LMFDB - Embedded newform 1856.4.a.bj.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, 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("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + \cdots + 14751072 $ x^12 - 2x^11 - 217x^10 + 520x^9 + 16022x^8 - 37368x^7 - 509640x^6 + 989168x^5 + 7106592x^4 - 7979328x^3 - 38912400x^2 + 1083456*x + 14751072
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.613335$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61334 q^{3} +10.5322 q^{5} +13.7418 q^{7} -24.3972 q^{9} +O(q^{10})$	$q-1.61334 q^{3} +10.5322 q^{5} +13.7418 q^{7} -24.3972 q^{9} +30.2948 q^{11} -19.9286 q^{13} -16.9919 q^{15} +105.498 q^{17} +79.5429 q^{19} -22.1701 q^{21} -14.8819 q^{23} -14.0732 q^{25} +82.9208 q^{27} -29.0000 q^{29} -61.9562 q^{31} -48.8756 q^{33} +144.731 q^{35} -6.43644 q^{37} +32.1516 q^{39} +351.682 q^{41} -473.240 q^{43} -256.955 q^{45} +492.253 q^{47} -154.163 q^{49} -170.203 q^{51} +668.911 q^{53} +319.070 q^{55} -128.329 q^{57} -12.7826 q^{59} -80.8937 q^{61} -335.261 q^{63} -209.892 q^{65} -529.153 q^{67} +24.0095 q^{69} -211.869 q^{71} +627.334 q^{73} +22.7048 q^{75} +416.305 q^{77} +915.349 q^{79} +524.944 q^{81} -411.330 q^{83} +1111.12 q^{85} +46.7867 q^{87} +1377.88 q^{89} -273.856 q^{91} +99.9561 q^{93} +837.761 q^{95} -649.398 q^{97} -739.106 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9}+O(q^{10}) $ 12 * q - 14 * q^3 + 10 * q^5 + 44 * q^7 + 130 * q^9	$ 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9} - 46 q^{11} + 34 q^{13} - 50 q^{15} + 36 q^{17} - 148 q^{19} + 92 q^{21} + 328 q^{23} + 486 q^{25} - 326 q^{27} - 348 q^{29} + 374 q^{31} + 710 q^{33} - 204 q^{35} + 340 q^{37} - 122 q^{39} + 32 q^{41} - 462 q^{43} + 1132 q^{45} + 434 q^{47} + 1508 q^{49} - 440 q^{51} - 610 q^{53} + 46 q^{55} - 932 q^{57} - 1240 q^{59} + 1228 q^{61} + 4240 q^{63} + 730 q^{65} - 1672 q^{67} + 528 q^{69} + 3220 q^{71} + 564 q^{73} - 6032 q^{75} - 644 q^{77} + 1862 q^{79} + 3040 q^{81} - 3736 q^{83} + 808 q^{85} + 406 q^{87} + 584 q^{89} - 4844 q^{91} + 3226 q^{93} + 2844 q^{95} + 904 q^{97} - 6832 q^{99}+O(q^{100}) $ 12 * q - 14 * q^3 + 10 * q^5 + 44 * q^7 + 130 * q^9 - 46 * q^11 + 34 * q^13 - 50 * q^15 + 36 * q^17 - 148 * q^19 + 92 * q^21 + 328 * q^23 + 486 * q^25 - 326 * q^27 - 348 * q^29 + 374 * q^31 + 710 * q^33 - 204 * q^35 + 340 * q^37 - 122 * q^39 + 32 * q^41 - 462 * q^43 + 1132 * q^45 + 434 * q^47 + 1508 * q^49 - 440 * q^51 - 610 * q^53 + 46 * q^55 - 932 * q^57 - 1240 * q^59 + 1228 * q^61 + 4240 * q^63 + 730 * q^65 - 1672 * q^67 + 528 * q^69 + 3220 * q^71 + 564 * q^73 - 6032 * q^75 - 644 * q^77 + 1862 * q^79 + 3040 * q^81 - 3736 * q^83 + 808 * q^85 + 406 * q^87 + 584 * q^89 - 4844 * q^91 + 3226 * q^93 + 2844 * q^95 + 904 * q^97 - 6832 * 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
$3$	−1.61334	−0.310486	−0.155243	−	0.987876i	$-0.549616\pi$
$3$	−1.61334	−0.310486	−0.155243	+	0.987876i	$0.549616\pi$
$4$	0	0
$5$	10.5322	0.942027	0.471013	−	0.882126i	$-0.343888\pi$
$5$	10.5322	0.942027	0.471013	+	0.882126i	$0.343888\pi$
$6$	0	0
$7$	13.7418	0.741988	0.370994	−	0.928635i	$-0.379017\pi$
$7$	13.7418	0.741988	0.370994	+	0.928635i	$0.379017\pi$
$8$	0	0
$9$	−24.3972	−0.903598
$10$	0	0
$11$	30.2948	0.830384	0.415192	−	0.909734i	$-0.363714\pi$
$11$	30.2948	0.830384	0.415192	+	0.909734i	$0.363714\pi$
$12$	0	0
$13$	−19.9286	−0.425170	−0.212585	−	0.977143i	$-0.568188\pi$
$13$	−19.9286	−0.425170	−0.212585	+	0.977143i	$0.568188\pi$
$14$	0	0
$15$	−16.9919	−0.292487
$16$	0	0
$17$	105.498	1.50512	0.752558	−	0.658526i	$-0.228820\pi$
$17$	105.498	1.50512	0.752558	+	0.658526i	$0.228820\pi$
$18$	0	0
$19$	79.5429	0.960442	0.480221	−	0.877147i	$-0.340556\pi$
$19$	79.5429	0.960442	0.480221	+	0.877147i	$0.340556\pi$
$20$	0	0
$21$	−22.1701	−0.230377
$22$	0	0
$23$	−14.8819	−0.134917	−0.0674586	−	0.997722i	$-0.521489\pi$
$23$	−14.8819	−0.134917	−0.0674586	+	0.997722i	$0.521489\pi$
$24$	0	0
$25$	−14.0732	−0.112586
$26$	0	0
$27$	82.9208	0.591041
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−61.9562	−0.358957	−0.179478	−	0.983762i	$-0.557441\pi$
$31$	−61.9562	−0.358957	−0.179478	+	0.983762i	$0.557441\pi$
$32$	0	0
$33$	−48.8756	−0.257823
$34$	0	0
$35$	144.731	0.698972
$36$	0	0
$37$	−6.43644	−0.0285985	−0.0142993	−	0.999898i	$-0.504552\pi$
$37$	−6.43644	−0.0285985	−0.0142993	+	0.999898i	$0.504552\pi$
$38$	0	0
$39$	32.1516	0.132010
$40$	0	0
$41$	351.682	1.33960	0.669799	−	0.742543i	$-0.266380\pi$
$41$	351.682	1.33960	0.669799	+	0.742543i	$0.266380\pi$
$42$	0	0
$43$	−473.240	−1.67834	−0.839169	−	0.543871i	$-0.816958\pi$
$43$	−473.240	−1.67834	−0.839169	+	0.543871i	$0.816958\pi$
$44$	0	0
$45$	−256.955	−0.851214
$46$	0	0
$47$	492.253	1.52771	0.763856	−	0.645387i	$-0.223304\pi$
$47$	492.253	1.52771	0.763856	+	0.645387i	$0.223304\pi$
$48$	0	0
$49$	−154.163	−0.449454
