	−121.572	−0.194917	−0.0974583	+	0.995240i	$0.531071\pi$
$74$	0	0
$75$	−1914.00	−2.94679
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	554.288	0.789396	0.394698	−	0.918811i	$-0.370849\pi$
$79$	554.288	0.789396	0.394698	+	0.918811i	$0.370849\pi$
$80$	0	0
$81$	−245.091	−0.336201
$82$	0	0
$83$	880.351	1.16423	0.582115	−	0.813106i	$-0.302225\pi$
$83$	880.351	1.16423	0.582115	+	0.813106i	$0.302225\pi$
$84$	0	0
$85$	698.530	0.891368
$86$	0	0
$87$	−945.182	−1.16476
$88$	0	0
$89$	−133.898	−0.159474	−0.0797371	−	0.996816i	$-0.525408\pi$
$89$	−133.898	−0.159474	−0.0797371	+	0.996816i	$0.525408\pi$
$90$	0	0
$91$	377.167	0.434481
$92$	0	0
$93$	−1294.59	−1.44347
$94$	0	0
$95$	−25.7893	−0.0278518
$96$	0	0
$97$	1379.84	1.44434	0.722171	−	0.691714i	$-0.243144\pi$
$97$	1379.84	1.44434	0.722171	+	0.691714i	$0.243144\pi$
$98$	0	0
$99$	−432.411	−0.438979
$100$	0	0
$101$	−106.020	−0.104449	−0.0522247	−	0.998635i	$-0.516631\pi$
$101$	−106.020	−0.104449	−0.0522247	+	0.998635i	$0.516631\pi$
$102$	0	0
$103$	224.181	0.214459	0.107229	−	0.994234i	$-0.465802\pi$
$103$	224.181	0.214459	0.107229	+	0.994234i	$0.465802\pi$
$104$	0	0
$105$	1081.60	1.00527
$106$	0	0
$107$	1966.08	1.77634	0.888171	−	0.459514i	$-0.151976\pi$
$107$	1966.08	1.77634	0.888171	+	0.459514i	$0.151976\pi$
$108$	0	0
$109$	−605.688	−0.532242	−0.266121	−	0.963940i	$-0.585742\pi$
$109$	−605.688	−0.532242	−0.266121	+	0.963940i	$0.585742\pi$
$110$	0	0
$111$	−1323.36	−1.13160
$112$	0	0
$113$	1299.94	1.08220	0.541099	−	0.840959i	$-0.318008\pi$
$113$	1299.94	1.08220	0.541099	+	0.840959i	$0.318008\pi$
$114$	0	0
$115$	−507.826	−0.411782
$116$	0	0
$117$	−2118.06	−1.67363
$118$	0	0
$119$	−257.694	−0.198511
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	3698.23	2.71104
$124$	0	0
$125$	2088.10	1.49412
$126$	0	0
$127$	1507.17	1.05307	0.526533	−	0.850155i	$-0.323492\pi$
$127$	1507.17	1.05307	0.526533	+	0.850155i	$0.323492\pi$
$128$	0	0
$129$	3128.15	2.13503
$130$	0	0
$131$	−355.149	−0.236866	−0.118433	−	0.992962i	$-0.537787\pi$
$131$	−355.149	−0.236866	−0.118433	+	0.992962i	$0.537787\pi$
$132$	0	0
$133$	9.51390	0.00620271
$134$	0	0
$135$	−1902.08	−1.21263
$136$	0	0
$137$	2833.61	1.76710	0.883548	−	0.468341i	$-0.155148\pi$
$137$	2833.61	1.76710	0.883548	+	0.468341i	$0.155148\pi$
$138$	0	0
$139$	−1711.61	−1.04444	−0.522220	−	0.852811i	$-0.674896\pi$
$139$	−1711.61	−1.04444	−0.522220	+	0.852811i	$0.674896\pi$
$140$	0	0
$141$	−2152.58	−1.28568
$142$	0	0
$143$	592.690	0.346596
$144$	0	0
$145$	2202.44	1.26140
$146$	0	0
$147$	−399.012	−0.223877
$148$	0	0
$149$	1368.34	0.752342	0.376171	−	0.926550i	$-0.377241\pi$
$149$	1368.34	0.752342	0.376171	+	0.926550i	$0.377241\pi$
$150$	0	0
$151$	−647.934	−0.349193	−0.174597	−	0.984640i	$-0.555862\pi$
$151$	−647.934	−0.349193	−0.174597	+	0.984640i	$0.555862\pi$
$152$	0	0
$153$	1447.14	0.764668
$154$	0	0
$155$	3016.63	1.56324
$156$	0	0
$157$	2143.67	1.08970	0.544851	−	0.838533i	$-0.316586\pi$
$157$	2143.67	1.08970	0.544851	+	0.838533i	$0.316586\pi$
$158$	0	0
$159$	−926.999	−0.462364
$160$	0	0
$161$	187.341	0.0917055
$162$	0	0
$163$	−2259.29	−1.08565	−0.542827	−	0.839845i	$-0.682646\pi$
$163$	−2259.29	−1.08565	−0.542827	+	0.839845i	$0.682646\pi$
$164$	0	0
$165$	1699.66	0.801928
$166$	0	0
$167$	555.905	0.257588	0.128794	−	0.991671i	$-0.458889\pi$
$167$	555.905	0.257588	0.128794	+	0.991671i	$0.458889\pi$
$168$	0	0
$169$	706.155	0.321418
$170$	0	0
$171$	−53.4274	−0.0238930
$172$	0	0
$173$	−2732.45	−1.20083	−0.600417	−	0.799687i	$-0.704999\pi$
$173$	−2732.45	−1.20083	−0.600417	+	0.799687i	$0.704999\pi$
$174$	0	0
$175$	−1645.32	−0.710711
$176$	0	0
$177$	−4906.92	−2.08377
$178$	0	0
$179$	1420.98	0.593348	0.296674	−	0.954979i	$-0.404123\pi$
$179$	1420.98	0.593348	0.296674	+	0.954979i	$0.404123\pi$
$180$	0	0
$181$	−1475.55	−0.605949	−0.302974	−	0.952999i	$-0.597980\pi$
$181$	−1475.55	−0.605949	−0.302974	+	0.952999i	$0.597980\pi$
$182$	0	0
$183$	5712.23	2.30743
$184$	0	0
$185$	3083.67	1.22549
$186$	0	0
$187$	−404.948	−0.158357
$188$	0	0
$189$	701.694	0.270057
$190$	0	0
$191$	4637.56	1.75687	0.878434	−	0.477863i	$-0.158589\pi$
