LMFDB - Embedded newform 1305.4.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,4,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1305.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1305.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:	$76.9974925575$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 49x^{4} + 27x^{3} + 692x^{2} - 82x - 2588 $ x^6 - x^5 - 49x^4 + 27x^3 + 692x^2 - 82x - 2588
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-5.05047$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.05047 q^{2} +17.5072 q^{4} +5.00000 q^{5} +19.3889 q^{7} +48.0160 q^{8} +25.2523 q^{10} -6.81451 q^{11} +36.9473 q^{13} +97.9229 q^{14} +102.445 q^{16} +71.5529 q^{17} +88.3064 q^{19} +87.5361 q^{20} -34.4164 q^{22} -185.418 q^{23} +25.0000 q^{25} +186.601 q^{26} +339.446 q^{28} -29.0000 q^{29} -120.393 q^{31} +133.269 q^{32} +361.376 q^{34} +96.9444 q^{35} -117.351 q^{37} +445.989 q^{38} +240.080 q^{40} +229.591 q^{41} +66.8319 q^{43} -119.303 q^{44} -936.448 q^{46} +42.0170 q^{47} +32.9286 q^{49} +126.262 q^{50} +646.845 q^{52} -9.42394 q^{53} -34.0725 q^{55} +930.976 q^{56} -146.464 q^{58} +232.484 q^{59} -546.541 q^{61} -608.042 q^{62} -146.492 q^{64} +184.737 q^{65} +953.049 q^{67} +1252.69 q^{68} +489.615 q^{70} -429.898 q^{71} +554.903 q^{73} -592.675 q^{74} +1546.00 q^{76} -132.126 q^{77} +965.168 q^{79} +512.226 q^{80} +1159.54 q^{82} -625.186 q^{83} +357.765 q^{85} +337.532 q^{86} -327.205 q^{88} +271.169 q^{89} +716.367 q^{91} -3246.16 q^{92} +212.206 q^{94} +441.532 q^{95} +1167.66 q^{97} +166.305 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + 51 q^{4} + 30 q^{5} + 47 q^{7} - 51 q^{8} - 5 q^{10} - 81 q^{11} + 169 q^{13} + 30 q^{14} + 131 q^{16} + q^{17} + 116 q^{19} + 255 q^{20} + 90 q^{22} + 52 q^{23} + 150 q^{25} - 294 q^{26}+ \cdots + 1433 q^{98}+O(q^{100}) $ 6 * q - q^2 + 51 * q^4 + 30 * q^5 + 47 * q^7 - 51 * q^8 - 5 * q^10 - 81 * q^11 + 169 * q^13 + 30 * q^14 + 131 * q^16 + q^17 + 116 * q^19 + 255 * q^20 + 90 * q^22 + 52 * q^23 + 150 * q^25 - 294 * q^26 + 344 * q^28 - 174 * q^29 + 340 * q^31 - 499 * q^32 + 920 * q^34 + 235 * q^35 + 332 * q^37 + 378 * q^38 - 255 * q^40 + 616 * q^41 + 334 * q^43 + 52 * q^44 - 158 * q^46 + 85 * q^47 + 879 * q^49 - 25 * q^50 + 2220 * q^52 + 850 * q^53 - 405 * q^55 + 624 * q^56 + 29 * q^58 + 758 * q^59 - 36 * q^61 + 152 * q^62 + 1795 * q^64 + 845 * q^65 + 939 * q^67 + 186 * q^68 + 150 * q^70 + 1388 * q^71 + 1708 * q^73 + 814 * q^74 + 566 * q^76 - 2585 * q^77 + 1250 * q^79 + 655 * q^80 + 1372 * q^82 + 748 * q^83 + 5 * q^85 + 800 * q^86 + 536 * q^88 - 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 4542 * q^94 + 580 * q^95 - 22 * q^97 + 1433 * q^98

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 $

$2$	5.05047	1.78561	0.892805	−	0.450443i	$-0.148734\pi$
$2$	5.05047	1.78561	0.892805	+	0.450443i	$0.148734\pi$
$3$	0	0
$3$	0	0
$4$	17.5072	2.18840
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	19.3889	1.04690	0.523451	−	0.852056i	$-0.324644\pi$
$7$	19.3889	1.04690	0.523451	+	0.852056i	$0.324644\pi$
$8$	48.0160	2.12203
$9$	0	0
$10$	25.2523	0.798549
$11$	−6.81451	−0.186786	−0.0933932	−	0.995629i	$-0.529771\pi$
$11$	−6.81451	−0.186786	−0.0933932	+	0.995629i	$0.529771\pi$
$12$	0	0
$13$	36.9473	0.788257	0.394128	−	0.919055i	$-0.371046\pi$
$13$	36.9473	0.788257	0.394128	+	0.919055i	$0.371046\pi$
$14$	97.9229	1.86936
$15$	0	0
$16$	102.445	1.60071
$17$	71.5529	1.02083	0.510416	−	0.859928i	$-0.329492\pi$
$17$	71.5529	1.02083	0.510416	+	0.859928i	$0.329492\pi$
$18$	0	0
$19$	88.3064	1.06626	0.533129	−	0.846034i	$-0.321016\pi$
$19$	88.3064	1.06626	0.533129	+	0.846034i	$0.321016\pi$
$20$	87.5361	0.978684
$21$	0	0
$22$	−34.4164	−0.333528
$23$	−185.418	−1.68097	−0.840486	−	0.541834i	$-0.817730\pi$
$23$	−185.418	−1.68097	−0.840486	+	0.541834i	$0.817730\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	186.601	1.40752
$27$	0	0
$28$	339.446	2.29104
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−120.393	−0.697524	−0.348762	−	0.937211i	$-0.613398\pi$
$31$	−120.393	−0.697524	−0.348762	+	0.937211i	$0.613398\pi$
$32$	133.269	0.736213
$33$	0	0
$34$	361.376	1.82281
$35$	96.9444	0.468188
$36$	0	0
$37$	−117.351	−0.521414	−0.260707	−	0.965418i	$-0.583956\pi$
$37$	−117.351	−0.521414	−0.260707	+	0.965418i	$0.583956\pi$
$38$	445.989	1.90392
$39$	0	0
$40$	240.080	0.948999
$41$	229.591	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$41$	229.591	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$42$	0	0
$43$	66.8319	0.237018	0.118509	−	0.992953i	$-0.462189\pi$
$43$	66.8319	0.237018	0.118509	+	0.992953i	$0.462189\pi$
$44$	−119.303	−0.408764
$45$	0	0
$46$	−936.448	−3.00156
$47$	42.0170	0.130400	0.0652001	−	0.997872i	$-0.479231\pi$
$47$	42.0170	0.130400	0.0652001	+	0.997872i	$0.479231\pi$
$48$	0	0
$49$	32.9286	0.0960018
$50$	126.262	0.357122
$51$	0	0
$52$	646.845	1.72502
$53$	−9.42394	−0.0244241	−0.0122121	−	0.999925i	$-0.503887\pi$
$53$	−9.42394	−0.0244241	−0.0122121	+	0.999925i	$0.503887\pi$
$54$	0	0
$55$	−34.0725	−0.0835334
$56$	930.976	2.22155