$50$	0	0
$51$	−170.203	−0.467318
$52$	0	0
$53$	668.911	1.73362	0.866811	−	0.498637i	$-0.166166\pi$
$53$	668.911	1.73362	0.866811	+	0.498637i	$0.166166\pi$
$54$	0	0
$55$	319.070	0.782244
$56$	0	0
$57$	−128.329	−0.298204
$58$	0	0
$59$	−12.7826	−0.0282060	−0.0141030	−	0.999901i	$-0.504489\pi$
$59$	−12.7826	−0.0282060	−0.0141030	+	0.999901i	$0.504489\pi$
$60$	0	0
$61$	−80.8937	−0.169793	−0.0848965	−	0.996390i	$-0.527056\pi$
$61$	−80.8937	−0.169793	−0.0848965	+	0.996390i	$0.527056\pi$
$62$	0	0
$63$	−335.261	−0.670459
$64$	0	0
$65$	−209.892	−0.400521
$66$	0	0
$67$	−529.153	−0.964870	−0.482435	−	0.875932i	$-0.660248\pi$
$67$	−529.153	−0.964870	−0.482435	+	0.875932i	$0.660248\pi$
$68$	0	0
$69$	24.0095	0.0418899
$70$	0	0
$71$	−211.869	−0.354144	−0.177072	−	0.984198i	$-0.556663\pi$
$71$	−211.869	−0.354144	−0.177072	+	0.984198i	$0.556663\pi$
$72$	0	0
$73$	627.334	1.00581	0.502903	−	0.864343i	$-0.332265\pi$
$73$	627.334	1.00581	0.502903	+	0.864343i	$0.332265\pi$
$74$	0	0
$75$	22.7048	0.0349563
$76$	0	0
$77$	416.305	0.616135
$78$	0	0
$79$	915.349	1.30361	0.651803	−	0.758389i	$-0.274013\pi$
$79$	915.349	1.30361	0.651803	+	0.758389i	$0.274013\pi$
$80$	0	0
$81$	524.944	0.720088
$82$	0	0
$83$	−411.330	−0.543967	−0.271984	−	0.962302i	$-0.587680\pi$
$83$	−411.330	−0.543967	−0.271984	+	0.962302i	$0.587680\pi$
$84$	0	0
$85$	1111.12	1.41786
$86$	0	0
$87$	46.7867	0.0576559
$88$	0	0
$89$	1377.88	1.64107	0.820536	−	0.571595i	$-0.193675\pi$
$89$	1377.88	1.64107	0.820536	+	0.571595i	$0.193675\pi$
$90$	0	0
$91$	−273.856	−0.315471
$92$	0	0
$93$	99.9561	0.111451
$94$	0	0
$95$	837.761	0.904762
$96$	0	0
$97$	−649.398	−0.679756	−0.339878	−	0.940470i	$-0.610386\pi$
$97$	−649.398	−0.679756	−0.339878	+	0.940470i	$0.610386\pi$
$98$	0	0
$99$	−739.106	−0.750333
$100$	0	0
$101$	851.483	0.838869	0.419434	−	0.907786i	$-0.362228\pi$
$101$	851.483	0.838869	0.419434	+	0.907786i	$0.362228\pi$
$102$	0	0
$103$	−1923.60	−1.84017	−0.920085	−	0.391719i	$-0.871881\pi$
$103$	−1923.60	−1.84017	−0.920085	+	0.391719i	$0.871881\pi$
$104$	0	0
$105$	−233.500	−0.217021
$106$	0	0
$107$	−1648.55	−1.48945	−0.744726	−	0.667370i	$-0.767420\pi$
$107$	−1648.55	−1.48945	−0.744726	+	0.667370i	$0.767420\pi$
$108$	0	0
$109$	−770.267	−0.676865	−0.338432	−	0.940991i	$-0.609897\pi$
$109$	−770.267	−0.676865	−0.338432	+	0.940991i	$0.609897\pi$
$110$	0	0
$111$	10.3841	0.00887945
$112$	0	0
$113$	1429.70	1.19022	0.595111	−	0.803643i	$-0.297108\pi$
$113$	1429.70	1.19022	0.595111	+	0.803643i	$0.297108\pi$
$114$	0	0
$115$	−156.739	−0.127096
$116$	0	0
$117$	486.202	0.384183
$118$	0	0
$119$	1449.73	1.11678
$120$	0	0
$121$	−413.226	−0.310463
$122$	0	0
$123$	−567.381	−0.415927
$124$	0	0
$125$	−1464.74	−1.04809
$126$	0	0
$127$	2102.32	1.46890	0.734451	−	0.678662i	$-0.237440\pi$
$127$	2102.32	1.46890	0.734451	+	0.678662i	$0.237440\pi$
$128$	0	0
$129$	763.495	0.521101
$130$	0	0
$131$	−1376.03	−0.917742	−0.458871	−	0.888503i	$-0.651746\pi$
$131$	−1376.03	−0.917742	−0.458871	+	0.888503i	$0.651746\pi$
$132$	0	0
$133$	1093.06	0.712637
$134$	0	0
$135$	873.337	0.556777
$136$	0	0
$137$	304.947	0.190171	0.0950853	−	0.995469i	$-0.469688\pi$
$137$	304.947	0.190171	0.0950853	+	0.995469i	$0.469688\pi$
$138$	0	0
$139$	364.348	0.222328	0.111164	−	0.993802i	$-0.464542\pi$
$139$	364.348	0.222328	0.111164	+	0.993802i	$0.464542\pi$
$140$	0	0
$141$	−794.169	−0.474334
$142$	0	0
$143$	−603.734	−0.353054
$144$	0	0
$145$	−305.433	−0.174930
$146$	0	0
$147$	248.716	0.139549
$148$	0	0
$149$	−200.018	−0.109974	−0.0549869	−	0.998487i	$-0.517512\pi$
$149$	−200.018	−0.109974	−0.0549869	+	0.998487i	$0.517512\pi$
$150$	0	0
$151$	−35.5937	−0.0191826	−0.00959130	−	0.999954i	$-0.503053\pi$
$151$	−35.5937	−0.0191826	−0.00959130	+	0.999954i	$0.503053\pi$
$152$	0	0
$153$	−2573.85	−1.36002
$154$	0	0
$155$	−652.534	−0.338147
$156$	0	0
$157$	−137.397	−0.0698437	−0.0349218	−	0.999390i	$-0.511118\pi$
$157$	−137.397	−0.0698437	−0.0349218	+	0.999390i	$0.511118\pi$
$158$	0	0
$159$	−1079.18	−0.538266
$160$	0	0
$161$	−204.504	−0.100107
$162$	0	0
$163$	867.825	0.417014	0.208507	−	0.978021i	$-0.433140\pi$
$163$	867.825	0.417014	0.208507	+	0.978021i	$0.433140\pi$
$164$	0	0
$165$	−514.767	−0.242876
$166$	0	0
$167$	−66.2847	−0.0307142	−0.0153571	−	0.999882i	$-0.504889\pi$
$167$	−66.2847	−0.0307142	−0.0153571	+	0.999882i	$0.504889\pi$
$168$	0	0
$169$	−1799.85	−0.819230
$170$	0	0
$171$	−1940.62	−0.867854
$172$	0	0
$173$	1798.76	0.790505	0.395253	−	0.918572i	$-0.370657\pi$
$173$	1798.76	0.790505	0.395253	+	0.918572i	$0.370657\pi$
$174$	0	0
$175$	−193.391	−0.0835372
$176$	0	0
$177$	20.6226	0.00875759
$178$	0	0
$179$	−1214.69	−0.507208	−0.253604	−	0.967308i	$-0.581616\pi$
$179$	−1214.69	−0.507208	−0.253604	+	0.967308i	$0.581616\pi$
$180$	0	0
$181$	−643.623	−0.264310	−0.132155	−	0.991229i	$-0.542190\pi$
$181$	−643.623	−0.264310	−0.132155	+	0.991229i	$0.542190\pi$