$191$	4637.56	1.75687	0.878434	+	0.477863i	$0.158589\pi$
$192$	0	0
$193$	−3062.59	−1.14223	−0.571115	−	0.820870i	$-0.693489\pi$
$193$	−3062.59	−1.14223	−0.571115	+	0.820870i	$0.693489\pi$
$194$	0	0
$195$	8325.37	3.05740
$196$	0	0
$197$	3995.67	1.44508	0.722538	−	0.691331i	$-0.242975\pi$
$197$	3995.67	1.44508	0.722538	+	0.691331i	$0.242975\pi$
$198$	0	0
$199$	−1715.61	−0.611137	−0.305568	−	0.952170i	$-0.598846\pi$
$199$	−1715.61	−0.611137	−0.305568	+	0.952170i	$0.598846\pi$
$200$	0	0
$201$	3443.05	1.20823
$202$	0	0
$203$	−812.501	−0.280918
$204$	0	0
$205$	−8617.53	−2.93597
$206$	0	0
$207$	−1052.06	−0.353252
$208$	0	0
$209$	14.9504	0.00494805
$210$	0	0
$211$	−2763.66	−0.901698	−0.450849	−	0.892600i	$-0.648879\pi$
$211$	−2763.66	−0.901698	−0.450849	+	0.892600i	$0.648879\pi$
$212$	0	0
$213$	6725.01	2.16333
$214$	0	0
$215$	−7289.14	−2.31217
$216$	0	0
$217$	−1112.86	−0.348138
$218$	0	0
$219$	989.972	0.305462
$220$	0	0
$221$	−1983.54	−0.603745
$222$	0	0
$223$	5992.23	1.79941	0.899707	−	0.436495i	$-0.143780\pi$
$223$	5992.23	1.79941	0.899707	+	0.436495i	$0.143780\pi$
$224$	0	0
$225$	9239.65	2.73768
$226$	0	0
$227$	5352.72	1.56508	0.782539	−	0.622602i	$-0.213925\pi$
$227$	5352.72	1.56508	0.782539	+	0.622602i	$0.213925\pi$
$228$	0	0
$229$	5721.40	1.65101	0.825505	−	0.564395i	$-0.190891\pi$
$229$	5721.40	1.65101	0.825505	+	0.564395i	$0.190891\pi$
$230$	0	0
$231$	−627.019	−0.178592
$232$	0	0
$233$	4894.91	1.37629	0.688146	−	0.725572i	$-0.258425\pi$
$233$	4894.91	1.37629	0.688146	+	0.725572i	$0.258425\pi$
$234$	0	0
$235$	5015.90	1.39235
$236$	0	0
$237$	−4513.63	−1.23709
$238$	0	0
$239$	3506.15	0.948928	0.474464	−	0.880275i	$-0.342642\pi$
$239$	3506.15	0.948928	0.474464	+	0.880275i	$0.342642\pi$
$240$	0	0
$241$	4986.49	1.33281	0.666407	−	0.745588i	$-0.267831\pi$
$241$	4986.49	1.33281	0.666407	+	0.745588i	$0.267831\pi$
$242$	0	0
$243$	4702.33	1.24138
$244$	0	0
$245$	929.768	0.242452
$246$	0	0
$247$	73.2311	0.0188647
$248$	0	0
$249$	−7168.79	−1.82451
$250$	0	0
$251$	6365.99	1.60087	0.800434	−	0.599421i	$-0.204603\pi$
$251$	6365.99	1.60087	0.800434	+	0.599421i	$0.204603\pi$
$252$	0	0
$253$	294.394	0.0731556
$254$	0	0
$255$	−5688.20	−1.39690
$256$	0	0
$257$	4979.04	1.20850	0.604250	−	0.796795i	$-0.293473\pi$
$257$	4979.04	1.20850	0.604250	+	0.796795i	$0.293473\pi$
$258$	0	0
$259$	−1137.59	−0.272921
$260$	0	0
$261$	4562.78	1.08210
$262$	0	0
$263$	−4623.46	−1.08401	−0.542005	−	0.840375i	$-0.682335\pi$
$263$	−4623.46	−1.08401	−0.542005	+	0.840375i	$0.682335\pi$
$264$	0	0
$265$	2160.07	0.500725
$266$	0	0
$267$	1090.35	0.249919
$268$	0	0
$269$	−6374.15	−1.44475	−0.722377	−	0.691500i	$-0.756950\pi$
$269$	−6374.15	−1.44475	−0.722377	+	0.691500i	$0.756950\pi$
$270$	0	0
$271$	742.727	0.166485	0.0832425	−	0.996529i	$-0.473472\pi$
$271$	742.727	0.166485	0.0832425	+	0.996529i	$0.473472\pi$
$272$	0	0
$273$	−3071.30	−0.680893
$274$	0	0
$275$	−2585.50	−0.566951
$276$	0	0
$277$	−1717.47	−0.372538	−0.186269	−	0.982499i	$-0.559640\pi$
$277$	−1717.47	−0.372538	−0.186269	+	0.982499i	$0.559640\pi$
$278$	0	0
$279$	6249.52	1.34104
$280$	0	0
$281$	3964.04	0.841547	0.420774	−	0.907166i	$-0.361759\pi$
$281$	3964.04	0.841547	0.420774	+	0.907166i	$0.361759\pi$
$282$	0	0
$283$	6457.54	1.35640	0.678200	−	0.734878i	$-0.262760\pi$
$283$	6457.54	1.35640	0.678200	+	0.734878i	$0.262760\pi$
$284$	0	0
$285$	210.005	0.0436477
$286$	0	0
$287$	3179.08	0.653852
$288$	0	0
$289$	−3557.77	−0.724154
$290$	0	0
$291$	−11236.2	−2.26349
$292$	0	0
$293$	−3898.02	−0.777217	−0.388608	−	0.921403i	$-0.627044\pi$
$293$	−3898.02	−0.777217	−0.388608	+	0.921403i	$0.627044\pi$
$294$	0	0
$295$	11434.0	2.25665
$296$	0	0
$297$	1102.66	0.215431
$298$	0	0
$299$	1442.02	0.278910
$300$	0	0
$301$	2689.03	0.514928
$302$	0	0
$303$	863.331	0.163687
$304$	0	0
$305$	−13310.5	−2.49888
$306$	0	0
$307$	7547.15	1.40306	0.701529	−	0.712641i	$-0.252501\pi$
$307$	7547.15	1.40306	0.701529	+	0.712641i	$0.252501\pi$
$308$	0	0
$309$	−1825.53	−0.336087
$310$	0	0
$311$	10294.4	1.87699	0.938493	−	0.345299i	$-0.112223\pi$
$311$	10294.4	1.87699	0.938493	+	0.345299i	$0.112223\pi$