$57$	0	0
$58$	−146.464	−0.331579
$59$	232.484	0.512996	0.256498	−	0.966545i	$-0.417431\pi$
$59$	232.484	0.512996	0.256498	+	0.966545i	$0.417431\pi$
$60$	0	0
$61$	−546.541	−1.14717	−0.573586	−	0.819146i	$-0.694448\pi$
$61$	−546.541	−1.14717	−0.573586	+	0.819146i	$0.694448\pi$
$62$	−608.042	−1.24551
$63$	0	0
$64$	−146.492	−0.286118
$65$	184.737	0.352519
$66$	0	0
$67$	953.049	1.73781	0.868907	−	0.494976i	$-0.164823\pi$
$67$	953.049	1.73781	0.868907	+	0.494976i	$0.164823\pi$
$68$	1252.69	2.23399
$69$	0	0
$70$	489.615	0.836002
$71$	−429.898	−0.718584	−0.359292	−	0.933225i	$-0.616982\pi$
$71$	−429.898	−0.718584	−0.359292	+	0.933225i	$0.616982\pi$
$72$	0	0
$73$	554.903	0.889678	0.444839	−	0.895611i	$-0.353261\pi$
$73$	554.903	0.889678	0.444839	+	0.895611i	$0.353261\pi$
$74$	−592.675	−0.931041
$75$	0	0
$76$	1546.00	2.33340
$77$	−132.126	−0.195547
$78$	0	0
$79$	965.168	1.37456	0.687278	−	0.726395i	$-0.258805\pi$
$79$	965.168	1.37456	0.687278	+	0.726395i	$0.258805\pi$
$80$	512.226	0.715858
$81$	0	0
$82$	1159.54	1.56158
$83$	−625.186	−0.826784	−0.413392	−	0.910553i	$-0.635656\pi$
$83$	−625.186	−0.826784	−0.413392	+	0.910553i	$0.635656\pi$
$84$	0	0
$85$	357.765	0.456530
$86$	337.532	0.423222
$87$	0	0
$88$	−327.205	−0.396366
$89$	271.169	0.322964	0.161482	−	0.986876i	$-0.448373\pi$
$89$	271.169	0.322964	0.161482	+	0.986876i	$0.448373\pi$
$90$	0	0
$91$	716.367	0.825227
$92$	−3246.16	−3.67864
$93$	0	0
$94$	212.206	0.232844
$95$	441.532	0.476845
$96$	0	0
$97$	1167.66	1.22225	0.611125	−	0.791534i	$-0.290717\pi$
$97$	1167.66	1.22225	0.611125	+	0.791534i	$0.290717\pi$
$98$	166.305	0.171422
$99$	0	0
$100$	437.681	0.437681
$101$	−1004.86	−0.989977	−0.494989	−	0.868899i	$-0.664828\pi$
$101$	−1004.86	−0.989977	−0.494989	+	0.868899i	$0.664828\pi$
$102$	0	0
$103$	704.225	0.673683	0.336842	−	0.941561i	$-0.390641\pi$
$103$	704.225	0.673683	0.336842	+	0.941561i	$0.390641\pi$
$104$	1774.06	1.67270
$105$	0	0
$106$	−47.5953	−0.0436119
$107$	−1116.49	−1.00874	−0.504371	−	0.863487i	$-0.668276\pi$
$107$	−1116.49	−1.00874	−0.504371	+	0.863487i	$0.668276\pi$
$108$	0	0
$109$	−1306.48	−1.14806	−0.574029	−	0.818835i	$-0.694621\pi$
$109$	−1306.48	−1.14806	−0.574029	+	0.818835i	$0.694621\pi$
$110$	−172.082	−0.149158
$111$	0	0
$112$	1986.30	1.67578
$113$	−256.295	−0.213364	−0.106682	−	0.994293i	$-0.534023\pi$
$113$	−256.295	−0.213364	−0.106682	+	0.994293i	$0.534023\pi$
$114$	0	0
$115$	−927.090	−0.751753
$116$	−507.710	−0.406376
$117$	0	0
$118$	1174.15	0.916011
$119$	1387.33	1.06871
$120$	0	0
$121$	−1284.56	−0.965111
$122$	−2760.29	−2.04840
$123$	0	0
$124$	−2107.75	−1.52646
$125$	125.000	0.0894427
$126$	0	0
$127$	−1201.30	−0.839355	−0.419677	−	0.907673i	$-0.637857\pi$
$127$	−1201.30	−0.839355	−0.419677	+	0.907673i	$0.637857\pi$
$128$	−1806.01	−1.24711
$129$	0	0
$130$	933.006	0.629462
$131$	−1049.57	−0.700008	−0.350004	−	0.936748i	$-0.613820\pi$
$131$	−1049.57	−0.700008	−0.350004	+	0.936748i	$0.613820\pi$
$132$	0	0
$133$	1712.16	1.11627
$134$	4813.34	3.10306
$135$	0	0
$136$	3435.68	2.16623
$137$	−2881.21	−1.79678	−0.898388	−	0.439202i	$-0.855261\pi$
$137$	−2881.21	−1.79678	−0.898388	+	0.439202i	$0.855261\pi$
$138$	0	0
$139$	2584.43	1.57704	0.788521	−	0.615008i	$-0.210847\pi$
$139$	2584.43	1.57704	0.788521	+	0.615008i	$0.210847\pi$
$140$	1697.23	1.02459
$141$	0	0
$142$	−2171.18	−1.28311
$143$	−251.778	−0.147236
$144$	0	0
$145$	−145.000	−0.0830455
$146$	2802.52	1.58862
$147$	0	0
$148$	−2054.48	−1.14106
$149$	1449.23	0.796813	0.398407	−	0.917209i	$-0.369563\pi$
$149$	1449.23	0.796813	0.398407	+	0.917209i	$0.369563\pi$
$150$	0	0
$151$	2760.46	1.48770	0.743851	−	0.668345i	$-0.232997\pi$
$151$	2760.46	1.48770	0.743851	+	0.668345i	$0.232997\pi$
$152$	4240.12	2.26262
$153$	0	0
$154$	−667.296	−0.349171
$155$	−601.966	−0.311942
$156$	0	0
$157$	766.465	0.389621	0.194811	−	0.980841i	$-0.437591\pi$
$157$	766.465	0.389621	0.194811	+	0.980841i	$0.437591\pi$
$158$	4874.55	2.45442
$159$	0	0
$160$	666.344	0.329244
$161$	−3595.05	−1.75981
$162$	0	0
$163$	−1871.11	−0.899119	−0.449560	−	0.893250i	$-0.648419\pi$
$163$	−1871.11	−0.899119	−0.449560	+	0.893250i	$0.648419\pi$
$164$	4019.50	1.91384
$165$	0	0
$166$	−3157.48	−1.47631
$167$	−2343.62	−1.08596	−0.542978	−	0.839747i	$-0.682703\pi$
$167$	−2343.62	−1.08596	−0.542978	+	0.839747i	$0.682703\pi$
$168$	0	0
$169$	−831.896	−0.378651
$170$	1806.88	0.815184
$171$	0	0
$172$	1170.04	0.518691
$173$	1485.73	0.652935	0.326468	−	0.945208i	$-0.394142\pi$
$173$	1485.73	0.652935	0.326468	+	0.945208i	$0.394142\pi$
$174$	0	0
$175$	484.722	0.209380
$176$	−698.114	−0.298990
$177$	0	0
$178$	1369.53	0.576689
$179$	−71.4418	−0.0298314	−0.0149157	−	0.999889i	$-0.504748\pi$
$179$	−71.4418	−0.0298314	−0.0149157	+	0.999889i	$0.504748\pi$
$180$	0	0
$181$	−1696.31	−0.696607	−0.348304	−	0.937382i	$-0.613242\pi$
$181$	−1696.31	−0.696607	−0.348304	+	0.937382i	$0.613242\pi$
$182$	3617.99	1.47353
$183$	0	0
$184$	−8903.03	−3.56706
$185$	−586.753	−0.233183
$186$	0	0
$187$	−487.598	−0.190677
$188$	735.601	0.285368
$189$	0	0
$190$	2229.94	0.851459