$182$	0	0
$183$	130.509	0.0527184
$184$	0	0
$185$	−67.7898	−0.0269406
$186$	0	0
$187$	3196.03	1.24982
$188$	0	0
$189$	1139.48	0.438546
$190$	0	0
$191$	4085.97	1.54791	0.773953	−	0.633243i	$-0.218277\pi$
$191$	4085.97	1.54791	0.773953	+	0.633243i	$0.218277\pi$
$192$	0	0
$193$	864.326	0.322360	0.161180	−	0.986925i	$-0.448470\pi$
$193$	864.326	0.322360	0.161180	+	0.986925i	$0.448470\pi$
$194$	0	0
$195$	338.626	0.124357
$196$	0	0
$197$	1935.81	0.700106	0.350053	−	0.936730i	$-0.386164\pi$
$197$	1935.81	0.700106	0.350053	+	0.936730i	$0.386164\pi$
$198$	0	0
$199$	3593.39	1.28005	0.640023	−	0.768356i	$-0.278925\pi$
$199$	3593.39	1.28005	0.640023	+	0.768356i	$0.278925\pi$
$200$	0	0
$201$	853.701	0.299579
$202$	0	0
$203$	−398.512	−0.137784
$204$	0	0
$205$	3703.98	1.26194
$206$	0	0
$207$	363.076	0.121911
$208$	0	0
$209$	2409.74	0.797536
$210$	0	0
$211$	−692.347	−0.225892	−0.112946	−	0.993601i	$-0.536029\pi$
$211$	−692.347	−0.225892	−0.112946	+	0.993601i	$0.536029\pi$
$212$	0	0
$213$	341.816	0.109957
$214$	0	0
$215$	−4984.25	−1.58104
$216$	0	0
$217$	−851.390	−0.266342
$218$	0	0
$219$	−1012.10	−0.312289
$220$	0	0
$221$	−2102.43	−0.639930
$222$	0	0
$223$	1770.55	0.531680	0.265840	−	0.964017i	$-0.414351\pi$
$223$	1770.55	0.531680	0.265840	+	0.964017i	$0.414351\pi$
$224$	0	0
$225$	343.346	0.101732
$226$	0	0
$227$	2996.07	0.876017	0.438008	−	0.898971i	$-0.355684\pi$
$227$	2996.07	0.876017	0.438008	+	0.898971i	$0.355684\pi$
$228$	0	0
$229$	3790.17	1.09372	0.546860	−	0.837224i	$-0.315823\pi$
$229$	3790.17	1.09372	0.546860	+	0.837224i	$0.315823\pi$
$230$	0	0
$231$	−671.640	−0.191301
$232$	0	0
$233$	6257.35	1.75937	0.879684	−	0.475558i	$-0.157754\pi$
$233$	6257.35	1.75937	0.879684	+	0.475558i	$0.157754\pi$
$234$	0	0
$235$	5184.50	1.43915
$236$	0	0
$237$	−1476.77	−0.404752
$238$	0	0
$239$	5893.50	1.59506	0.797529	−	0.603280i	$-0.206140\pi$
$239$	5893.50	1.59506	0.797529	+	0.603280i	$0.206140\pi$
$240$	0	0
$241$	−418.978	−0.111986	−0.0559932	−	0.998431i	$-0.517833\pi$
$241$	−418.978	−0.111986	−0.0559932	+	0.998431i	$0.517833\pi$
$242$	0	0
$243$	−3085.77	−0.814619
$244$	0	0
$245$	−1623.67	−0.423398
$246$	0	0
$247$	−1585.18	−0.408351
$248$	0	0
$249$	663.612	0.168894
$250$	0	0
$251$	−7444.38	−1.87205	−0.936026	−	0.351932i	$-0.885525\pi$
$251$	−7444.38	−1.87205	−0.936026	+	0.351932i	$0.885525\pi$
$252$	0	0
$253$	−450.844	−0.112033
$254$	0	0
$255$	−1792.61	−0.440226
$256$	0	0
$257$	5332.46	1.29428	0.647140	−	0.762371i	$-0.275965\pi$
$257$	5332.46	1.29428	0.647140	+	0.762371i	$0.275965\pi$
$258$	0	0
$259$	−88.4484	−0.0212197
$260$	0	0
$261$	707.517	0.167794
$262$	0	0
$263$	832.925	0.195286	0.0976432	−	0.995221i	$-0.468870\pi$
$263$	832.925	0.195286	0.0976432	+	0.995221i	$0.468870\pi$
$264$	0	0
$265$	7045.09	1.63312
$266$	0	0
$267$	−2222.99	−0.509531
$268$	0	0
$269$	4291.22	0.972640	0.486320	−	0.873781i	$-0.338339\pi$
$269$	4291.22	0.972640	0.486320	+	0.873781i	$0.338339\pi$
$270$	0	0
$271$	−5967.70	−1.33768	−0.668841	−	0.743405i	$-0.733209\pi$
$271$	−5967.70	−1.33768	−0.668841	+	0.743405i	$0.733209\pi$
$272$	0	0
$273$	441.821	0.0979495
$274$	0	0
$275$	−426.345	−0.0934893
$276$	0	0
$277$	2234.76	0.484742	0.242371	−	0.970184i	$-0.422075\pi$
$277$	2234.76	0.484742	0.242371	+	0.970184i	$0.422075\pi$
$278$	0	0
$279$	1511.55	0.324353
$280$	0	0
$281$	−1672.59	−0.355083	−0.177542	−	0.984113i	$-0.556814\pi$
$281$	−1672.59	−0.355083	−0.177542	+	0.984113i	$0.556814\pi$
$282$	0	0
$283$	4837.14	1.01604	0.508018	−	0.861347i	$-0.330379\pi$
$283$	4837.14	1.01604	0.508018	+	0.861347i	$0.330379\pi$
$284$	0	0
$285$	−1351.59	−0.280916
$286$	0	0
$287$	4832.75	0.993965
$288$	0	0
$289$	6216.79	1.26538
$290$	0	0
$291$	1047.70	0.211055
$292$	0	0
$293$	8937.62	1.78205	0.891027	−	0.453951i	$-0.149986\pi$
$293$	8937.62	1.78205	0.891027	+	0.453951i	$0.149986\pi$
$294$	0	0
$295$	−134.629	−0.0265708
$296$	0	0
$297$	2512.07	0.490791
$298$	0	0
$299$	296.576	0.0573627
$300$	0	0
$301$	−6503.18	−1.24531
$302$	0	0
$303$	−1373.73	−0.260457
$304$	0	0
$305$	−851.987	−0.159950
$306$	0	0
$307$	−4175.09	−0.776173	−0.388086	−	0.921623i	$-0.626864\pi$
$307$	−4175.09	−0.776173	−0.388086	+	0.921623i	$0.626864\pi$
$308$	0	0
$309$	3103.40	0.571348
$310$	0	0
$311$	6986.98	1.27394	0.636970	−	0.770889i	$-0.280187\pi$
$311$	6986.98	1.27394	0.636970	+	0.770889i	$0.280187\pi$
$312$	0	0
$313$	−7458.22	−1.34685	−0.673425	−	0.739256i	$-0.735177\pi$
$313$	−7458.22	−1.34685	−0.673425	+	0.739256i	$0.735177\pi$
$314$	0	0
$315$	−3531.03	−0.631590
$316$	0	0
$317$	9362.43	1.65882	0.829411	−	0.558640i	$-0.188676\pi$
$317$	9362.43	1.65882	0.829411	+	0.558640i	$0.188676\pi$
$318$	0	0
$319$	−878.549	−0.154198
$320$	0	0
$321$	2659.67	0.462455
$322$	0	0
$323$	8391.61	1.44558
$324$	0	0
$325$	280.460	0.0478680
$326$	0	0
$327$	1242.70	0.210157
$328$	0	0
$329$	6764.45	1.13354