$312$	0	0
$313$	−4608.96	−0.832312	−0.416156	−	0.909293i	$-0.636623\pi$
$313$	−4608.96	−0.832312	−0.416156	+	0.909293i	$0.636623\pi$
$314$	0	0
$315$	−5221.32	−0.933931
$316$	0	0
$317$	−9011.70	−1.59668	−0.798340	−	0.602207i	$-0.794288\pi$
$317$	−9011.70	−1.59668	−0.798340	+	0.602207i	$0.794288\pi$
$318$	0	0
$319$	−1276.79	−0.224095
$320$	0	0
$321$	−16010.0	−2.78378
$322$	0	0
$323$	−50.0342	−0.00861913
$324$	0	0
$325$	−12664.5	−2.16153
$326$	0	0
$327$	4932.18	0.834098
$328$	0	0
$329$	−1850.41	−0.310081
$330$	0	0
$331$	−7592.35	−1.26076	−0.630382	−	0.776285i	$-0.717102\pi$
$331$	−7592.35	−1.26076	−0.630382	+	0.776285i	$0.717102\pi$
$332$	0	0
$333$	6388.40	1.05130
$334$	0	0
$335$	−8022.91	−1.30847
$336$	0	0
$337$	6806.77	1.10026	0.550131	−	0.835078i	$-0.314578\pi$
$337$	6806.77	1.10026	0.550131	+	0.835078i	$0.314578\pi$
$338$	0	0
$339$	−10585.6	−1.69596
$340$	0	0
$341$	−1748.78	−0.277718
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	4135.27	0.645321
$346$	0	0
$347$	−9174.37	−1.41933	−0.709663	−	0.704541i	$-0.751153\pi$
$347$	−9174.37	−1.41933	−0.709663	+	0.704541i	$0.751153\pi$
$348$	0	0
$349$	−6193.51	−0.949946	−0.474973	−	0.880000i	$-0.657542\pi$
$349$	−6193.51	−0.949946	−0.474973	+	0.880000i	$0.657542\pi$
$350$	0	0
$351$	5401.14	0.821343
$352$	0	0
$353$	8187.08	1.23443	0.617216	−	0.786794i	$-0.288261\pi$
$353$	8187.08	1.23443	0.617216	+	0.786794i	$0.288261\pi$
$354$	0	0
$355$	−15670.5	−2.34282
$356$	0	0
$357$	2098.43	0.311094
$358$	0	0
$359$	9145.13	1.34446	0.672231	−	0.740342i	$-0.265336\pi$
$359$	9145.13	1.34446	0.672231	+	0.740342i	$0.265336\pi$
$360$	0	0
$361$	−6857.15	−0.999731
$362$	0	0
$363$	−985.315	−0.142467
$364$	0	0
$365$	−2306.81	−0.330805
$366$	0	0
$367$	−1830.84	−0.260407	−0.130203	−	0.991487i	$-0.541563\pi$
$367$	−1830.84	−0.260407	−0.130203	+	0.991487i	$0.541563\pi$
$368$	0	0
$369$	−17852.9	−2.51865
$370$	0	0
$371$	−796.870	−0.111513
$372$	0	0
$373$	−4555.19	−0.632330	−0.316165	−	0.948704i	$-0.602395\pi$
$373$	−4555.19	−0.632330	−0.316165	+	0.948704i	$0.602395\pi$
$374$	0	0
$375$	−17003.6	−2.34150
$376$	0	0
$377$	−6254.04	−0.854376
$378$	0	0
$379$	5407.86	0.732937	0.366468	−	0.930431i	$-0.380567\pi$
$379$	5407.86	0.732937	0.366468	+	0.930431i	$0.380567\pi$
$380$	0	0
$381$	−12273.0	−1.65030
$382$	0	0
$383$	4749.82	0.633693	0.316846	−	0.948477i	$-0.397376\pi$
$383$	4749.82	0.633693	0.316846	+	0.948477i	$0.397376\pi$
$384$	0	0
$385$	1461.06	0.193410
$386$	0	0
$387$	−15100.9	−1.98351
$388$	0	0
$389$	−637.179	−0.0830495	−0.0415248	−	0.999137i	$-0.513222\pi$
$389$	−637.179	−0.0830495	−0.0415248	+	0.999137i	$0.513222\pi$
$390$	0	0
$391$	−985.241	−0.127432
$392$	0	0
$393$	2892.01	0.371203
$394$	0	0
$395$	10517.5	1.33973
$396$	0	0
$397$	4758.83	0.601609	0.300804	−	0.953686i	$-0.402745\pi$
$397$	4758.83	0.601609	0.300804	+	0.953686i	$0.402745\pi$
$398$	0	0
$399$	−77.4726	−0.00972051
$400$	0	0
$401$	7437.61	0.926225	0.463113	−	0.886299i	$-0.346733\pi$
$401$	7437.61	0.926225	0.463113	+	0.886299i	$0.346733\pi$
$402$	0	0
$403$	−8566.00	−1.05882
$404$	0	0
$405$	−4650.56	−0.570588
$406$	0	0
$407$	−1787.65	−0.217716
$408$	0	0
$409$	11562.8	1.39790	0.698952	−	0.715169i	$-0.253650\pi$
$409$	11562.8	1.39790	0.698952	+	0.715169i	$0.253650\pi$
$410$	0	0
$411$	−23074.4	−2.76929
$412$	0	0
$413$	−4218.10	−0.502565
$414$	0	0
$415$	16704.6	1.97589
$416$	0	0
$417$	13937.9	1.63679
$418$	0	0
$419$	2161.76	0.252050	0.126025	−	0.992027i	$-0.459778\pi$
$419$	2161.76	0.252050	0.126025	+	0.992027i	$0.459778\pi$
$420$	0	0
$421$	−10348.2	−1.19796	−0.598981	−	0.800763i	$-0.704427\pi$
$421$	−10348.2	−1.19796	−0.598981	+	0.800763i	$0.704427\pi$
$422$	0	0
$423$	10391.4	1.19444
$424$	0	0
$425$	8652.84	0.987586
$426$	0	0
$427$	4910.36	0.556509
$428$	0	0
$429$	−4826.34	−0.543165
$430$	0	0
$431$	−7772.37	−0.868635	−0.434317	−	0.900760i	$-0.643010\pi$
$431$	−7772.37	−0.868635	−0.434317	+	0.900760i	$0.643010\pi$
$432$	0	0
$433$	6478.56	0.719029	0.359515	−	0.933139i	$-0.382942\pi$
$433$	6478.56	0.719029	0.359515	+	0.933139i	$0.382942\pi$
$434$	0	0
$435$	−17934.7	−1.97679
$436$	0	0
$437$	36.3745	0.00398175