$191$	3657.80	1.38570	0.692852	−	0.721080i	$-0.256354\pi$
$191$	3657.80	1.38570	0.692852	+	0.721080i	$0.256354\pi$
$192$	0	0
$193$	−3156.73	−1.17734	−0.588669	−	0.808374i	$-0.700348\pi$
$193$	−3156.73	−1.17734	−0.588669	+	0.808374i	$0.700348\pi$
$194$	5897.25	2.18246
$195$	0	0
$196$	576.489	0.210091
$197$	969.523	0.350638	0.175319	−	0.984512i	$-0.443904\pi$
$197$	969.523	0.350638	0.175319	+	0.984512i	$0.443904\pi$
$198$	0	0
$199$	5423.02	1.93180	0.965898	−	0.258921i	$-0.0833670\pi$
$199$	5423.02	1.93180	0.965898	+	0.258921i	$0.0833670\pi$
$200$	1200.40	0.424405
$201$	0	0
$202$	−5075.03	−1.76771
$203$	−562.277	−0.194405
$204$	0	0
$205$	1147.95	0.391106
$206$	3556.67	1.20294
$207$	0	0
$208$	3785.08	1.26177
$209$	−601.765	−0.199162
$210$	0	0
$211$	−2848.51	−0.929382	−0.464691	−	0.885473i	$-0.653835\pi$
$211$	−2848.51	−0.929382	−0.464691	+	0.885473i	$0.653835\pi$
$212$	−164.987	−0.0534498
$213$	0	0
$214$	−5638.81	−1.80122
$215$	334.160	0.105998
$216$	0	0
$217$	−2334.29	−0.730239
$218$	−6598.35	−2.04999
$219$	0	0
$220$	−596.516	−0.182805
$221$	2643.69	0.804678
$222$	0	0
$223$	502.488	0.150893	0.0754465	−	0.997150i	$-0.475962\pi$
$223$	502.488	0.150893	0.0754465	+	0.997150i	$0.475962\pi$
$224$	2583.93	0.770742
$225$	0	0
$226$	−1294.41	−0.380985
$227$	−1336.83	−0.390876	−0.195438	−	0.980716i	$-0.562613\pi$
$227$	−1336.83	−0.390876	−0.195438	+	0.980716i	$0.562613\pi$
$228$	0	0
$229$	−1131.26	−0.326445	−0.163223	−	0.986589i	$-0.552189\pi$
$229$	−1131.26	−0.326445	−0.163223	+	0.986589i	$0.552189\pi$
$230$	−4682.24	−1.34234
$231$	0	0
$232$	−1392.46	−0.394050
$233$	−1271.63	−0.357541	−0.178770	−	0.983891i	$-0.557212\pi$
$233$	−1271.63	−0.357541	−0.178770	+	0.983891i	$0.557212\pi$
$234$	0	0
$235$	210.085	0.0583168
$236$	4070.14	1.12264
$237$	0	0
$238$	7006.67	1.90830
$239$	−1507.40	−0.407973	−0.203987	−	0.978974i	$-0.565390\pi$
$239$	−1507.40	−0.407973	−0.203987	+	0.978974i	$0.565390\pi$
$240$	0	0
$241$	4251.32	1.13631	0.568156	−	0.822921i	$-0.307657\pi$
$241$	4251.32	1.13631	0.568156	+	0.822921i	$0.307657\pi$
$242$	−6487.64	−1.72331
$243$	0	0
$244$	−9568.42	−2.51047
$245$	164.643	0.0429333
$246$	0	0
$247$	3262.69	0.840485
$248$	−5780.79	−1.48016
$249$	0	0
$250$	631.309	0.159710
$251$	−7219.41	−1.81548	−0.907739	−	0.419535i	$-0.862193\pi$
$251$	−7219.41	−1.81548	−0.907739	+	0.419535i	$0.862193\pi$
$252$	0	0
$253$	1263.53	0.313983
$254$	−6067.12	−1.49876
$255$	0	0
$256$	−7949.23	−1.94073
$257$	−3436.04	−0.833986	−0.416993	−	0.908910i	$-0.636916\pi$
$257$	−3436.04	−0.833986	−0.416993	+	0.908910i	$0.636916\pi$
$258$	0	0
$259$	−2275.29	−0.545868
$260$	3234.23	0.771454
$261$	0	0
$262$	−5300.80	−1.24994
$263$	−1494.14	−0.350314	−0.175157	−	0.984540i	$-0.556043\pi$
$263$	−1494.14	−0.350314	−0.175157	+	0.984540i	$0.556043\pi$
$264$	0	0
$265$	−47.1197	−0.0109228
$266$	8647.22	1.99322
$267$	0	0
$268$	16685.2	3.80304
$269$	1692.06	0.383520	0.191760	−	0.981442i	$-0.438581\pi$
$269$	1692.06	0.383520	0.191760	+	0.981442i	$0.438581\pi$
$270$	0	0
$271$	3147.75	0.705581	0.352790	−	0.935702i	$-0.385233\pi$
$271$	3147.75	0.705581	0.352790	+	0.935702i	$0.385233\pi$
$272$	7330.26	1.63405
$273$	0	0
$274$	−14551.5	−3.20834
$275$	−170.363	−0.0373573
$276$	0	0
$277$	−3873.16	−0.840129	−0.420065	−	0.907494i	$-0.637993\pi$
$277$	−3873.16	−0.840129	−0.420065	+	0.907494i	$0.637993\pi$
$278$	13052.6	2.81598
$279$	0	0
$280$	4654.88	0.993508
$281$	4951.42	1.05116	0.525582	−	0.850743i	$-0.323848\pi$
$281$	4951.42	1.05116	0.525582	+	0.850743i	$0.323848\pi$
$282$	0	0
$283$	−4895.30	−1.02825	−0.514126	−	0.857715i	$-0.671884\pi$
$283$	−4895.30	−1.02825	−0.514126	+	0.857715i	$0.671884\pi$
$284$	−7526.32	−1.57255
$285$	0	0
$286$	−1271.60	−0.262906
$287$	4451.51	0.915555
$288$	0	0
$289$	206.822	0.0420969
$290$	−732.318	−0.148287
$291$	0	0
$292$	9714.82	1.94697
$293$	8978.98	1.79030	0.895150	−	0.445766i	$-0.147068\pi$
$293$	8978.98	1.79030	0.895150	+	0.445766i	$0.147068\pi$
$294$	0	0
$295$	1162.42	0.229419
$296$	−5634.70	−1.10645
$297$	0	0
$298$	7319.27	1.42280
$299$	−6850.70	−1.32504
$300$	0	0
$301$	1295.80	0.248134
$302$	13941.6	2.65646
$303$	0	0
$304$	9046.57	1.70676
$305$	−2732.71	−0.513031
$306$	0	0
$307$	7282.96	1.35394	0.676972	−	0.736009i	$-0.263292\pi$
$307$	7282.96	1.35394	0.676972	+	0.736009i	$0.263292\pi$
$308$	−2313.15	−0.427936
$309$	0	0
$310$	−3040.21	−0.557007
$311$	2424.98	0.442149	0.221074	−	0.975257i	$-0.429044\pi$
$311$	2424.98	0.442149	0.221074	+	0.975257i	$0.429044\pi$
$312$	0	0
$313$	−6511.66	−1.17591	−0.587957	−	0.808892i	$-0.700068\pi$
$313$	−6511.66	−1.17591	−0.587957	+	0.808892i	$0.700068\pi$
$314$	3871.01	0.695712
$315$	0	0
$316$	16897.4	3.00808
$317$	−6634.14	−1.17543	−0.587713	−	0.809069i	$-0.699972\pi$
$317$	−6634.14	−1.17543	−0.587713	+	0.809069i	$0.699972\pi$
$318$	0	0
$319$	197.621	0.0346854
$320$	−732.462	−0.127956
$321$	0	0
$322$	−18156.7	−3.14234
$323$	6318.58	1.08847
$324$	0	0
$325$	923.683	0.157651
$326$	−9449.97	−1.60548
$327$	0	0
$328$	11024.0	1.85579
$329$	814.663	0.136516
$330$	0	0
$331$	−6727.98	−1.11723	−0.558615	−	0.829427i	$-0.688667\pi$