$330$	0	0
$331$	7034.25	1.16809	0.584044	−	0.811722i	$-0.301469\pi$
$331$	7034.25	1.16809	0.584044	+	0.811722i	$0.301469\pi$
$332$	0	0
$333$	157.031	0.0258416
$334$	0	0
$335$	−5573.13	−0.908933
$336$	0	0
$337$	3071.54	0.496490	0.248245	−	0.968697i	$-0.420146\pi$
$337$	3071.54	0.496490	0.248245	+	0.968697i	$0.420146\pi$
$338$	0	0
$339$	−2306.59	−0.369548
$340$	0	0
$341$	−1876.95	−0.298072
$342$	0	0
$343$	−6831.91	−1.07548
$344$	0	0
$345$	252.872	0.0394614
$346$	0	0
$347$	3918.32	0.606186	0.303093	−	0.952961i	$-0.401981\pi$
$347$	3918.32	0.606186	0.303093	+	0.952961i	$0.401981\pi$
$348$	0	0
$349$	−4880.00	−0.748482	−0.374241	−	0.927331i	$-0.622097\pi$
$349$	−4880.00	−0.748482	−0.374241	+	0.927331i	$0.622097\pi$
$350$	0	0
$351$	−1652.50	−0.251293
$352$	0	0
$353$	−1386.41	−0.209041	−0.104520	−	0.994523i	$-0.533331\pi$
$353$	−1386.41	−0.209041	−0.104520	+	0.994523i	$0.533331\pi$
$354$	0	0
$355$	−2231.44	−0.333613
$356$	0	0
$357$	−2338.90	−0.346745
$358$	0	0
$359$	1798.75	0.264441	0.132220	−	0.991220i	$-0.457789\pi$
$359$	1798.75	0.264441	0.132220	+	0.991220i	$0.457789\pi$
$360$	0	0
$361$	−531.919	−0.0775506
$362$	0	0
$363$	666.673	0.0963946
$364$	0	0
$365$	6607.19	0.947497
$366$	0	0
$367$	−8826.24	−1.25538	−0.627692	−	0.778462i	$-0.716000\pi$
$367$	−8826.24	−1.25538	−0.627692	+	0.778462i	$0.716000\pi$
$368$	0	0
$369$	−8580.04	−1.21046
$370$	0	0
$371$	9192.04	1.28633
$372$	0	0
$373$	5324.16	0.739074	0.369537	−	0.929216i	$-0.379516\pi$
$373$	5324.16	0.739074	0.369537	+	0.929216i	$0.379516\pi$
$374$	0	0
$375$	2363.12	0.325416
$376$	0	0
$377$	577.930	0.0789521
$378$	0	0
$379$	486.170	0.0658915	0.0329458	−	0.999457i	$-0.489511\pi$
$379$	486.170	0.0658915	0.0329458	+	0.999457i	$0.489511\pi$
$380$	0	0
$381$	−3391.74	−0.456074
$382$	0	0
$383$	−10274.8	−1.37081	−0.685405	−	0.728162i	$-0.740375\pi$
$383$	−10274.8	−1.37081	−0.685405	+	0.728162i	$0.740375\pi$
$384$	0	0
$385$	4384.60	0.580415
$386$	0	0
$387$	11545.7	1.51654
$388$	0	0
$389$	8823.81	1.15009	0.575045	−	0.818122i	$-0.304985\pi$
$389$	8823.81	1.15009	0.575045	+	0.818122i	$0.304985\pi$
$390$	0	0
$391$	−1570.01	−0.203066
$392$	0	0
$393$	2220.00	0.284946
$394$	0	0
$395$	9640.62	1.22803
$396$	0	0
$397$	7624.66	0.963906	0.481953	−	0.876197i	$-0.339928\pi$
$397$	7624.66	0.963906	0.481953	+	0.876197i	$0.339928\pi$
$398$	0	0
$399$	−1763.48	−0.221264
$400$	0	0
$401$	5192.75	0.646667	0.323333	−	0.946285i	$-0.395196\pi$
$401$	5192.75	0.646667	0.323333	+	0.946285i	$0.395196\pi$
$402$	0	0
$403$	1234.70	0.152618
$404$	0	0
$405$	5528.80	0.678342
$406$	0	0
$407$	−194.991	−0.0237477
$408$	0	0
$409$	15423.3	1.86463	0.932316	−	0.361645i	$-0.117785\pi$
$409$	15423.3	1.86463	0.932316	+	0.361645i	$0.117785\pi$
$410$	0	0
$411$	−491.981	−0.0590454
$412$	0	0
$413$	−175.656	−0.0209285
$414$	0	0
$415$	−4332.20	−0.512432
$416$	0	0
$417$	−587.815	−0.0690298
$418$	0	0
$419$	2112.52	0.246308	0.123154	−	0.992388i	$-0.460699\pi$
$419$	2112.52	0.246308	0.123154	+	0.992388i	$0.460699\pi$
$420$	0	0
$421$	−2010.09	−0.232698	−0.116349	−	0.993208i	$-0.537119\pi$
$421$	−2010.09	−0.232698	−0.116349	+	0.993208i	$0.537119\pi$
$422$	0	0
$423$	−12009.6	−1.38044
$424$	0	0
$425$	−1484.69	−0.169454
$426$	0	0
$427$	−1111.63	−0.125984
$428$	0	0
$429$	974.025	0.109619
$430$	0	0
$431$	−10433.5	−1.16604	−0.583021	−	0.812457i	$-0.698129\pi$
$431$	−10433.5	−1.16604	−0.583021	+	0.812457i	$0.698129\pi$
$432$	0	0
$433$	−6973.11	−0.773917	−0.386959	−	0.922097i	$-0.626474\pi$
$433$	−6973.11	−0.773917	−0.386959	+	0.922097i	$0.626474\pi$
$434$	0	0
$435$	492.766	0.0543134
$436$	0	0
$437$	−1183.75	−0.129580
$438$	0	0
$439$	−3247.81	−0.353096	−0.176548	−	0.984292i	$-0.556493\pi$
$439$	−3247.81	−0.353096	−0.176548	+	0.984292i	$0.556493\pi$
$440$	0	0
$441$	3761.13	0.406126
$442$	0	0
$443$	−8227.84	−0.882430	−0.441215	−	0.897401i	$-0.645452\pi$
$443$	−8227.84	−0.882430	−0.441215	+	0.897401i	$0.645452\pi$
$444$	0	0
$445$	14512.1	1.54593
$446$	0	0
$447$	322.696	0.0341454
$448$	0	0
$449$	−8823.99	−0.927460	−0.463730	−	0.885976i	$-0.653489\pi$
$449$	−8823.99	−0.927460	−0.463730	+	0.885976i	$0.653489\pi$
$450$	0	0
$451$	10654.1	1.11238
$452$	0	0
$453$	57.4245	0.00595594
$454$	0	0
$455$	−2884.30	−0.297182
$456$	0	0
$457$	−1162.68	−0.119010	−0.0595052	−	0.998228i	$-0.518952\pi$
$457$	−1162.68	−0.119010	−0.0595052	+	0.998228i	$0.518952\pi$
$458$	0	0
$459$	8747.97	0.889586
$460$	0	0
$461$	−13275.8	−1.34124	−0.670622	−	0.741799i	$-0.733973\pi$
$461$	−13275.8	−1.34124	−0.670622	+	0.741799i	$0.733973\pi$
$462$	0	0
$463$	17666.1	1.77325	0.886625	−	0.462488i	$-0.153043\pi$
$463$	17666.1	1.77325	0.886625	+	0.462488i	$0.153043\pi$
$464$	0	0
$465$	1052.76	0.104990
$466$	0	0
$467$	−19505.3	−1.93276	−0.966380	−	0.257117i	$-0.917227\pi$
$467$	−19505.3	−1.93276	−0.966380	+	0.257117i	$0.917227\pi$
$468$	0	0
$469$	−7271.52	−0.715922