$438$	0	0
$439$	5911.78	0.642719	0.321360	−	0.946957i	$-0.395860\pi$
$439$	5911.78	0.642719	0.321360	+	0.946957i	$0.395860\pi$
$440$	0	0
$441$	1926.19	0.207990
$442$	0	0
$443$	−10767.2	−1.15477	−0.577385	−	0.816472i	$-0.695927\pi$
$443$	−10767.2	−1.15477	−0.577385	+	0.816472i	$0.695927\pi$
$444$	0	0
$445$	−2540.71	−0.270654
$446$	0	0
$447$	−11142.5	−1.17902
$448$	0	0
$449$	14146.3	1.48687	0.743436	−	0.668807i	$-0.233195\pi$
$449$	14146.3	1.48687	0.743436	+	0.668807i	$0.233195\pi$
$450$	0	0
$451$	4995.70	0.521593
$452$	0	0
$453$	5276.19	0.547235
$454$	0	0
$455$	7156.68	0.737386
$456$	0	0
$457$	2008.34	0.205571	0.102786	−	0.994704i	$-0.467224\pi$
$457$	2008.34	0.205571	0.102786	+	0.994704i	$0.467224\pi$
$458$	0	0
$459$	−3690.26	−0.375265
$460$	0	0
$461$	12197.8	1.23234	0.616169	−	0.787614i	$-0.288684\pi$
$461$	12197.8	1.23234	0.616169	+	0.787614i	$0.288684\pi$
$462$	0	0
$463$	−4406.36	−0.442291	−0.221146	−	0.975241i	$-0.570980\pi$
$463$	−4406.36	−0.442291	−0.221146	+	0.975241i	$0.570980\pi$
$464$	0	0
$465$	−24564.7	−2.44981
$466$	0	0
$467$	−4904.56	−0.485987	−0.242993	−	0.970028i	$-0.578129\pi$
$467$	−4904.56	−0.485987	−0.242993	+	0.970028i	$0.578129\pi$
$468$	0	0
$469$	2959.72	0.291402
$470$	0	0
$471$	−17456.1	−1.70772
$472$	0	0
$473$	4225.62	0.410770
$474$	0	0
$475$	−319.457	−0.0308583
$476$	0	0
$477$	4475.00	0.429552
$478$	0	0
$479$	1144.86	0.109207	0.0546034	−	0.998508i	$-0.482611\pi$
$479$	1144.86	0.109207	0.0546034	+	0.998508i	$0.482611\pi$
$480$	0	0
$481$	−8756.36	−0.830054
$482$	0	0
$483$	−1525.54	−0.143715
$484$	0	0
$485$	26182.2	2.45129
$486$	0	0
$487$	12394.1	1.15324	0.576622	−	0.817011i	$-0.304370\pi$
$487$	12394.1	1.15324	0.576622	+	0.817011i	$0.304370\pi$
$488$	0	0
$489$	18397.7	1.70137
$490$	0	0
$491$	−19092.1	−1.75481	−0.877406	−	0.479748i	$-0.840728\pi$
$491$	−19092.1	−1.75481	−0.877406	+	0.479748i	$0.840728\pi$
$492$	0	0
$493$	4272.99	0.390357
$494$	0	0
$495$	−8204.93	−0.745019
$496$	0	0
$497$	5780.98	0.521755
$498$	0	0
$499$	−8243.92	−0.739576	−0.369788	−	0.929116i	$-0.620570\pi$
$499$	−8243.92	−0.739576	−0.369788	+	0.929116i	$0.620570\pi$
$500$	0	0
$501$	−4526.79	−0.403677
$502$	0	0
$503$	−13961.1	−1.23757	−0.618784	−	0.785561i	$-0.712374\pi$
$503$	−13961.1	−1.23757	−0.618784	+	0.785561i	$0.712374\pi$
$504$	0	0
$505$	−2011.72	−0.177268
$506$	0	0
$507$	−5750.29	−0.503707
$508$	0	0
$509$	3893.22	0.339025	0.169513	−	0.985528i	$-0.445781\pi$
$509$	3893.22	0.339025	0.169513	+	0.985528i	$0.445781\pi$
$510$	0	0
$511$	851.003	0.0736716
$512$	0	0
$513$	136.242	0.0117256
$514$	0	0
$515$	4253.81	0.363971
$516$	0	0
$517$	−2907.79	−0.247359
$518$	0	0
$519$	22250.6	1.88188
$520$	0	0
$521$	22225.0	1.86889	0.934446	−	0.356104i	$-0.115895\pi$
$521$	22225.0	1.86889	0.934446	+	0.356104i	$0.115895\pi$
$522$	0	0
$523$	−16966.7	−1.41855	−0.709274	−	0.704932i	$-0.750977\pi$
$523$	−16966.7	−1.41855	−0.709274	+	0.704932i	$0.750977\pi$
$524$	0	0
$525$	13398.0	1.11378
$526$	0	0
$527$	5852.61	0.483764
$528$	0	0
$529$	−11450.7	−0.941131
$530$	0	0
$531$	23687.7	1.93589
$532$	0	0
$533$	24470.3	1.98860
$534$	0	0
$535$	37306.2	3.01474
$536$	0	0
$537$	−11571.2	−0.929859
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	1639.18	0.130266	0.0651328	−	0.997877i	$-0.479253\pi$
$541$	1639.18	0.130266	0.0651328	+	0.997877i	$0.479253\pi$
$542$	0	0
$543$	12015.5	0.949606
$544$	0	0
$545$	−11492.8	−0.903302
$546$	0	0
$547$	16057.3	1.25514	0.627570	−	0.778560i	$-0.284050\pi$
$547$	16057.3	1.25514	0.627570	+	0.778560i	$0.284050\pi$
$548$	0	0
$549$	−27575.2	−2.14368
$550$	0	0
$551$	−157.756	−0.0121972
$552$	0	0
$553$	−3880.02	−0.298364
$554$	0	0
$555$	−25110.6	−1.92051
$556$	0	0
$557$	−8096.11	−0.615876	−0.307938	−	0.951406i	$-0.599639\pi$
$557$	−8096.11	−0.615876	−0.307938	+	0.951406i	$0.599639\pi$
$558$	0	0
$559$	20698.2	1.56608
$560$	0	0
$561$	3297.53	0.248167
$562$	0	0
$563$	−9347.28	−0.699717	−0.349859	−	0.936802i	$-0.613770\pi$
$563$	−9347.28	−0.699717	−0.349859	+	0.936802i	$0.613770\pi$
$564$	0	0
$565$	24666.3	1.83667
$566$	0	0
$567$	1715.63	0.127072
$568$	0	0