$331$	−6727.98	−1.11723	−0.558615	+	0.829427i	$0.688667\pi$
$332$	−10945.3	−1.80934
$333$	0	0
$334$	−11836.4	−1.93909
$335$	4765.25	0.777174
$336$	0	0
$337$	−7683.33	−1.24195	−0.620976	−	0.783830i	$-0.713264\pi$
$337$	−7683.33	−1.24195	−0.620976	+	0.783830i	$0.713264\pi$
$338$	−4201.46	−0.676123
$339$	0	0
$340$	6263.47	0.999071
$341$	820.420	0.130288
$342$	0	0
$343$	−6011.94	−0.946397
$344$	3209.00	0.502958
$345$	0	0
$346$	7503.62	1.16589
$347$	−1401.55	−0.216827	−0.108413	−	0.994106i	$-0.534577\pi$
$347$	−1401.55	−0.216827	−0.108413	+	0.994106i	$0.534577\pi$
$348$	0	0
$349$	−10253.9	−1.57272	−0.786362	−	0.617766i	$-0.788038\pi$
$349$	−10253.9	−1.57272	−0.786362	+	0.617766i	$0.788038\pi$
$350$	2448.07	0.373871
$351$	0	0
$352$	−908.160	−0.137515
$353$	3309.99	0.499074	0.249537	−	0.968365i	$-0.419722\pi$
$353$	3309.99	0.499074	0.249537	+	0.968365i	$0.419722\pi$
$354$	0	0
$355$	−2149.49	−0.321361
$356$	4747.41	0.706777
$357$	0	0
$358$	−360.815	−0.0532672
$359$	−10427.4	−1.53297	−0.766483	−	0.642264i	$-0.777995\pi$
$359$	−10427.4	−1.53297	−0.766483	+	0.642264i	$0.777995\pi$
$360$	0	0
$361$	939.026	0.136904
$362$	−8567.17	−1.24387
$363$	0	0
$364$	12541.6	1.80593
$365$	2774.52	0.397876
$366$	0	0
$367$	−6667.61	−0.948356	−0.474178	−	0.880429i	$-0.657255\pi$
$367$	−6667.61	−0.948356	−0.474178	+	0.880429i	$0.657255\pi$
$368$	−18995.2	−2.69074
$369$	0	0
$370$	−2963.37	−0.416374
$371$	−182.720	−0.0255696
$372$	0	0
$373$	6100.55	0.846849	0.423424	−	0.905931i	$-0.360828\pi$
$373$	6100.55	0.846849	0.423424	+	0.905931i	$0.360828\pi$
$374$	−2462.60	−0.340476
$375$	0	0
$376$	2017.49	0.276713
$377$	−1071.47	−0.146376
$378$	0	0
$379$	6291.88	0.852750	0.426375	−	0.904546i	$-0.359790\pi$
$379$	6291.88	0.852750	0.426375	+	0.904546i	$0.359790\pi$
$380$	7730.00	1.04353
$381$	0	0
$382$	18473.6	2.47433
$383$	−10349.3	−1.38074	−0.690369	−	0.723457i	$-0.742552\pi$
$383$	−10349.3	−1.38074	−0.690369	+	0.723457i	$0.742552\pi$
$384$	0	0
$385$	−660.628	−0.0874512
$386$	−15943.0	−2.10227
$387$	0	0
$388$	20442.6	2.67478
$389$	2361.08	0.307742	0.153871	−	0.988091i	$-0.450826\pi$
$389$	2361.08	0.307742	0.153871	+	0.988091i	$0.450826\pi$
$390$	0	0
$391$	−13267.2	−1.71599
$392$	1581.10	0.203718
$393$	0	0
$394$	4896.55	0.626103
$395$	4825.84	0.614720
$396$	0	0
$397$	6963.12	0.880274	0.440137	−	0.897931i	$-0.354930\pi$
$397$	6963.12	0.880274	0.440137	+	0.897931i	$0.354930\pi$
$398$	27388.8	3.44944
$399$	0	0
$400$	2561.13	0.320141
$401$	1419.32	0.176752	0.0883759	−	0.996087i	$-0.471832\pi$
$401$	1419.32	0.176752	0.0883759	+	0.996087i	$0.471832\pi$
$402$	0	0
$403$	−4448.20	−0.549828
$404$	−17592.4	−2.16647
$405$	0	0
$406$	−2839.76	−0.347131
$407$	799.686	0.0973930
$408$	0	0
$409$	−5199.93	−0.628655	−0.314327	−	0.949315i	$-0.601779\pi$
$409$	−5199.93	−0.628655	−0.314327	+	0.949315i	$0.601779\pi$
$410$	5797.71	0.698362
$411$	0	0
$412$	12329.0	1.47429
$413$	4507.60	0.537056
$414$	0	0
$415$	−3125.93	−0.369749
$416$	4923.92	0.580325
$417$	0	0
$418$	−3039.19	−0.355626
$419$	−4476.06	−0.521885	−0.260943	−	0.965354i	$-0.584033\pi$
$419$	−4476.06	−0.521885	−0.260943	+	0.965354i	$0.584033\pi$
$420$	0	0
$421$	2924.54	0.338558	0.169279	−	0.985568i	$-0.445856\pi$
$421$	2924.54	0.338558	0.169279	+	0.985568i	$0.445856\pi$
$422$	−14386.3	−1.65951
$423$	0	0
$424$	−452.499	−0.0518286
$425$	1788.82	0.204166
$426$	0	0
$427$	−10596.8	−1.20097
$428$	−19546.7	−2.20754
$429$	0	0
$430$	1687.66	0.189270
$431$	−1488.98	−0.166408	−0.0832039	−	0.996533i	$-0.526515\pi$
$431$	−1488.98	−0.166408	−0.0832039	+	0.996533i	$0.526515\pi$
$432$	0	0
$433$	−8411.84	−0.933597	−0.466798	−	0.884364i	$-0.654593\pi$
$433$	−8411.84	−0.933597	−0.466798	+	0.884364i	$0.654593\pi$
$434$	−11789.3	−1.30392
$435$	0	0
$436$	−22872.9	−2.51242
$437$	−16373.6	−1.79235
$438$	0	0
$439$	−10584.7	−1.15076	−0.575378	−	0.817887i	$-0.695145\pi$
$439$	−10584.7	−1.15076	−0.575378	+	0.817887i	$0.695145\pi$
$440$	−1636.02	−0.177260
$441$	0	0
$442$	13351.9	1.43684
$443$	10335.6	1.10848	0.554242	−	0.832356i	$-0.313008\pi$
$443$	10335.6	1.10848	0.554242	+	0.832356i	$0.313008\pi$
$444$	0	0
$445$	1355.84	0.144434
$446$	2537.80	0.269436
$447$	0	0
$448$	−2840.32	−0.299537
$449$	−16626.7	−1.74758	−0.873788	−	0.486306i	$-0.838344\pi$
$449$	−16626.7	−1.74758	−0.873788	+	0.486306i	$0.838344\pi$
$450$	0	0
$451$	−1564.55	−0.163352
$452$	−4487.01	−0.466927
$453$	0	0
$454$	−6751.63	−0.697951
$455$	3581.83	0.369053
$456$	0	0
$457$	−9008.26	−0.922076	−0.461038	−	0.887380i	$-0.652523\pi$
$457$	−9008.26	−0.922076	−0.461038	+	0.887380i	$0.652523\pi$
$458$	−5713.41	−0.582904
$459$	0	0
$460$	−16230.8	−1.64514
$461$	8097.37	0.818074	0.409037	−	0.912518i	$-0.365865\pi$
$461$	8097.37	0.818074	0.409037	+	0.912518i	$0.365865\pi$
$462$	0	0
$463$	4897.89	0.491629	0.245815	−	0.969317i	$-0.420945\pi$
$463$	4897.89	0.491629	0.245815	+	0.969317i	$0.420945\pi$
$464$	−2970.91	−0.297244
$465$	0	0
$466$	−6422.31	−0.638429
$467$	3331.22	0.330087	0.165043	−	0.986286i	$-0.447224\pi$
$467$	3331.22	0.330087	0.165043	+	0.986286i	$0.447224\pi$
$468$	0	0