$470$	0	0
$471$	221.667	0.0216855
$472$	0	0
$473$	−14336.7	−1.39366
$474$	0	0
$475$	−1119.42	−0.108132
$476$	0	0
$477$	−16319.5	−1.56650
$478$	0	0
$479$	9224.14	0.879878	0.439939	−	0.898028i	$-0.355000\pi$
$479$	9224.14	0.879878	0.439939	+	0.898028i	$0.355000\pi$
$480$	0	0
$481$	128.270	0.0121592
$482$	0	0
$483$	329.934	0.0310818
$484$	0	0
$485$	−6839.57	−0.640348
$486$	0	0
$487$	−9009.75	−0.838338	−0.419169	−	0.907908i	$-0.637679\pi$
$487$	−9009.75	−0.838338	−0.419169	+	0.907908i	$0.637679\pi$
$488$	0	0
$489$	−1400.09	−0.129477
$490$	0	0
$491$	2822.52	0.259426	0.129713	−	0.991552i	$-0.458594\pi$
$491$	2822.52	0.259426	0.129713	+	0.991552i	$0.458594\pi$
$492$	0	0
$493$	−3059.44	−0.279493
$494$	0	0
$495$	−7784.40	−0.706834
$496$	0	0
$497$	−2911.46	−0.262771
$498$	0	0
$499$	−5407.39	−0.485106	−0.242553	−	0.970138i	$-0.577985\pi$
$499$	−5407.39	−0.485106	−0.242553	+	0.970138i	$0.577985\pi$
$500$	0	0
$501$	106.939	0.00953633
$502$	0	0
$503$	−757.352	−0.0671345	−0.0335673	−	0.999436i	$-0.510687\pi$
$503$	−757.352	−0.0671345	−0.0335673	+	0.999436i	$0.510687\pi$
$504$	0	0
$505$	8967.97	0.790237
$506$	0	0
$507$	2903.76	0.254360
$508$	0	0
$509$	−2637.36	−0.229664	−0.114832	−	0.993385i	$-0.536633\pi$
$509$	−2637.36	−0.229664	−0.114832	+	0.993385i	$0.536633\pi$
$510$	0	0
$511$	8620.70	0.746296
$512$	0	0
$513$	6595.77	0.567661
$514$	0	0
$515$	−20259.7	−1.73349
$516$	0	0
$517$	14912.7	1.26859
$518$	0	0
$519$	−2902.01	−0.245441
$520$	0	0
$521$	−16495.4	−1.38709	−0.693546	−	0.720413i	$-0.743952\pi$
$521$	−16495.4	−1.38709	−0.693546	+	0.720413i	$0.743952\pi$
$522$	0	0
$523$	713.955	0.0596923	0.0298462	−	0.999555i	$-0.490498\pi$
$523$	713.955	0.0596923	0.0298462	+	0.999555i	$0.490498\pi$
$524$	0	0
$525$	312.005	0.0259372
$526$	0	0
$527$	−6536.24	−0.540272
$528$	0	0
$529$	−11945.5	−0.981797
$530$	0	0
$531$	311.859	0.0254869
$532$	0	0
$533$	−7008.54	−0.569557
$534$	0	0
$535$	−17362.8	−1.40310
$536$	0	0
$537$	1959.70	0.157481
$538$	0	0
$539$	−4670.32	−0.373219
$540$	0	0
$541$	−21514.0	−1.70972	−0.854861	−	0.518857i	$-0.826358\pi$
$541$	−21514.0	−1.70972	−0.854861	+	0.518857i	$0.826358\pi$
$542$	0	0
$543$	1038.38	0.0820648
$544$	0	0
$545$	−8112.60	−0.637625
$546$	0	0
$547$	−6782.38	−0.530153	−0.265077	−	0.964227i	$-0.585397\pi$
$547$	−6782.38	−0.530153	−0.265077	+	0.964227i	$0.585397\pi$
$548$	0	0
$549$	1973.58	0.153425
$550$	0	0
$551$	−2306.75	−0.178350
$552$	0	0
$553$	12578.6	0.967260
$554$	0	0
$555$	109.368	0.00836468
$556$	0	0
$557$	11559.5	0.879337	0.439668	−	0.898160i	$-0.355096\pi$
$557$	11559.5	0.879337	0.439668	+	0.898160i	$0.355096\pi$
$558$	0	0
$559$	9431.04	0.713579
$560$	0	0
$561$	−5156.27	−0.388053
$562$	0	0
$563$	11151.0	0.834736	0.417368	−	0.908737i	$-0.362953\pi$
$563$	11151.0	0.834736	0.417368	+	0.908737i	$0.362953\pi$
$564$	0	0
$565$	15057.9	1.12122
$566$	0	0
$567$	7213.68	0.534296
$568$	0	0
$569$	11955.9	0.880873	0.440436	−	0.897784i	$-0.354824\pi$
$569$	11955.9	0.880873	0.440436	+	0.897784i	$0.354824\pi$
$570$	0	0
$571$	18691.6	1.36991	0.684956	−	0.728585i	$-0.259822\pi$
$571$	18691.6	1.36991	0.684956	+	0.728585i	$0.259822\pi$
$572$	0	0
$573$	−6592.03	−0.480604
$574$	0	0
$575$	209.436	0.0151897
$576$	0	0
$577$	−11793.2	−0.850876	−0.425438	−	0.904987i	$-0.639880\pi$
$577$	−11793.2	−0.850876	−0.425438	+	0.904987i	$0.639880\pi$
$578$	0	0
$579$	−1394.45	−0.100089
$580$	0	0
$581$	−5652.41	−0.403617
$582$	0	0
$583$	20264.5	1.43957
$584$	0	0
$585$	5120.77	0.361910
$586$	0	0
$587$	759.837	0.0534273	0.0267137	−	0.999643i	$-0.491496\pi$
$587$	759.837	0.0534273	0.0267137	+	0.999643i	$0.491496\pi$
$588$	0	0
$589$	−4928.18	−0.344757
$590$	0	0
$591$	−3123.11	−0.217373
$592$	0	0
$593$	−1286.39	−0.0890825	−0.0445412	−	0.999008i	$-0.514183\pi$
$593$	−1286.39	−0.0890825	−0.0445412	+	0.999008i	$0.514183\pi$
$594$	0	0
$595$	15268.8	1.05203
$596$	0	0
$597$	−5797.35	−0.397437
$598$	0	0
$599$	4463.66	0.304474	0.152237	−	0.988344i	$-0.451352\pi$
$599$	4463.66	0.304474	0.152237	+	0.988344i	$0.451352\pi$
$600$	0	0
$601$	−13078.7	−0.887673	−0.443837	−	0.896108i	$-0.646383\pi$
$601$	−13078.7	−0.887673	−0.443837	+	0.896108i	$0.646383\pi$
$602$	0	0
$603$	12909.8	0.871855
$604$	0	0
$605$	−4352.17	−0.292465
$606$	0	0
$607$	−475.770	−0.0318137	−0.0159069	−	0.999873i	$-0.505064\pi$
$607$	−475.770	−0.0318137	−0.0159069	+	0.999873i	$0.505064\pi$
$608$	0	0
$609$	642.934	0.0427800
$610$	0	0
$611$	−9809.93	−0.649537
$612$	0	0
$613$	−12137.1	−0.799697	−0.399849	−	0.916581i	$-0.630937\pi$
$613$	−12137.1	−0.799697	−0.399849	+	0.916581i	$0.630937\pi$
$614$	0	0
$615$	−5975.76	−0.391814
$616$	0	0
$617$	3751.56	0.244785	0.122392	−	0.992482i	$-0.460943\pi$
$617$	3751.56	0.244785	0.122392	+	0.992482i	$0.460943\pi$
$618$	0	0
$619$	14965.7	0.971763	0.485882	−	0.874025i	$-0.338499\pi$
$619$	14965.7	0.971763	0.485882	+	0.874025i	$0.338499\pi$