$569$	13078.4	0.963580	0.481790	−	0.876287i	$-0.339987\pi$
$569$	13078.4	0.963580	0.481790	+	0.876287i	$0.339987\pi$
$570$	0	0
$571$	−1978.17	−0.144981	−0.0724904	−	0.997369i	$-0.523095\pi$
$571$	−1978.17	−0.144981	−0.0724904	+	0.997369i	$0.523095\pi$
$572$	0	0
$573$	−37764.1	−2.75326
$574$	0	0
$575$	−6290.54	−0.456232
$576$	0	0
$577$	−2917.19	−0.210475	−0.105238	−	0.994447i	$-0.533560\pi$
$577$	−2917.19	−0.210475	−0.105238	+	0.994447i	$0.533560\pi$
$578$	0	0
$579$	24939.0	1.79003
$580$	0	0
$581$	−6162.46	−0.440038
$582$	0	0
$583$	−1252.22	−0.0889569
$584$	0	0
$585$	−40190.0	−2.84043
$586$	0	0
$587$	−5633.72	−0.396130	−0.198065	−	0.980189i	$-0.563466\pi$
$587$	−5633.72	−0.396130	−0.198065	+	0.980189i	$0.563466\pi$
$588$	0	0
$589$	−216.075	−0.0151158
$590$	0	0
$591$	−32537.2	−2.26464
$592$	0	0
$593$	23515.7	1.62846	0.814228	−	0.580545i	$-0.197161\pi$
$593$	23515.7	1.62846	0.814228	+	0.580545i	$0.197161\pi$
$594$	0	0
$595$	−4889.71	−0.336905
$596$	0	0
$597$	13970.4	0.957737
$598$	0	0
$599$	−1092.06	−0.0744917	−0.0372459	−	0.999306i	$-0.511858\pi$
$599$	−1092.06	−0.0744917	−0.0372459	+	0.999306i	$0.511858\pi$
$600$	0	0
$601$	−8044.41	−0.545987	−0.272994	−	0.962016i	$-0.588014\pi$
$601$	−8044.41	−0.545987	−0.272994	+	0.962016i	$0.588014\pi$
$602$	0	0
$603$	−16621.0	−1.12249
$604$	0	0
$605$	2295.96	0.154288
$606$	0	0
$607$	3675.93	0.245801	0.122901	−	0.992419i	$-0.460780\pi$
$607$	3675.93	0.245801	0.122901	+	0.992419i	$0.460780\pi$
$608$	0	0
$609$	6616.28	0.440238
$610$	0	0
$611$	−14243.1	−0.943069
$612$	0	0
$613$	14219.5	0.936901	0.468451	−	0.883490i	$-0.344812\pi$
$613$	14219.5	0.936901	0.468451	+	0.883490i	$0.344812\pi$
$614$	0	0
$615$	70173.4	4.60108
$616$	0	0
$617$	−25418.7	−1.65854	−0.829269	−	0.558849i	$-0.811243\pi$
$617$	−25418.7	−1.65854	−0.829269	+	0.558849i	$0.811243\pi$
$618$	0	0
$619$	27238.5	1.76867	0.884335	−	0.466852i	$-0.154612\pi$
$619$	27238.5	1.76867	0.884335	+	0.466852i	$0.154612\pi$
$620$	0	0
$621$	2682.79	0.173360
$622$	0	0
$623$	937.289	0.0602756
$624$	0	0
$625$	10240.7	0.655405
$626$	0	0
$627$	−121.743	−0.00775428
$628$	0	0
$629$	5982.67	0.379244
$630$	0	0
$631$	−11788.5	−0.743726	−0.371863	−	0.928288i	$-0.621281\pi$
$631$	−11788.5	−0.743726	−0.371863	+	0.928288i	$0.621281\pi$
$632$	0	0
$633$	22504.8	1.41309
$634$	0	0
$635$	28598.3	1.78722
$636$	0	0
$637$	−2640.17	−0.164218
$638$	0	0
$639$	−32464.4	−2.00981
$640$	0	0
$641$	−30705.7	−1.89205	−0.946025	−	0.324095i	$-0.894940\pi$
$641$	−30705.7	−1.89205	−0.946025	+	0.324095i	$0.894940\pi$
$642$	0	0
$643$	−6528.20	−0.400384	−0.200192	−	0.979757i	$-0.564157\pi$
$643$	−6528.20	−0.400384	−0.200192	+	0.979757i	$0.564157\pi$
$644$	0	0
$645$	59356.2	3.62349
$646$	0	0
$647$	−5248.59	−0.318923	−0.159462	−	0.987204i	$-0.550976\pi$
$647$	−5248.59	−0.318923	−0.159462	+	0.987204i	$0.550976\pi$
$648$	0	0
$649$	−6628.45	−0.400908
$650$	0	0
$651$	9062.14	0.545581
$652$	0	0
$653$	23643.6	1.41691	0.708456	−	0.705755i	$-0.249392\pi$
$653$	23643.6	1.41691	0.708456	+	0.705755i	$0.249392\pi$
$654$	0	0
$655$	−6738.90	−0.402001
$656$	0	0
$657$	−4779.00	−0.283785
$658$	0	0
$659$	−10204.1	−0.603182	−0.301591	−	0.953437i	$-0.597518\pi$
$659$	−10204.1	−0.603182	−0.301591	+	0.953437i	$0.597518\pi$
$660$	0	0
$661$	−25945.0	−1.52669	−0.763345	−	0.645991i	$-0.776444\pi$
$661$	−25945.0	−1.52669	−0.763345	+	0.645991i	$0.776444\pi$
$662$	0	0
$663$	16152.2	0.946152
$664$	0	0
$665$	180.525	0.0105270
$666$	0	0
$667$	−3106.43	−0.180332
$668$	0	0
$669$	−48795.3	−2.81993
$670$	0	0
$671$	7716.29	0.443940
$672$	0	0
$673$	−10785.0	−0.617729	−0.308864	−	0.951106i	$-0.599949\pi$
$673$	−10785.0	−0.617729	−0.308864	+	0.951106i	$0.599949\pi$
$674$	0	0
$675$	−23561.4	−1.34353
$676$	0	0
$677$	−4226.92	−0.239961	−0.119981	−	0.992776i	$-0.538283\pi$
$677$	−4226.92	−0.239961	−0.119981	+	0.992776i	$0.538283\pi$
$678$	0	0
$679$	−9658.86	−0.545910
$680$	0	0
$681$	−43587.7	−2.45270
$682$	0	0
$683$	21754.6	1.21877	0.609383	−	0.792876i	$-0.291417\pi$
$683$	21754.6	1.21877	0.609383	+	0.792876i	$0.291417\pi$
$684$	0	0
$685$	53767.5	2.99905
$686$	0	0
$687$	−46590.0	−2.58736
$688$	0	0