$469$	18478.6	1.81932
$470$	1061.03	0.104131
$471$	0	0
$472$	11162.9	1.08859
$473$	−455.426	−0.0442717
$474$	0	0
$475$	2207.66	0.213251
$476$	24288.3	2.33877
$477$	0	0
$478$	−7613.08	−0.728482
$479$	13606.3	1.29788	0.648942	−	0.760838i	$-0.275212\pi$
$479$	13606.3	1.29788	0.648942	+	0.760838i	$0.275212\pi$
$480$	0	0
$481$	−4335.79	−0.411008
$482$	21471.1	2.02901
$483$	0	0
$484$	−22489.1	−2.11205
$485$	5838.32	0.546607
$486$	0	0
$487$	−15460.9	−1.43861	−0.719304	−	0.694696i	$-0.755539\pi$
$487$	−15460.9	−1.43861	−0.719304	+	0.694696i	$0.755539\pi$
$488$	−26242.7	−2.43433
$489$	0	0
$490$	831.525	0.0766621
$491$	19016.4	1.74785	0.873927	−	0.486057i	$-0.161565\pi$
$491$	19016.4	1.74785	0.873927	+	0.486057i	$0.161565\pi$
$492$	0	0
$493$	−2075.03	−0.189564
$494$	16478.1	1.50078
$495$	0	0
$496$	−12333.7	−1.11653
$497$	−8335.23	−0.752286
$498$	0	0
$499$	−13675.0	−1.22681	−0.613406	−	0.789768i	$-0.710201\pi$
$499$	−13675.0	−1.22681	−0.613406	+	0.789768i	$0.710201\pi$
$500$	2188.40	0.195737
$501$	0	0
$502$	−36461.4	−3.24174
$503$	1456.45	0.129105	0.0645525	−	0.997914i	$-0.479438\pi$
$503$	1456.45	0.129105	0.0645525	+	0.997914i	$0.479438\pi$
$504$	0	0
$505$	−5024.32	−0.442731
$506$	6381.43	0.560651
$507$	0	0
$508$	−21031.4	−1.83685
$509$	−15219.7	−1.32535	−0.662674	−	0.748908i	$-0.730578\pi$
$509$	−15219.7	−1.32535	−0.662674	+	0.748908i	$0.730578\pi$
$510$	0	0
$511$	10758.9	0.931405
$512$	−25699.3	−2.21828
$513$	0	0
$514$	−17353.6	−1.48917
$515$	3521.13	0.301280
$516$	0	0
$517$	−286.325	−0.0243570
$518$	−11491.3	−0.974708
$519$	0	0
$520$	8870.30	0.748055
$521$	−19988.9	−1.68086	−0.840432	−	0.541917i	$-0.817699\pi$
$521$	−19988.9	−1.68086	−0.840432	+	0.541917i	$0.817699\pi$
$522$	0	0
$523$	9728.36	0.813368	0.406684	−	0.913569i	$-0.366685\pi$
$523$	9728.36	0.813368	0.406684	+	0.913569i	$0.366685\pi$
$524$	−18375.0	−1.53190
$525$	0	0
$526$	−7546.11	−0.625525
$527$	−8614.48	−0.712055
$528$	0	0
$529$	22212.9	1.82566
$530$	−237.976	−0.0195038
$531$	0	0
$532$	29975.2	2.44284
$533$	8482.77	0.689361
$534$	0	0
$535$	−5582.47	−0.451123
$536$	45761.6	3.68768
$537$	0	0
$538$	8545.70	0.684817
$539$	−224.392	−0.0179318
$540$	0	0
$541$	15290.2	1.21511	0.607556	−	0.794277i	$-0.292150\pi$
$541$	15290.2	1.21511	0.607556	+	0.794277i	$0.292150\pi$
$542$	15897.6	1.25989
$543$	0	0
$544$	9535.77	0.751549
$545$	−6532.42	−0.513428
$546$	0	0
$547$	−9101.20	−0.711406	−0.355703	−	0.934599i	$-0.615759\pi$
$547$	−9101.20	−0.711406	−0.355703	+	0.934599i	$0.615759\pi$
$548$	−50442.0	−3.93207
$549$	0	0
$550$	−860.411	−0.0667055
$551$	−2560.89	−0.197999
$552$	0	0
$553$	18713.5	1.43902
$554$	−19561.3	−1.50014
$555$	0	0
$556$	45246.3	3.45120
$557$	3752.41	0.285448	0.142724	−	0.989763i	$-0.454414\pi$
$557$	3752.41	0.285448	0.142724	+	0.989763i	$0.454414\pi$
$558$	0	0
$559$	2469.26	0.186831
$560$	9931.49	0.749432
$561$	0	0
$562$	25007.0	1.87697
$563$	−18097.1	−1.35471	−0.677356	−	0.735656i	$-0.736874\pi$
$563$	−18097.1	−1.35471	−0.677356	+	0.735656i	$0.736874\pi$
$564$	0	0
$565$	−1281.47	−0.0954194
$566$	−24723.6	−1.83606
$567$	0	0
$568$	−20641.9	−1.52485
$569$	14101.9	1.03898	0.519492	−	0.854475i	$-0.326121\pi$
$569$	14101.9	1.03898	0.519492	+	0.854475i	$0.326121\pi$
$570$	0	0
$571$	2874.72	0.210689	0.105344	−	0.994436i	$-0.466406\pi$
$571$	2874.72	0.210689	0.105344	+	0.994436i	$0.466406\pi$
$572$	−4407.93	−0.322211
$573$	0	0
$574$	22482.2	1.63482
$575$	−4635.45	−0.336194
$576$	0	0
$577$	11659.8	0.841256	0.420628	−	0.907233i	$-0.361810\pi$
$577$	11659.8	0.841256	0.420628	+	0.907233i	$0.361810\pi$
$578$	1044.55	0.0751686
$579$	0	0
$580$	−2538.55	−0.181737
$581$	−12121.6	−0.865561
$582$	0	0
$583$	64.2195	0.00456209
$584$	26644.2	1.88792
$585$	0	0
$586$	45348.1	3.19678
$587$	14565.9	1.02419	0.512095	−	0.858929i	$-0.328870\pi$
$587$	14565.9	1.02419	0.512095	+	0.858929i	$0.328870\pi$
$588$	0	0
$589$	−10631.5	−0.743740
$590$	5870.75	0.409653
$591$	0	0
$592$	−12022.0	−0.834630
$593$	18632.6	1.29030	0.645152	−	0.764054i	$-0.276794\pi$
$593$	18632.6	1.29030	0.645152	+	0.764054i	$0.276794\pi$
$594$	0	0
$595$	6936.66	0.477941
$596$	25371.9	1.74375
$597$	0	0
$598$	−34599.2	−2.36600
$599$	11080.2	0.755802	0.377901	−	0.925846i	$-0.376646\pi$
$599$	11080.2	0.755802	0.377901	+	0.925846i	$0.376646\pi$
$600$	0	0
$601$	−26806.7	−1.81941	−0.909706	−	0.415254i	$-0.863693\pi$
$601$	−26806.7	−1.81941	−0.909706	+	0.415254i	$0.863693\pi$
$602$	6544.38	0.443071
$603$	0	0
$604$	48328.0	3.25569
$605$	−6422.81	−0.431611
$606$	0	0
$607$	2408.52	0.161052	0.0805262	−	0.996752i	$-0.474340\pi$
$607$	2408.52	0.161052	0.0805262	+	0.996752i	$0.474340\pi$
$608$	11768.5	0.784992
$609$	0	0
$610$	−13801.4	−0.916073
$611$	1552.42	0.102789
$612$	0	0
$613$	24558.8	1.61814	0.809071	−	0.587710i	$-0.199971\pi$
$613$	24558.8	1.61814	0.809071	+	0.587710i	$0.199971\pi$
$614$	36782.4	2.41762
$615$	0	0
$616$	−6344.14	−0.414956
$617$	6367.34	0.415461	0.207731	−	0.978186i	$-0.433392\pi$
$617$	6367.34	0.415461	0.207731	+	0.978186i	$0.433392\pi$
$618$	0	0
$619$	25893.6	1.68134	0.840671	−	0.541546i	$-0.182161\pi$