$620$	0	0
$621$	−1234.02	−0.0797416
$622$	0	0
$623$	18934.6	1.21766
$624$	0	0
$625$	−13667.8	−0.874739
$626$	0	0
$627$	−3887.71	−0.247624
$628$	0	0
$629$	−679.031	−0.0430441
$630$	0	0
$631$	4591.19	0.289655	0.144828	−	0.989457i	$-0.453737\pi$
$631$	4591.19	0.289655	0.144828	+	0.989457i	$0.453737\pi$
$632$	0	0
$633$	1116.99	0.0701363
$634$	0	0
$635$	22142.0	1.38375
$636$	0	0
$637$	3072.25	0.191094
$638$	0	0
$639$	5169.00	0.320004
$640$	0	0
$641$	25387.9	1.56437	0.782187	−	0.623044i	$-0.214104\pi$
$641$	25387.9	1.56437	0.782187	+	0.623044i	$0.214104\pi$
$642$	0	0
$643$	−25270.1	−1.54986	−0.774928	−	0.632050i	$-0.782214\pi$
$643$	−25270.1	−1.54986	−0.774928	+	0.632050i	$0.782214\pi$
$644$	0	0
$645$	8041.27	0.490891
$646$	0	0
$647$	18802.7	1.14252	0.571259	−	0.820770i	$-0.306455\pi$
$647$	18802.7	1.14252	0.571259	+	0.820770i	$0.306455\pi$
$648$	0	0
$649$	−387.246	−0.0234218
$650$	0	0
$651$	1373.58	0.0826955
$652$	0	0
$653$	16345.7	0.979564	0.489782	−	0.871845i	$-0.337076\pi$
$653$	16345.7	0.979564	0.489782	+	0.871845i	$0.337076\pi$
$654$	0	0
$655$	−14492.6	−0.864537
$656$	0	0
$657$	−15305.2	−0.908845
$658$	0	0
$659$	−25185.3	−1.48874	−0.744370	−	0.667768i	$-0.767250\pi$
$659$	−25185.3	−1.48874	−0.744370	+	0.667768i	$0.767250\pi$
$660$	0	0
$661$	−17987.6	−1.05845	−0.529227	−	0.848480i	$-0.677518\pi$
$661$	−17987.6	−1.05845	−0.529227	+	0.848480i	$0.677518\pi$
$662$	0	0
$663$	3391.92	0.198690
$664$	0	0
$665$	11512.3	0.671323
$666$	0	0
$667$	431.576	0.0250535
$668$	0	0
$669$	−2856.49	−0.165079
$670$	0	0
$671$	−2450.66	−0.140993
$672$	0	0
$673$	10249.1	0.587033	0.293516	−	0.955954i	$-0.405174\pi$
$673$	10249.1	0.587033	0.293516	+	0.955954i	$0.405174\pi$
$674$	0	0
$675$	−1166.96	−0.0665428
$676$	0	0
$677$	4149.01	0.235538	0.117769	−	0.993041i	$-0.462426\pi$
$677$	4149.01	0.235538	0.117769	+	0.993041i	$0.462426\pi$
$678$	0	0
$679$	−8923.90	−0.504371
$680$	0	0
$681$	−4833.66	−0.271991
$682$	0	0
$683$	−33571.1	−1.88076	−0.940382	−	0.340121i	$-0.889532\pi$
$683$	−33571.1	−1.88076	−0.940382	+	0.340121i	$0.889532\pi$
$684$	0	0
$685$	3211.75	0.179146
$686$	0	0
$687$	−6114.82	−0.339585
$688$	0	0
$689$	−13330.5	−0.737084
$690$	0	0
$691$	27656.6	1.52258	0.761292	−	0.648409i	$-0.224565\pi$
$691$	27656.6	1.52258	0.761292	+	0.648409i	$0.224565\pi$
$692$	0	0
$693$	−10156.7	−0.556738
$694$	0	0
$695$	3837.37	0.209439
$696$	0	0
$697$	37101.7	2.01625
$698$	0	0
$699$	−10095.2	−0.546260
$700$	0	0
$701$	14493.7	0.780912	0.390456	−	0.920622i	$-0.372317\pi$
$701$	14493.7	0.780912	0.390456	+	0.920622i	$0.372317\pi$
$702$	0	0
$703$	−511.974	−0.0274672
$704$	0	0
$705$	−8364.33	−0.446835
$706$	0	0
$707$	11700.9	0.622431
$708$	0	0
$709$	8279.81	0.438582	0.219291	−	0.975659i	$-0.429626\pi$
$709$	8279.81	0.438582	0.219291	+	0.975659i	$0.429626\pi$
$710$	0	0
$711$	−22331.9	−1.17794
$712$	0	0
$713$	922.027	0.0484294
$714$	0	0
$715$	−6358.63	−0.332586
$716$	0	0
$717$	−9508.19	−0.495244
$718$	0	0
$719$	−3180.45	−0.164966	−0.0824832	−	0.996592i	$-0.526285\pi$
$719$	−3180.45	−0.164966	−0.0824832	+	0.996592i	$0.526285\pi$
$720$	0	0
$721$	−26433.7	−1.36538
$722$	0	0
$723$	675.952	0.0347703
$724$	0	0
$725$	408.123	0.0209066
$726$	0	0
$727$	11749.6	0.599409	0.299704	−	0.954032i	$-0.403112\pi$
$727$	11749.6	0.599409	0.299704	+	0.954032i	$0.403112\pi$
$728$	0	0
$729$	−9195.10	−0.467160
$730$	0	0
$731$	−49925.8	−2.52609
$732$	0	0
$733$	15005.6	0.756130	0.378065	−	0.925779i	$-0.376590\pi$
$733$	15005.6	0.756130	0.378065	+	0.925779i	$0.376590\pi$
$734$	0	0
$735$	2619.52	0.131459
$736$	0	0
$737$	−16030.6	−0.801212
$738$	0	0
$739$	−5555.46	−0.276537	−0.138269	−	0.990395i	$-0.544154\pi$
$739$	−5555.46	−0.276537	−0.138269	+	0.990395i	$0.544154\pi$
$740$	0	0
$741$	2557.43	0.126788
$742$	0	0
$743$	−18358.9	−0.906493	−0.453246	−	0.891385i	$-0.649734\pi$
$743$	−18358.9	−0.906493	−0.453246	+	0.891385i	$0.649734\pi$
$744$	0	0
$745$	−2106.63	−0.103598
$746$	0	0
$747$	10035.3	0.491528
$748$	0	0
$749$	−22654.1	−1.10516
$750$	0	0
$751$	−8818.75	−0.428496	−0.214248	−	0.976779i	$-0.568730\pi$
$751$	−8818.75	−0.428496	−0.214248	+	0.976779i	$0.568730\pi$
$752$	0	0
$753$	12010.3	0.581247
$754$	0	0
$755$	−374.879	−0.0180705
$756$	0	0
$757$	1462.27	0.0702073	0.0351037	−	0.999384i	$-0.488824\pi$
$757$	1462.27	0.0702073	0.0351037	+	0.999384i	$0.488824\pi$
$758$	0	0
$759$	727.363	0.0347847
$760$	0	0
$761$	15232.2	0.725581	0.362790	−	0.931871i	$-0.381824\pi$
$761$	15232.2	0.725581	0.362790	+	0.931871i	$0.381824\pi$
$762$	0	0
$763$	−10584.9	−0.502225
$764$	0	0
$765$	−27108.2	−1.28118
$766$	0	0
$767$	254.740	0.0119923
$768$	0	0
$769$	−32110.5	−1.50577	−0.752883	−	0.658155i	$-0.771337\pi$
$769$	−32110.5	−1.50577	−0.752883	+	0.658155i	$0.771337\pi$
$770$	0	0
$771$	−8603.05	−0.401856
$772$	0	0
$773$	13012.4	0.605465	0.302733	−	0.953076i	$-0.402101\pi$
$773$	13012.4	0.605465	0.302733	+	0.953076i	$0.402101\pi$