$689$	−6133.73	−0.339153
$690$	0	0
$691$	−21019.4	−1.15718	−0.578592	−	0.815617i	$-0.696398\pi$
$691$	−21019.4	−1.15718	−0.578592	+	0.815617i	$0.696398\pi$
$692$	0	0
$693$	3026.87	0.165918
$694$	0	0
$695$	−32477.7	−1.77259
$696$	0	0
$697$	−16719.0	−0.908576
$698$	0	0
$699$	−39859.7	−2.15684
$700$	0	0
$701$	−2567.14	−0.138316	−0.0691579	−	0.997606i	$-0.522031\pi$
$701$	−2567.14	−0.138316	−0.0691579	+	0.997606i	$0.522031\pi$
$702$	0	0
$703$	−220.876	−0.0118499
$704$	0	0
$705$	−40845.0	−2.18200
$706$	0	0
$707$	742.140	0.0394781
$708$	0	0
$709$	−21507.8	−1.13927	−0.569636	−	0.821897i	$-0.692916\pi$
$709$	−21507.8	−1.13927	−0.569636	+	0.821897i	$0.692916\pi$
$710$	0	0
$711$	21789.1	1.14930
$712$	0	0
$713$	−4254.80	−0.223483
$714$	0	0
$715$	11246.2	0.588230
$716$	0	0
$717$	−28550.9	−1.48710
$718$	0	0
$719$	4886.60	0.253462	0.126731	−	0.991937i	$-0.459551\pi$
$719$	4886.60	0.253462	0.126731	+	0.991937i	$0.459551\pi$
$720$	0	0
$721$	−1569.27	−0.0810577
$722$	0	0
$723$	−40605.5	−2.08871
$724$	0	0
$725$	27282.1	1.39756
$726$	0	0
$727$	15713.6	0.801630	0.400815	−	0.916159i	$-0.368727\pi$
$727$	15713.6	0.801630	0.400815	+	0.916159i	$0.368727\pi$
$728$	0	0
$729$	−31674.1	−1.60921
$730$	0	0
$731$	−14141.8	−0.715531
$732$	0	0
$733$	14696.9	0.740578	0.370289	−	0.928917i	$-0.379259\pi$
$733$	14696.9	0.740578	0.370289	+	0.928917i	$0.379259\pi$
$734$	0	0
$735$	−7571.20	−0.379956
$736$	0	0
$737$	4651.00	0.232458
$738$	0	0
$739$	4564.65	0.227217	0.113609	−	0.993526i	$-0.463759\pi$
$739$	4564.65	0.227217	0.113609	+	0.993526i	$0.463759\pi$
$740$	0	0
$741$	−596.328	−0.0295637
$742$	0	0
$743$	−13840.5	−0.683388	−0.341694	−	0.939811i	$-0.611001\pi$
$743$	−13840.5	−0.683388	−0.341694	+	0.939811i	$0.611001\pi$
$744$	0	0
$745$	25964.1	1.27685
$746$	0	0
$747$	34606.7	1.69504
$748$	0	0
$749$	−13762.6	−0.671394
$750$	0	0
$751$	−8644.69	−0.420039	−0.210019	−	0.977697i	$-0.567353\pi$
$751$	−8644.69	−0.420039	−0.210019	+	0.977697i	$0.567353\pi$
$752$	0	0
$753$	−51838.9	−2.50878
$754$	0	0
$755$	−12294.5	−0.592638
$756$	0	0
$757$	−21461.5	−1.03043	−0.515213	−	0.857062i	$-0.672287\pi$
$757$	−21461.5	−1.03043	−0.515213	+	0.857062i	$0.672287\pi$
$758$	0	0
$759$	−2397.28	−0.114645
$760$	0	0
$761$	5830.04	0.277712	0.138856	−	0.990313i	$-0.455657\pi$
$761$	5830.04	0.277712	0.138856	+	0.990313i	$0.455657\pi$
$762$	0	0
$763$	4239.82	0.201169
$764$	0	0
$765$	27459.3	1.29777
$766$	0	0
$767$	−32467.9	−1.52849
$768$	0	0
$769$	−812.174	−0.0380855	−0.0190427	−	0.999819i	$-0.506062\pi$
$769$	−812.174	−0.0380855	−0.0190427	+	0.999819i	$0.506062\pi$
$770$	0	0
$771$	−40544.9	−1.89389
$772$	0	0
$773$	23852.3	1.10984	0.554921	−	0.831903i	$-0.312748\pi$
$773$	23852.3	1.10984	0.554921	+	0.831903i	$0.312748\pi$
$774$	0	0
$775$	37367.6	1.73198
$776$	0	0
$777$	9263.53	0.427705
$778$	0	0
$779$	617.255	0.0283895
$780$	0	0
$781$	9084.40	0.416217
$782$	0	0
$783$	−11635.3	−0.531047
$784$	0	0
$785$	40675.8	1.84940
$786$	0	0
$787$	20889.3	0.946155	0.473077	−	0.881021i	$-0.343143\pi$
$787$	20889.3	0.946155	0.473077	+	0.881021i	$0.343143\pi$
$788$	0	0
$789$	37649.3	1.69880
$790$	0	0
$791$	−9099.61	−0.409033
$792$	0	0
$793$	37796.4	1.69255
$794$	0	0
$795$	−17589.7	−0.784707
$796$	0	0
$797$	−15023.5	−0.667704	−0.333852	−	0.942625i	$-0.608349\pi$
$797$	−15023.5	−0.667704	−0.333852	+	0.942625i	$0.608349\pi$
$798$	0	0
$799$	9731.43	0.430880
$800$	0	0
$801$	−5263.56	−0.232183
$802$	0	0
$803$	1337.29	0.0587696
$804$	0	0
$805$	3554.78	0.155639
$806$	0	0
$807$	51905.3	2.26413
$808$	0	0
$809$	3189.41	0.138608	0.0693039	−	0.997596i	$-0.477922\pi$
$809$	3189.41	0.138608	0.0693039	+	0.997596i	$0.477922\pi$
$810$	0	0
$811$	−10841.9	−0.469435	−0.234717	−	0.972064i	$-0.575416\pi$
$811$	−10841.9	−0.469435	−0.234717	+	0.972064i	$0.575416\pi$
$812$	0	0
$813$	−6048.10	−0.260905
$814$	0	0
$815$	−42869.8	−1.84253
$816$	0	0
$817$	522.106	0.0223576
$818$	0	0
$819$	14826.4	0.632574
$820$	0	0
$821$	1482.72	0.0630294	0.0315147	−	0.999503i	$-0.489967\pi$
$821$	1482.72	0.0630294	0.0315147	+	0.999503i	$0.489967\pi$
$822$	0	0
$823$	−6435.15	−0.272558	−0.136279	−	0.990670i	$-0.543514\pi$