$619$	25893.6	1.68134	0.840671	+	0.541546i	$0.182161\pi$
$620$	−10538.8	−0.682656
$621$	0	0
$622$	12247.3	0.789505
$623$	5257.66	0.338112
$624$	0	0
$625$	625.000	0.0400000
$626$	−32887.0	−2.09972
$627$	0	0
$628$	13418.7	0.852649
$629$	−8396.77	−0.532275
$630$	0	0
$631$	−1864.31	−0.117618	−0.0588089	−	0.998269i	$-0.518730\pi$
$631$	−1864.31	−0.117618	−0.0588089	+	0.998269i	$0.518730\pi$
$632$	46343.5	2.91684
$633$	0	0
$634$	−33505.5	−2.09885
$635$	−6006.50	−0.375371
$636$	0	0
$637$	1216.62	0.0756741
$638$	998.077	0.0619345
$639$	0	0
$640$	−9030.03	−0.557724
$641$	25891.1	1.59538	0.797689	−	0.603069i	$-0.206056\pi$
$641$	25891.1	1.59538	0.797689	+	0.603069i	$0.206056\pi$
$642$	0	0
$643$	−19494.1	−1.19560	−0.597801	−	0.801645i	$-0.703959\pi$
$643$	−19494.1	−1.19560	−0.597801	+	0.801645i	$0.703959\pi$
$644$	−62939.3	−3.85118
$645$	0	0
$646$	31911.8	1.94358
$647$	17987.6	1.09299	0.546495	−	0.837463i	$-0.315962\pi$
$647$	17987.6	1.09299	0.546495	+	0.837463i	$0.315962\pi$
$648$	0	0
$649$	−1584.26	−0.0958207
$650$	4665.03	0.281504
$651$	0	0
$652$	−32757.9	−1.96764
$653$	−2069.77	−0.124037	−0.0620186	−	0.998075i	$-0.519754\pi$
$653$	−2069.77	−0.124037	−0.0620186	+	0.998075i	$0.519754\pi$
$654$	0	0
$655$	−5247.83	−0.313053
$656$	23520.5	1.39988
$657$	0	0
$658$	4114.43	0.243765
$659$	15787.8	0.933239	0.466619	−	0.884458i	$-0.345472\pi$
$659$	15787.8	0.933239	0.466619	+	0.884458i	$0.345472\pi$
$660$	0	0
$661$	32239.9	1.89711	0.948554	−	0.316615i	$-0.102546\pi$
$661$	32239.9	1.89711	0.948554	+	0.316615i	$0.102546\pi$
$662$	−33979.4	−1.99494
$663$	0	0
$664$	−30018.9	−1.75446
$665$	8560.81	0.499209
$666$	0	0
$667$	5377.12	0.312149
$668$	−41030.2	−2.37651
$669$	0	0
$670$	24066.7	1.38773
$671$	3724.41	0.214276
$672$	0	0
$673$	1039.67	0.0595488	0.0297744	−	0.999557i	$-0.490521\pi$
$673$	1039.67	0.0595488	0.0297744	+	0.999557i	$0.490521\pi$
$674$	−38804.4	−2.21764
$675$	0	0
$676$	−14564.2	−0.828641
$677$	12781.5	0.725604	0.362802	−	0.931866i	$-0.381820\pi$
$677$	12781.5	0.725604	0.362802	+	0.931866i	$0.381820\pi$
$678$	0	0
$679$	22639.7	1.27958
$680$	17178.4	0.968768
$681$	0	0
$682$	4143.50	0.232644
$683$	−176.490	−0.00988755	−0.00494378	−	0.999988i	$-0.501574\pi$
$683$	−176.490	−0.00988755	−0.00494378	+	0.999988i	$0.501574\pi$
$684$	0	0
$685$	−14406.1	−0.803543
$686$	−30363.1	−1.68990
$687$	0	0
$688$	6846.61	0.379396
$689$	−348.189	−0.0192525
$690$	0	0
$691$	28767.9	1.58377	0.791884	−	0.610672i	$-0.209101\pi$
$691$	28767.9	1.58377	0.791884	+	0.610672i	$0.209101\pi$
$692$	26011.0	1.42889
$693$	0	0
$694$	−7078.46	−0.387168
$695$	12922.2	0.705275
$696$	0	0
$697$	16427.9	0.892757
$698$	−51787.2	−2.80827
$699$	0	0
$700$	8486.14	0.458208
$701$	27326.9	1.47236	0.736179	−	0.676787i	$-0.236628\pi$
$701$	27326.9	1.47236	0.736179	+	0.676787i	$0.236628\pi$
$702$	0	0
$703$	−10362.8	−0.555961
$704$	998.274	0.0534430
$705$	0	0
$706$	16717.0	0.891152
$707$	−19483.2	−1.03641
$708$	0	0
$709$	3138.76	0.166261	0.0831303	−	0.996539i	$-0.473508\pi$
$709$	3138.76	0.166261	0.0831303	+	0.996539i	$0.473508\pi$
$710$	−10855.9	−0.573825
$711$	0	0
$712$	13020.4	0.685339
$713$	22323.1	1.17252
$714$	0	0
$715$	−1258.89	−0.0658458
$716$	−1250.75	−0.0652830
$717$	0	0
$718$	−52663.1	−2.73728
$719$	−21026.6	−1.09062	−0.545312	−	0.838233i	$-0.683589\pi$
$719$	−21026.6	−1.09062	−0.545312	+	0.838233i	$0.683589\pi$
$720$	0	0
$721$	13654.1	0.705280
$722$	4742.52	0.244458
$723$	0	0
$724$	−29697.7	−1.52446
$725$	−725.000	−0.0371391
$726$	0	0
$727$	4458.08	0.227429	0.113715	−	0.993513i	$-0.463725\pi$
$727$	4458.08	0.227429	0.113715	+	0.993513i	$0.463725\pi$
$728$	34397.0	1.75115
$729$	0	0
$730$	14012.6	0.710452
$731$	4782.02	0.241955
$732$	0	0
$733$	−28933.7	−1.45797	−0.728984	−	0.684531i	$-0.760007\pi$
$733$	−28933.7	−1.45797	−0.728984	+	0.684531i	$0.760007\pi$
$734$	−33674.6	−1.69339
$735$	0	0
$736$	−24710.4	−1.23755
$737$	−6494.56	−0.324600
$738$	0	0
$739$	−9135.10	−0.454723	−0.227361	−	0.973810i	$-0.573010\pi$
$739$	−9135.10	−0.454723	−0.227361	+	0.973810i	$0.573010\pi$
$740$	−10272.4	−0.510299
$741$	0	0
$742$	−922.819	−0.0456574
$743$	26380.8	1.30258	0.651290	−	0.758829i	$-0.274228\pi$
$743$	26380.8	1.30258	0.651290	+	0.758829i	$0.274228\pi$
$744$	0	0
$745$	7246.13	0.356346
$746$	30810.6	1.51214
$747$	0	0
$748$	−8536.49	−0.417279
$749$	−21647.6	−1.05605
$750$	0	0
$751$	18205.6	0.884597	0.442298	−	0.896868i	$-0.354163\pi$
$751$	18205.6	0.884597	0.442298	+	0.896868i	$0.354163\pi$
$752$	4304.44	0.208733
$753$	0	0
$754$	−5411.44	−0.261370
$755$	13802.3	0.665321
$756$	0	0
$757$	14889.8	0.714902	0.357451	−	0.933932i	$-0.383646\pi$
$757$	14889.8	0.714902	0.357451	+	0.933932i	$0.383646\pi$
$758$	31777.0	1.52268
$759$	0	0
$760$	21200.6	1.01188
$761$	−4359.81	−0.207678	−0.103839	−	0.994594i	$-0.533113\pi$
$761$	−4359.81	−0.207678	−0.103839	+	0.994594i	$0.533113\pi$
$762$	0	0
$763$	−25331.2	−1.20190
$764$	64038.0	3.03248
$765$	0	0
$766$	−52268.7	−2.46546
$767$	8589.64	0.404373
$768$	0	0
$769$	−23823.7	−1.11717	−0.558585	−	0.829448i	$-0.688655\pi$
$769$	−23823.7	−1.11717	−0.558585	+	0.829448i	$0.688655\pi$