$774$	0	0
$775$	871.922	0.0404134
$776$	0	0
$777$	142.697	0.00658844
$778$	0	0
$779$	27973.8	1.28661
$780$	0	0
$781$	−6418.52	−0.294075
$782$	0	0
$783$	−2404.70	−0.109754
$784$	0	0
$785$	−1447.09	−0.0657946
$786$	0	0
$787$	2738.98	0.124059	0.0620293	−	0.998074i	$-0.480243\pi$
$787$	2738.98	0.124059	0.0620293	+	0.998074i	$0.480243\pi$
$788$	0	0
$789$	−1343.79	−0.0606338
$790$	0	0
$791$	19646.7	0.883131
$792$	0	0
$793$	1612.10	0.0721909
$794$	0	0
$795$	−11366.1	−0.507061
$796$	0	0
$797$	−43359.0	−1.92704	−0.963522	−	0.267628i	$-0.913760\pi$
$797$	−43359.0	−1.92704	−0.963522	+	0.267628i	$0.913760\pi$
$798$	0	0
$799$	51931.6	2.29938
$800$	0	0
$801$	−33616.4	−1.48287
$802$	0	0
$803$	19004.9	0.835205
$804$	0	0
$805$	−2153.88	−0.0943033
$806$	0	0
$807$	−6923.17	−0.301992
$808$	0	0
$809$	400.518	0.0174060	0.00870300	−	0.999962i	$-0.497230\pi$
$809$	400.518	0.0174060	0.00870300	+	0.999962i	$0.497230\pi$
$810$	0	0
$811$	−37589.7	−1.62756	−0.813782	−	0.581170i	$-0.802595\pi$
$811$	−37589.7	−1.62756	−0.813782	+	0.581170i	$0.802595\pi$
$812$	0	0
$813$	9627.90	0.415332
$814$	0	0
$815$	9140.09	0.392839
$816$	0	0
$817$	−37642.9	−1.61195
$818$	0	0
$819$	6681.29	0.285059
$820$	0	0
$821$	−17756.6	−0.754821	−0.377411	−	0.926046i	$-0.623185\pi$
$821$	−17756.6	−0.754821	−0.377411	+	0.926046i	$0.623185\pi$
$822$	0	0
$823$	38533.6	1.63207	0.816036	−	0.578001i	$-0.196167\pi$
$823$	38533.6	1.63207	0.816036	+	0.578001i	$0.196167\pi$
$824$	0	0
$825$	687.837	0.0290272
$826$	0	0
$827$	−45875.2	−1.92895	−0.964473	−	0.264182i	$-0.914898\pi$
$827$	−45875.2	−1.92895	−0.964473	+	0.264182i	$0.914898\pi$
$828$	0	0
$829$	−22725.1	−0.952082	−0.476041	−	0.879423i	$-0.657929\pi$
$829$	−22725.1	−0.952082	−0.476041	+	0.879423i	$0.657929\pi$
$830$	0	0
$831$	−3605.41	−0.150506
$832$	0	0
$833$	−16263.8	−0.676480
$834$	0	0
$835$	−698.123	−0.0289336
$836$	0	0
$837$	−5137.46	−0.212158
$838$	0	0
$839$	31480.4	1.29538	0.647690	−	0.761904i	$-0.275735\pi$
$839$	31480.4	1.29538	0.647690	+	0.761904i	$0.275735\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	2698.45	0.110249
$844$	0	0
$845$	−18956.3	−0.771737
$846$	0	0
$847$	−5678.48	−0.230360
$848$	0	0
$849$	−7803.92	−0.315465
$850$	0	0
$851$	95.7866	0.00385843
$852$	0	0
$853$	−14863.4	−0.596617	−0.298309	−	0.954470i	$-0.596422\pi$
$853$	−14863.4	−0.596617	−0.298309	+	0.954470i	$0.596422\pi$
$854$	0	0
$855$	−20439.0	−0.817542
$856$	0	0
$857$	30070.7	1.19859	0.599297	−	0.800527i	$-0.295447\pi$
$857$	30070.7	1.19859	0.599297	+	0.800527i	$0.295447\pi$
$858$	0	0
$859$	16015.7	0.636146	0.318073	−	0.948066i	$-0.396964\pi$
$859$	16015.7	0.636146	0.318073	+	0.948066i	$0.396964\pi$
$860$	0	0
$861$	−7796.84	−0.308613
$862$	0	0
$863$	4726.12	0.186418	0.0932092	−	0.995647i	$-0.470287\pi$
$863$	4726.12	0.186418	0.0932092	+	0.995647i	$0.470287\pi$
$864$	0	0
$865$	18944.9	0.744677
$866$	0	0
$867$	−10029.8	−0.392882
$868$	0	0
$869$	27730.3	1.08249
$870$	0	0
$871$	10545.3	0.410234
$872$	0	0
$873$	15843.5	0.614226
$874$	0	0
$875$	−20128.2	−0.777667
$876$	0	0
$877$	−35681.6	−1.37387	−0.686933	−	0.726720i	$-0.741044\pi$
$877$	−35681.6	−1.37387	−0.686933	+	0.726720i	$0.741044\pi$
$878$	0	0
$879$	−14419.4	−0.553303
$880$	0	0
$881$	−14945.7	−0.571548	−0.285774	−	0.958297i	$-0.592251\pi$
$881$	−14945.7	−0.571548	−0.285774	+	0.958297i	$0.592251\pi$
$882$	0	0
$883$	−39264.7	−1.49645	−0.748223	−	0.663447i	$-0.769093\pi$
$883$	−39264.7	−1.49645	−0.748223	+	0.663447i	$0.769093\pi$
$884$	0	0
$885$	217.201	0.00824988
$886$	0	0
$887$	19020.1	0.719991	0.359995	−	0.932954i	$-0.382778\pi$
$887$	19020.1	0.719991	0.359995	+	0.932954i	$0.382778\pi$
$888$	0	0
$889$	28889.7	1.08991
$890$	0	0
$891$	15903.1	0.597949
$892$	0	0
$893$	39155.3	1.46728
$894$	0	0
$895$	−12793.3	−0.477803
$896$	0	0
$897$	−478.477	−0.0178103
$898$	0	0
$899$	1796.73	0.0666566
$900$	0	0
$901$	70568.6	2.60930
$902$	0	0
$903$	10491.8	0.386651
$904$	0	0
$905$	−6778.76	−0.248987
$906$	0	0
$907$	−24945.4	−0.913228	−0.456614	−	0.889665i	$-0.650938\pi$
$907$	−24945.4	−0.913228	−0.456614	+	0.889665i	$0.650938\pi$
$908$	0	0
$909$	−20773.8	−0.758000
$910$	0	0
$911$	2868.98	0.104340	0.0521698	−	0.998638i	$-0.483386\pi$
$911$	2868.98	0.104340	0.0521698	+	0.998638i	$0.483386\pi$
$912$	0	0
$913$	−12461.1	−0.451702
$914$	0	0
$915$	1374.54	0.0496622
$916$	0	0
$917$	−18909.1	−0.680953
$918$	0	0
$919$	−20331.1	−0.729774	−0.364887	−	0.931052i	$-0.618892\pi$
$919$	−20331.1	−0.729774	−0.364887	+	0.931052i	$0.618892\pi$
$920$	0	0
$921$	6735.82	0.240991
$922$	0	0
$923$	4222.26	0.150571
$924$	0	0
$925$	90.5814	0.00321978
$926$	0	0
$927$	46930.2	1.66277
$928$	0	0
$929$	2069.37	0.0730827	0.0365413	−	0.999332i	$-0.488366\pi$
$929$	2069.37	0.0730827	0.0365413	+	0.999332i	$0.488366\pi$
$930$	0	0
$931$	−12262.6	−0.431675
$932$	0	0
$933$	−11272.3	−0.395541
$934$	0	0