$823$	−6435.15	−0.272558	−0.136279	+	0.990670i	$0.543514\pi$
$824$	0	0
$825$	21054.0	0.888492
$826$	0	0
$827$	28986.6	1.21882	0.609408	−	0.792857i	$-0.291407\pi$
$827$	28986.6	1.21882	0.609408	+	0.792857i	$0.291407\pi$
$828$	0	0
$829$	9922.32	0.415701	0.207851	−	0.978161i	$-0.433353\pi$
$829$	9922.32	0.415701	0.207851	+	0.978161i	$0.433353\pi$
$830$	0	0
$831$	13985.6	0.583819
$832$	0	0
$833$	1803.86	0.0750300
$834$	0	0
$835$	10548.2	0.437169
$836$	0	0
$837$	−15936.5	−0.658120
$838$	0	0
$839$	20723.7	0.852753	0.426377	−	0.904546i	$-0.359790\pi$
$839$	20723.7	0.852753	0.426377	+	0.904546i	$0.359790\pi$
$840$	0	0
$841$	−10916.4	−0.447595
$842$	0	0
$843$	−32279.6	−1.31882
$844$	0	0
$845$	13399.2	0.545499
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	−52584.4	−2.12567
$850$	0	0
$851$	−4349.35	−0.175198
$852$	0	0
$853$	45125.3	1.81132	0.905662	−	0.424000i	$-0.139374\pi$
$853$	45125.3	1.81132	0.905662	+	0.424000i	$0.139374\pi$
$854$	0	0
$855$	−1013.78	−0.0405503
$856$	0	0
$857$	12979.2	0.517342	0.258671	−	0.965965i	$-0.416715\pi$
$857$	12979.2	0.517342	0.258671	+	0.965965i	$0.416715\pi$
$858$	0	0
$859$	6365.16	0.252825	0.126412	−	0.991978i	$-0.459654\pi$
$859$	6365.16	0.252825	0.126412	+	0.991978i	$0.459654\pi$
$860$	0	0
$861$	−25887.6	−1.02468
$862$	0	0
$863$	−17586.1	−0.693669	−0.346835	−	0.937926i	$-0.612743\pi$
$863$	−17586.1	−0.693669	−0.346835	+	0.937926i	$0.612743\pi$
$864$	0	0
$865$	−51847.9	−2.03801
$866$	0	0
$867$	28971.3	1.13485
$868$	0	0
$869$	−6097.17	−0.238012
$870$	0	0
$871$	22781.8	0.886260
$872$	0	0
$873$	54241.5	2.10286
$874$	0	0
$875$	−14616.7	−0.564725
$876$	0	0
$877$	33269.5	1.28099	0.640496	−	0.767961i	$-0.278729\pi$
$877$	33269.5	1.28099	0.640496	+	0.767961i	$0.278729\pi$
$878$	0	0
$879$	31741.9	1.21801
$880$	0	0
$881$	37859.8	1.44782	0.723909	−	0.689895i	$-0.242343\pi$
$881$	37859.8	1.44782	0.723909	+	0.689895i	$0.242343\pi$
$882$	0	0
$883$	−46458.6	−1.77062	−0.885309	−	0.465004i	$-0.846053\pi$
$883$	−46458.6	−1.77062	−0.885309	+	0.465004i	$0.846053\pi$
$884$	0	0
$885$	−93108.2	−3.53649
$886$	0	0
$887$	−15718.1	−0.594999	−0.297499	−	0.954722i	$-0.596153\pi$
$887$	−15718.1	−0.594999	−0.297499	+	0.954722i	$0.596153\pi$
$888$	0	0
$889$	−10550.2	−0.398021
$890$	0	0
$891$	2696.00	0.101368
$892$	0	0
$893$	−359.278	−0.0134634
$894$	0	0
$895$	26963.0	1.00701
$896$	0	0
$897$	−11742.5	−0.437091
$898$	0	0
$899$	18453.1	0.684588
$900$	0	0
$901$	4190.79	0.154956
$902$	0	0
$903$	−21897.1	−0.806964
$904$	0	0
$905$	−27998.3	−1.02839
$906$	0	0
$907$	10470.9	0.383331	0.191665	−	0.981460i	$-0.438611\pi$
$907$	10470.9	0.383331	0.191665	+	0.981460i	$0.438611\pi$
$908$	0	0
$909$	−4167.65	−0.152071
$910$	0	0
$911$	34206.6	1.24403	0.622017	−	0.783004i	$-0.286314\pi$
$911$	34206.6	1.24403	0.622017	+	0.783004i	$0.286314\pi$
$912$	0	0
$913$	−9683.87	−0.351029
$914$	0	0
$915$	108389.	3.91609
$916$	0	0
$917$	2486.04	0.0895271
$918$	0	0
$919$	−48866.0	−1.75402	−0.877008	−	0.480475i	$-0.840464\pi$
$919$	−48866.0	−1.75402	−0.877008	+	0.480475i	$0.840464\pi$
$920$	0	0
$921$	−61457.2	−2.19879
$922$	0	0
$923$	44497.8	1.58685
$924$	0	0
$925$	38198.0	1.35778
$926$	0	0
$927$	8812.58	0.312236
$928$	0	0
$929$	−48164.7	−1.70100	−0.850502	−	0.525971i	$-0.823702\pi$
$929$	−48164.7	−1.70100	−0.850502	+	0.525971i	$0.823702\pi$
$930$	0	0
$931$	−66.5973	−0.00234440
$932$	0	0
$933$	−83828.4	−2.94150
$934$	0	0
$935$	−7683.83	−0.268757
$936$	0	0
$937$	23855.2	0.831712	0.415856	−	0.909430i	$-0.363482\pi$
$937$	23855.2	0.831712	0.415856	+	0.909430i	$0.363482\pi$
$938$	0	0
$939$	37531.2	1.30435
$940$	0	0
$941$	−24494.4	−0.848561	−0.424280	−	0.905531i	$-0.639473\pi$
$941$	−24494.4	−0.848561	−0.424280	+	0.905531i	$0.639473\pi$
$942$	0	0
$943$	12154.6	0.419732
$944$	0	0
$945$	13314.6	0.458331
$946$	0	0
$947$	−22907.3	−0.786048	−0.393024	−	0.919528i	$-0.628571\pi$
$947$	−22907.3	−0.786048	−0.393024	+	0.919528i	$0.628571\pi$
$948$	0	0
$949$	6550.41	0.224062
$950$	0	0
$951$	73383.2	2.50222
$952$	0	0
$953$	−20901.0	−0.710440	−0.355220	−	0.934783i	$-0.615594\pi$
$953$	−20901.0	−0.710440	−0.355220	+	0.934783i	$0.615594\pi$