$770$	−3336.48	−0.156154
$771$	0	0
$772$	−55265.6	−2.57649
$773$	−18653.0	−0.867922	−0.433961	−	0.900932i	$-0.642884\pi$
$773$	−18653.0	−0.867922	−0.433961	+	0.900932i	$0.642884\pi$
$774$	0	0
$775$	−3009.83	−0.139505
$776$	56066.5	2.59365
$777$	0	0
$778$	11924.6	0.549508
$779$	20274.4	0.932483
$780$	0	0
$781$	2929.54	0.134222
$782$	−67005.6	−3.06409
$783$	0	0
$784$	3373.38	0.153671
$785$	3832.32	0.174244
$786$	0	0
$787$	16426.1	0.743998	0.371999	−	0.928233i	$-0.378672\pi$
$787$	16426.1	0.743998	0.371999	+	0.928233i	$0.378672\pi$
$788$	16973.7	0.767337
$789$	0	0
$790$	24372.8	1.09765
$791$	−4969.26	−0.223371
$792$	0	0
$793$	−20193.2	−0.904266
$794$	35167.0	1.57183
$795$	0	0
$796$	94942.0	4.22755
$797$	8480.05	0.376887	0.188443	−	0.982084i	$-0.439656\pi$
$797$	8480.05	0.376887	0.188443	+	0.982084i	$0.439656\pi$
$798$	0	0
$799$	3006.44	0.133117
$800$	3331.72	0.147243
$801$	0	0
$802$	7168.23	0.315610
$803$	−3781.39	−0.166180
$804$	0	0
$805$	−17975.2	−0.787011
$806$	−22465.5	−0.981779
$807$	0	0
$808$	−48249.5	−2.10076
$809$	28294.8	1.22965	0.614827	−	0.788662i	$-0.289226\pi$
$809$	28294.8	1.22965	0.614827	+	0.788662i	$0.289226\pi$
$810$	0	0
$811$	−28383.8	−1.22896	−0.614482	−	0.788931i	$-0.710635\pi$
$811$	−28383.8	−1.22896	−0.614482	+	0.788931i	$0.710635\pi$
$812$	−9843.92	−0.425436
$813$	0	0
$814$	4038.79	0.173906
$815$	−9355.54	−0.402098
$816$	0	0
$817$	5901.69	0.252722
$818$	−26262.1	−1.12253
$819$	0	0
$820$	20097.5	0.855897
$821$	−27288.2	−1.16001	−0.580003	−	0.814615i	$-0.696948\pi$
$821$	−27288.2	−1.16001	−0.580003	+	0.814615i	$0.696948\pi$
$822$	0	0
$823$	−37261.0	−1.57818	−0.789088	−	0.614281i	$-0.789446\pi$
$823$	−37261.0	−1.57818	−0.789088	+	0.614281i	$0.789446\pi$
$824$	33814.1	1.42957
$825$	0	0
$826$	22765.5	0.958973
$827$	−2964.91	−0.124668	−0.0623338	−	0.998055i	$-0.519854\pi$
$827$	−2964.91	−0.124668	−0.0623338	+	0.998055i	$0.519854\pi$
$828$	0	0
$829$	35023.4	1.46733	0.733664	−	0.679513i	$-0.237809\pi$
$829$	35023.4	1.46733	0.733664	+	0.679513i	$0.237809\pi$
$830$	−15787.4	−0.660227
$831$	0	0
$832$	−5412.50	−0.225535
$833$	2356.14	0.0980016
$834$	0	0
$835$	−11718.1	−0.485654
$836$	−10535.2	−0.435848
$837$	0	0
$838$	−22606.2	−0.931884
$839$	5316.37	0.218762	0.109381	−	0.994000i	$-0.465113\pi$
$839$	5316.37	0.218762	0.109381	+	0.994000i	$0.465113\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	14770.3	0.604533
$843$	0	0
$844$	−49869.5	−2.03386
$845$	−4159.48	−0.169338
$846$	0	0
$847$	−24906.2	−1.01038
$848$	−965.437	−0.0390958
$849$	0	0
$850$	9034.39	0.364561
$851$	21758.9	0.876481
$852$	0	0
$853$	−40627.7	−1.63079	−0.815396	−	0.578904i	$-0.803481\pi$
$853$	−40627.7	−1.63079	−0.815396	+	0.578904i	$0.803481\pi$
$854$	−53518.9	−2.14447
$855$	0	0
$856$	−53609.5	−2.14058
$857$	−22175.9	−0.883914	−0.441957	−	0.897036i	$-0.645716\pi$
$857$	−22175.9	−0.883914	−0.441957	+	0.897036i	$0.645716\pi$
$858$	0	0
$859$	−39450.8	−1.56699	−0.783495	−	0.621399i	$-0.786565\pi$
$859$	−39450.8	−1.56699	−0.783495	+	0.621399i	$0.786565\pi$
$860$	5850.21	0.231966
$861$	0	0
$862$	−7520.06	−0.297139
$863$	−5285.21	−0.208471	−0.104236	−	0.994553i	$-0.533240\pi$
$863$	−5285.21	−0.208471	−0.104236	+	0.994553i	$0.533240\pi$
$864$	0	0
$865$	7428.64	0.292002
$866$	−42483.7	−1.66704
$867$	0	0
$868$	−40866.9	−1.59806
$869$	−6577.14	−0.256748
$870$	0	0
$871$	35212.6	1.36984
$872$	−62732.0	−2.43621
$873$	0	0
$874$	−82694.4	−3.20043
$875$	2423.61	0.0936377
$876$	0	0
$877$	17785.8	0.684815	0.342407	−	0.939552i	$-0.388758\pi$
$877$	17785.8	0.684815	0.342407	+	0.939552i	$0.388758\pi$
$878$	−53457.9	−2.05480
$879$	0	0
$880$	−3490.57	−0.133713
$881$	−19542.5	−0.747338	−0.373669	−	0.927562i	$-0.621900\pi$
$881$	−19542.5	−0.747338	−0.373669	+	0.927562i	$0.621900\pi$
$882$	0	0
$883$	−1625.38	−0.0619460	−0.0309730	−	0.999520i	$-0.509861\pi$
$883$	−1625.38	−0.0619460	−0.0309730	+	0.999520i	$0.509861\pi$
$884$	46283.7	1.76096
$885$	0	0
$886$	52199.5	1.97932
$887$	48250.9	1.82650	0.913251	−	0.407396i	$-0.133563\pi$
$887$	48250.9	1.82650	0.913251	+	0.407396i	$0.133563\pi$
$888$	0	0
$889$	−23291.8	−0.878722
$890$	6847.65	0.257903
$891$	0	0
$892$	8797.18	0.330215
$893$	3710.37	0.139040
$894$	0	0
$895$	−357.209	−0.0133410
$896$	−35016.4	−1.30560
$897$	0	0
$898$	−83972.6	−3.12049
$899$	3491.40	0.129527
$900$	0	0
$901$	−674.310	−0.0249329
$902$	−7901.70	−0.291683
$903$	0	0
$904$	−12306.2	−0.452764
$905$	−8481.56	−0.311532
$906$	0	0
$907$	20513.2	0.750971	0.375486	−	0.926828i	$-0.377476\pi$
$907$	20513.2	0.750971	0.375486	+	0.926828i	$0.377476\pi$
$908$	−23404.2	−0.855393
$909$	0	0
$910$	18089.9	0.658984
$911$	33692.7	1.22534	0.612672	−	0.790338i	$-0.290095\pi$
$911$	33692.7	1.22534	0.612672	+	0.790338i	$0.290095\pi$
$912$	0	0
$913$	4260.33	0.154432
$914$	−45495.9	−1.64647
$915$	0	0
$916$	−19805.3	−0.714394
$917$	−20349.9	−0.732839
$918$	0	0
$919$	36764.9	1.31966	0.659828	−	0.751417i	$-0.270629\pi$
$919$	36764.9	1.31966	0.659828	+	0.751417i	$0.270629\pi$
$920$	−44515.1	−1.59524
$921$	0	0
$922$	40895.5	1.46076
$923$	−15883.6	−0.566429
$924$	0	0