$935$	33661.2	1.17737
$936$	0	0
$937$	−27162.9	−0.947038	−0.473519	−	0.880784i	$-0.657016\pi$
$937$	−27162.9	−0.947038	−0.473519	+	0.880784i	$0.657016\pi$
$938$	0	0
$939$	12032.6	0.418178
$940$	0	0
$941$	−8127.05	−0.281546	−0.140773	−	0.990042i	$-0.544959\pi$
$941$	−8127.05	−0.281546	−0.140773	+	0.990042i	$0.544959\pi$
$942$	0	0
$943$	−5233.70	−0.180735
$944$	0	0
$945$	12001.2	0.413122
$946$	0	0
$947$	−9170.78	−0.314689	−0.157344	−	0.987544i	$-0.550293\pi$
$947$	−9170.78	−0.314689	−0.157344	+	0.987544i	$0.550293\pi$
$948$	0	0
$949$	−12501.9	−0.427639
$950$	0	0
$951$	−15104.7	−0.515042
$952$	0	0
$953$	−25938.9	−0.881683	−0.440841	−	0.897585i	$-0.645320\pi$
$953$	−25938.9	−0.881683	−0.440841	+	0.897585i	$0.645320\pi$
$954$	0	0
$955$	43034.1	1.45817
$956$	0	0
$957$	1417.39	0.0478765
$958$	0	0
$959$	4190.52	0.141104
$960$	0	0
$961$	−25952.4	−0.871150
$962$	0	0
$963$	40220.0	1.34587
$964$	0	0
$965$	9103.24	0.303672
$966$	0	0
$967$	−6987.35	−0.232366	−0.116183	−	0.993228i	$-0.537066\pi$
$967$	−6987.35	−0.232366	−0.116183	+	0.993228i	$0.537066\pi$
$968$	0	0
$969$	−13538.5	−0.448832
$970$	0	0
$971$	−37865.2	−1.25144	−0.625722	−	0.780046i	$-0.715196\pi$
$971$	−37865.2	−1.25144	−0.625722	+	0.780046i	$0.715196\pi$
$972$	0	0
$973$	5006.80	0.164965
$974$	0	0
$975$	−452.476	−0.0148624
$976$	0	0
$977$	−47682.8	−1.56142	−0.780711	−	0.624893i	$-0.785143\pi$
$977$	−47682.8	−1.56142	−0.780711	+	0.624893i	$0.785143\pi$
$978$	0	0
$979$	41742.7	1.36272
$980$	0	0
$981$	18792.3	0.611614
$982$	0	0
$983$	10728.5	0.348103	0.174052	−	0.984737i	$-0.444314\pi$
$983$	10728.5	0.348103	0.174052	+	0.984737i	$0.444314\pi$
$984$	0	0
$985$	20388.3	0.659518
$986$	0	0
$987$	−10913.3	−0.351950
$988$	0	0
$989$	7042.72	0.226436
$990$	0	0
$991$	−33130.6	−1.06198	−0.530992	−	0.847377i	$-0.678181\pi$
$991$	−33130.6	−1.06198	−0.530992	+	0.847377i	$0.678181\pi$
$992$	0	0
$993$	−11348.6	−0.362676
$994$	0	0
$995$	37846.3	1.20584
$996$	0	0
$997$	19283.2	0.612543	0.306272	−	0.951944i	$-0.400918\pi$
$997$	19283.2	0.612543	0.306272	+	0.951944i	$0.400918\pi$
$998$	0	0
$999$	−533.715	−0.0169029

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bj.1.6		12
4.3	odd	2		1856.4.a.bl.1.7		12
8.3	odd	2		928.4.a.h.1.6	✓	12
8.5	even	2		928.4.a.j.1.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.h.1.6	✓	12	8.3	odd	2
928.4.a.j.1.7	yes	12	8.5	even	2
1856.4.a.bj.1.6		12	1.1	even	1	trivial
1856.4.a.bl.1.7		12	4.3	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61334 q^{3} +10.5322 q^{5} +13.7418 q^{7} -24.3972 q^{9} +O(q^{10})\)	\(q-1.61334 q^{3} +10.5322 q^{5} +13.7418 q^{7} -24.3972 q^{9} +30.2948 q^{11} -19.9286 q^{13} -16.9919 q^{15} +105.498 q^{17} +79.5429 q^{19} -22.1701 q^{21} -14.8819 q^{23} -14.0732 q^{25} +82.9208 q^{27} -29.0000 q^{29} -61.9562 q^{31} -48.8756 q^{33} +144.731 q^{35} -6.43644 q^{37} +32.1516 q^{39} +351.682 q^{41} -473.240 q^{43} -256.955 q^{45} +492.253 q^{47} -154.163 q^{49} -170.203 q^{51} +668.911 q^{53} +319.070 q^{55} -128.329 q^{57} -12.7826 q^{59} -80.8937 q^{61} -335.261 q^{63} -209.892 q^{65} -529.153 q^{67} +24.0095 q^{69} -211.869 q^{71} +627.334 q^{73} +22.7048 q^{75} +416.305 q^{77} +915.349 q^{79} +524.944 q^{81} -411.330 q^{83} +1111.12 q^{85} +46.7867 q^{87} +1377.88 q^{89} -273.856 q^{91} +99.9561 q^{93} +837.761 q^{95} -649.398 q^{97} -739.106 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9}+O(q^{10}) \) 12 * q - 14 * q^3 + 10 * q^5 + 44 * q^7 + 130 * q^9	\( 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9} - 46 q^{11} + 34 q^{13} - 50 q^{15} + 36 q^{17} - 148 q^{19} + 92 q^{21} + 328 q^{23} + 486 q^{25} - 326 q^{27} - 348 q^{29} + 374 q^{31} + 710 q^{33} - 204 q^{35} + 340 q^{37} - 122 q^{39} + 32 q^{41} - 462 q^{43} + 1132 q^{45} + 434 q^{47} + 1508 q^{49} - 440 q^{51} - 610 q^{53} + 46 q^{55} - 932 q^{57} - 1240 q^{59} + 1228 q^{61} + 4240 q^{63} + 730 q^{65} - 1672 q^{67} + 528 q^{69} + 3220 q^{71} + 564 q^{73} - 6032 q^{75} - 644 q^{77} + 1862 q^{79} + 3040 q^{81} - 3736 q^{83} + 808 q^{85} + 406 q^{87} + 584 q^{89} - 4844 q^{91} + 3226 q^{93} + 2844 q^{95} + 904 q^{97} - 6832 q^{99}+O(q^{100}) \) 12 * q - 14 * q^3 + 10 * q^5 + 44 * q^7 + 130 * q^9 - 46 * q^11 + 34 * q^13 - 50 * q^15 + 36 * q^17 - 148 * q^19 + 92 * q^21 + 328 * q^23 + 486 * q^25 - 326 * q^27 - 348 * q^29 + 374 * q^31 + 710 * q^33 - 204 * q^35 + 340 * q^37 - 122 * q^39 + 32 * q^41 - 462 * q^43 + 1132 * q^45 + 434 * q^47 + 1508 * q^49 - 440 * q^51 - 610 * q^53 + 46 * q^55 - 932 * q^57 - 1240 * q^59 + 1228 * q^61 + 4240 * q^63 + 730 * q^65 - 1672 * q^67 + 528 * q^69 + 3220 * q^71 + 564 * q^73 - 6032 * q^75 - 644 * q^77 + 1862 * q^79 + 3040 * q^81 - 3736 * q^83 + 808 * q^85 + 406 * q^87 + 584 * q^89 - 4844 * q^91 + 3226 * q^93 + 2844 * q^95 + 904 * q^97 - 6832 * q^99

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