$954$	0	0
$955$	87997.0	2.98169
$956$	0	0
$957$	10397.0	0.351189
$958$	0	0
$959$	−19835.3	−0.667899
$960$	0	0
$961$	−4516.29	−0.151599
$962$	0	0
$963$	77286.9	2.58623
$964$	0	0
$965$	−58112.3	−1.93855
$966$	0	0
$967$	−4280.30	−0.142342	−0.0711712	−	0.997464i	$-0.522674\pi$
$967$	−4280.30	−0.142342	−0.0711712	+	0.997464i	$0.522674\pi$
$968$	0	0
$969$	407.434	0.0135074
$970$	0	0
$971$	−12181.1	−0.402584	−0.201292	−	0.979531i	$-0.564514\pi$
$971$	−12181.1	−0.402584	−0.201292	+	0.979531i	$0.564514\pi$
$972$	0	0
$973$	11981.3	0.394762
$974$	0	0
$975$	103128.	3.38743
$976$	0	0
$977$	35733.1	1.17012	0.585058	−	0.810992i	$-0.301072\pi$
$977$	35733.1	1.17012	0.585058	+	0.810992i	$0.301072\pi$
$978$	0	0
$979$	1472.88	0.0480833
$980$	0	0
$981$	−23809.6	−0.774906
$982$	0	0
$983$	−57409.8	−1.86276	−0.931378	−	0.364053i	$-0.881393\pi$
$983$	−57409.8	−1.86276	−0.931378	+	0.364053i	$0.881393\pi$
$984$	0	0
$985$	75817.4	2.45253
$986$	0	0
$987$	15068.1	0.485940
$988$	0	0
$989$	10281.0	0.330552
$990$	0	0
$991$	−18567.0	−0.595156	−0.297578	−	0.954697i	$-0.596179\pi$
$991$	−18567.0	−0.595156	−0.297578	+	0.954697i	$0.596179\pi$
$992$	0	0
$993$	61825.2	1.97580
$994$	0	0
$995$	−32553.4	−1.03720
$996$	0	0
$997$	−5253.29	−0.166874	−0.0834369	−	0.996513i	$-0.526590\pi$
$997$	−5253.29	−0.166874	−0.0834369	+	0.996513i	$0.526590\pi$
$998$	0	0
$999$	−16290.7	−0.515929

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.bc.1.2		7
4.3	odd	2		616.4.a.k.1.6	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.k.1.6	✓	7	4.3	odd	2
1232.4.a.bc.1.2		7	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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q-8.14310 q^{3} +18.9749 q^{5} -7.00000 q^{7} +39.3101 q^{9} +O(q^{10})\)	\(q-8.14310 q^{3} +18.9749 q^{5} -7.00000 q^{7} +39.3101 q^{9} -11.0000 q^{11} -53.8809 q^{13} -154.514 q^{15} +36.8135 q^{17} -1.35913 q^{19} +57.0017 q^{21} -26.7631 q^{23} +235.046 q^{25} -100.242 q^{27} +116.072 q^{29} +158.980 q^{31} +89.5741 q^{33} -132.824 q^{35} +162.513 q^{37} +438.758 q^{39} -454.155 q^{41} -384.147 q^{43} +745.903 q^{45} +264.345 q^{47} +49.0000 q^{49} -299.776 q^{51} +113.839 q^{53} -208.724 q^{55} +11.0675 q^{57} +602.586 q^{59} -701.481 q^{61} -275.170 q^{63} -1022.38 q^{65} -422.818 q^{67} +217.934 q^{69} -825.854 q^{71} -121.572 q^{73} -1914.00 q^{75} +77.0000 q^{77} +554.288 q^{79} -245.091 q^{81} +880.351 q^{83} +698.530 q^{85} -945.182 q^{87} -133.898 q^{89} +377.167 q^{91} -1294.59 q^{93} -25.7893 q^{95} +1379.84 q^{97} -432.411 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} + 11 q^{5} - 49 q^{7} + 108 q^{9}+O(q^{10}) \) 7 * q - 3 * q^3 + 11 * q^5 - 49 * q^7 + 108 * q^9	\( 7 q - 3 q^{3} + 11 q^{5} - 49 q^{7} + 108 q^{9} - 77 q^{11} + 26 q^{13} - 67 q^{15} + 44 q^{17} - 34 q^{19} + 21 q^{21} + 29 q^{23} + 182 q^{25} - 99 q^{27} + 94 q^{29} + 173 q^{31} + 33 q^{33} - 77 q^{35} + 255 q^{37} + 60 q^{39} + 508 q^{41} - 656 q^{43} + 466 q^{45} + 18 q^{47} + 343 q^{49} - 850 q^{51} + 1806 q^{53} - 121 q^{55} + 1154 q^{57} - 665 q^{59} + 608 q^{61} - 756 q^{63} + 588 q^{65} - 669 q^{67} + 2067 q^{69} - 1169 q^{71} + 380 q^{73} - 1254 q^{75} + 539 q^{77} + 110 q^{79} + 2847 q^{81} + 496 q^{83} + 3446 q^{85} + 582 q^{87} + 1321 q^{89} - 182 q^{91} + 1493 q^{93} + 50 q^{95} + 3927 q^{97} - 1188 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 + 11 * q^5 - 49 * q^7 + 108 * q^9 - 77 * q^11 + 26 * q^13 - 67 * q^15 + 44 * q^17 - 34 * q^19 + 21 * q^21 + 29 * q^23 + 182 * q^25 - 99 * q^27 + 94 * q^29 + 173 * q^31 + 33 * q^33 - 77 * q^35 + 255 * q^37 + 60 * q^39 + 508 * q^41 - 656 * q^43 + 466 * q^45 + 18 * q^47 + 343 * q^49 - 850 * q^51 + 1806 * q^53 - 121 * q^55 + 1154 * q^57 - 665 * q^59 + 608 * q^61 - 756 * q^63 + 588 * q^65 - 669 * q^67 + 2067 * q^69 - 1169 * q^71 + 380 * q^73 - 1254 * q^75 + 539 * q^77 + 110 * q^79 + 2847 * q^81 + 496 * q^83 + 3446 * q^85 + 582 * q^87 + 1321 * q^89 - 182 * q^91 + 1493 * q^93 + 50 * q^95 + 3927 * q^97 - 1188 * q^99

Introduction

L-functions

Modular forms

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