$925$	−2933.76	−0.104283
$926$	24736.6	0.877858
$927$	0	0
$928$	−3864.79	−0.136711
$929$	−13533.8	−0.477965	−0.238982	−	0.971024i	$-0.576814\pi$
$929$	−13533.8	−0.477965	−0.238982	+	0.971024i	$0.576814\pi$
$930$	0	0
$931$	2907.81	0.102363
$932$	−22262.7	−0.782444
$933$	0	0
$934$	16824.2	0.589406
$935$	−2437.99	−0.0852736
$936$	0	0
$937$	−54144.0	−1.88773	−0.943867	−	0.330325i	$-0.892842\pi$
$937$	−54144.0	−1.88773	−0.943867	+	0.330325i	$0.892842\pi$
$938$	93325.3	3.24859
$939$	0	0
$940$	3678.01	0.127621
$941$	8397.60	0.290918	0.145459	−	0.989364i	$-0.453534\pi$
$941$	8397.60	0.290918	0.145459	+	0.989364i	$0.453534\pi$
$942$	0	0
$943$	−42570.3	−1.47007
$944$	23816.8	0.821157
$945$	0	0
$946$	−2300.12	−0.0790521
$947$	4931.24	0.169212	0.0846059	−	0.996414i	$-0.473037\pi$
$947$	4931.24	0.169212	0.0846059	+	0.996414i	$0.473037\pi$
$948$	0	0
$949$	20502.2	0.701295
$950$	11149.7	0.380784
$951$	0	0
$952$	66614.0	2.26783
$953$	21509.1	0.731111	0.365555	−	0.930790i	$-0.380879\pi$
$953$	21509.1	0.731111	0.365555	+	0.930790i	$0.380879\pi$
$954$	0	0
$955$	18289.0	0.619705
$956$	−26390.4	−0.892811
$957$	0	0
$958$	68718.0	2.31751
$959$	−55863.4	−1.88105
$960$	0	0
$961$	−15296.5	−0.513460
$962$	−21897.7	−0.733900
$963$	0	0
$964$	74428.8	2.48671
$965$	−15783.6	−0.526522
$966$	0	0
$967$	56174.6	1.86810	0.934051	−	0.357140i	$-0.116248\pi$
$967$	56174.6	1.86810	0.934051	+	0.357140i	$0.116248\pi$
$968$	−61679.5	−2.04799
$969$	0	0
$970$	29486.2	0.976027
$971$	−59470.2	−1.96549	−0.982745	−	0.184966i	$-0.940783\pi$
$971$	−59470.2	−1.96549	−0.982745	+	0.184966i	$0.940783\pi$
$972$	0	0
$973$	50109.3	1.65101
$974$	−78084.9	−2.56879
$975$	0	0
$976$	−55990.5	−1.83628
$977$	54861.4	1.79649	0.898245	−	0.439494i	$-0.144842\pi$
$977$	54861.4	1.79649	0.898245	+	0.439494i	$0.144842\pi$
$978$	0	0
$979$	−1847.88	−0.0603254
$980$	2882.44	0.0939554
$981$	0	0
$982$	96041.5	3.12099
$983$	20759.3	0.673568	0.336784	−	0.941582i	$-0.390661\pi$
$983$	20759.3	0.673568	0.336784	+	0.941582i	$0.390661\pi$
$984$	0	0
$985$	4847.62	0.156810
$986$	−10479.9	−0.338487
$987$	0	0
$988$	57120.6	1.83932
$989$	−12391.8	−0.398420
$990$	0	0
$991$	−25662.6	−0.822603	−0.411301	−	0.911499i	$-0.634926\pi$
$991$	−25662.6	−0.822603	−0.411301	+	0.911499i	$0.634926\pi$
$992$	−16044.6	−0.513526
$993$	0	0
$994$	−42096.8	−1.34329
$995$	27115.1	0.863926
$996$	0	0
$997$	43697.7	1.38808	0.694042	−	0.719935i	$-0.255828\pi$
$997$	43697.7	1.38808	0.694042	+	0.719935i	$0.255828\pi$
$998$	−69065.4	−2.19061
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.h.1.6		6
3.2	odd	2		435.4.a.h.1.1	✓	6
15.14	odd	2		2175.4.a.k.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.h.1.1	✓	6	3.2	odd	2
1305.4.a.h.1.6		6	1.1	even	1	trivial
2175.4.a.k.1.6		6	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q+5.05047 q^{2} +17.5072 q^{4} +5.00000 q^{5} +19.3889 q^{7} +48.0160 q^{8} +25.2523 q^{10} -6.81451 q^{11} +36.9473 q^{13} +97.9229 q^{14} +102.445 q^{16} +71.5529 q^{17} +88.3064 q^{19} +87.5361 q^{20} -34.4164 q^{22} -185.418 q^{23} +25.0000 q^{25} +186.601 q^{26} +339.446 q^{28} -29.0000 q^{29} -120.393 q^{31} +133.269 q^{32} +361.376 q^{34} +96.9444 q^{35} -117.351 q^{37} +445.989 q^{38} +240.080 q^{40} +229.591 q^{41} +66.8319 q^{43} -119.303 q^{44} -936.448 q^{46} +42.0170 q^{47} +32.9286 q^{49} +126.262 q^{50} +646.845 q^{52} -9.42394 q^{53} -34.0725 q^{55} +930.976 q^{56} -146.464 q^{58} +232.484 q^{59} -546.541 q^{61} -608.042 q^{62} -146.492 q^{64} +184.737 q^{65} +953.049 q^{67} +1252.69 q^{68} +489.615 q^{70} -429.898 q^{71} +554.903 q^{73} -592.675 q^{74} +1546.00 q^{76} -132.126 q^{77} +965.168 q^{79} +512.226 q^{80} +1159.54 q^{82} -625.186 q^{83} +357.765 q^{85} +337.532 q^{86} -327.205 q^{88} +271.169 q^{89} +716.367 q^{91} -3246.16 q^{92} +212.206 q^{94} +441.532 q^{95} +1167.66 q^{97} +166.305 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 51 q^{4} + 30 q^{5} + 47 q^{7} - 51 q^{8} - 5 q^{10} - 81 q^{11} + 169 q^{13} + 30 q^{14} + 131 q^{16} + q^{17} + 116 q^{19} + 255 q^{20} + 90 q^{22} + 52 q^{23} + 150 q^{25} - 294 q^{26}+ \cdots + 1433 q^{98}+O(q^{100}) \) 6 * q - q^2 + 51 * q^4 + 30 * q^5 + 47 * q^7 - 51 * q^8 - 5 * q^10 - 81 * q^11 + 169 * q^13 + 30 * q^14 + 131 * q^16 + q^17 + 116 * q^19 + 255 * q^20 + 90 * q^22 + 52 * q^23 + 150 * q^25 - 294 * q^26 + 344 * q^28 - 174 * q^29 + 340 * q^31 - 499 * q^32 + 920 * q^34 + 235 * q^35 + 332 * q^37 + 378 * q^38 - 255 * q^40 + 616 * q^41 + 334 * q^43 + 52 * q^44 - 158 * q^46 + 85 * q^47 + 879 * q^49 - 25 * q^50 + 2220 * q^52 + 850 * q^53 - 405 * q^55 + 624 * q^56 + 29 * q^58 + 758 * q^59 - 36 * q^61 + 152 * q^62 + 1795 * q^64 + 845 * q^65 + 939 * q^67 + 186 * q^68 + 150 * q^70 + 1388 * q^71 + 1708 * q^73 + 814 * q^74 + 566 * q^76 - 2585 * q^77 + 1250 * q^79 + 655 * q^80 + 1372 * q^82 + 748 * q^83 + 5 * q^85 + 800 * q^86 + 536 * q^88 - 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 4542 * q^94 + 580 * q^95 - 22 * q^97 + 1433 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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