LMFDB - Embedded newform 325.6.b.h.274.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$325 = 5^{2} \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	325.b (of order $2$ , degree $1$ , not minimal)

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:	no
Analytic conductor:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{18} + 378 x^{16} + 59175 x^{14} + 4956036 x^{12} + 239061247 x^{10} + 6629044458 x^{8} + \cdots + 134659641600$ x^18 + 378x^16 + 59175x^14 + 4956036x^12 + 239061247x^10 + 6629044458x^8 + 98240243849x^6 + 625586966104x^4 + 584919029904x^2 + 134659641600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{7}\cdot 5^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.15
Root			$6.50844i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.h.274.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.50844i q^{2} -17.3551i q^{3} -24.3767 q^{4} +130.310 q^{6} -149.455i q^{7} +57.2388i q^{8} -58.2000 q^{9} -492.989 q^{11} +423.061i q^{12} +169.000i q^{13} +1122.18 q^{14} -1209.83 q^{16} +1824.24i q^{17} -436.991i q^{18} -424.035 q^{19} -2593.81 q^{21} -3701.58i q^{22} +3660.46i q^{23} +993.387 q^{24} -1268.93 q^{26} -3207.23i q^{27} +3643.23i q^{28} +8193.74 q^{29} +8113.24 q^{31} -7252.30i q^{32} +8555.87i q^{33} -13697.2 q^{34} +1418.73 q^{36} -6297.67i q^{37} -3183.84i q^{38} +2933.01 q^{39} +4298.36 q^{41} -19475.5i q^{42} -5822.12i q^{43} +12017.5 q^{44} -27484.4 q^{46} -8502.70i q^{47} +20996.7i q^{48} -5529.90 q^{49} +31660.0 q^{51} -4119.67i q^{52} -7049.74i q^{53} +24081.3 q^{54} +8554.65 q^{56} +7359.17i q^{57} +61522.2i q^{58} +21509.2 q^{59} +35743.1 q^{61} +60917.8i q^{62} +8698.30i q^{63} +15738.9 q^{64} -64241.3 q^{66} +15239.3i q^{67} -44469.1i q^{68} +63527.7 q^{69} +69104.1 q^{71} -3331.30i q^{72} +4134.60i q^{73} +47285.7 q^{74} +10336.6 q^{76} +73679.8i q^{77} +22022.4i q^{78} +16065.3 q^{79} -69804.4 q^{81} +32274.0i q^{82} +115739. i q^{83} +63228.7 q^{84} +43715.0 q^{86} -142203. i q^{87} -28218.1i q^{88} +35250.6 q^{89} +25258.0 q^{91} -89230.1i q^{92} -140806. i q^{93} +63842.1 q^{94} -125864. q^{96} +32169.5i q^{97} -41520.9i q^{98} +28691.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39}+ \cdots + 1515552 q^{99}+O(q^{100})$ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$ .

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	7.50844i	1.32732i	0.748035	+	0.663659i	$0.230997\pi$
$2$	7.50844i	1.32732i	−0.748035	+	0.663659i	$0.769003\pi$
$3$	− 17.3551i	− 1.11333i	−0.830737	−	0.556666i	$-0.812081\pi$
$3$	− 17.3551i	− 1.11333i	0.830737	−	0.556666i	$-0.187919\pi$
$4$	−24.3767	−0.761773
$5$	0	0
$5$	0	0
$6$	130.310	1.47774
$7$	− 149.455i	− 1.15283i	−0.817156	−	0.576416i	$-0.804451\pi$
$7$	− 149.455i	− 1.15283i	0.817156	−	0.576416i	$-0.195549\pi$
$8$	57.2388i	0.316203i
$9$	−58.2000	−0.239506
$10$	0	0
$11$	−492.989	−1.22844	−0.614222	−	0.789133i	$-0.710530\pi$
$11$	−492.989	−1.22844	−0.614222	+	0.789133i	$0.710530\pi$
$12$	423.061i	0.848106i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	1122.18	1.53018
$15$	0	0
$16$	−1209.83	−1.18147
$17$	1824.24i	1.53095i	0.643466	+	0.765475i	$0.277496\pi$
$17$	1824.24i	1.53095i	−0.643466	+	0.765475i	$0.722504\pi$
$18$	− 436.991i	− 0.317901i
$19$	−424.035	−0.269474	−0.134737	−	0.990881i	$-0.543019\pi$
$19$	−424.035	−0.269474	−0.134737	+	0.990881i	$0.543019\pi$
$20$	0	0
$21$	−2593.81	−1.28348
$22$	− 3701.58i	− 1.63054i
$23$	3660.46i	1.44283i	0.692501	+	0.721417i	$0.256509\pi$
$23$	3660.46i	1.44283i	−0.692501	+	0.721417i	$0.743491\pi$
$24$	993.387	0.352039
$25$	0	0
$26$	−1268.93	−0.368132
$27$	− 3207.23i	− 0.846682i
$28$	3643.23i	0.878197i
$29$	8193.74	1.80920	0.904601	−	0.426259i	$-0.140169\pi$
$29$	8193.74	1.80920	0.904601	+	0.426259i	$0.140169\pi$
$30$	0	0
$31$	8113.24	1.51632	0.758158	−	0.652070i	$-0.226099\pi$
$31$	8113.24	1.51632	0.758158	+	0.652070i	$0.226099\pi$
$32$	− 7252.30i	− 1.25199i
$33$	8555.87i	1.36766i
$34$	−13697.2	−2.03206
$35$	0	0
$36$	1418.73	0.182449
$37$	− 6297.67i	− 0.756268i	−0.925751	−	0.378134i	$-0.876566\pi$
$37$	− 6297.67i	− 0.756268i	0.925751	−	0.378134i	$-0.123434\pi$
$38$	− 3183.84i	− 0.357678i
$39$	2933.01	0.308782
$40$	0	0
$41$	4298.36	0.399341	0.199670	−	0.979863i	$-0.436013\pi$
$41$	4298.36	0.399341	0.199670	+	0.979863i	$0.436013\pi$
$42$	− 19475.5i	− 1.70359i
$43$	− 5822.12i	− 0.480186i	−0.970750	−	0.240093i	$-0.922822\pi$
$43$	− 5822.12i	− 0.480186i	0.970750	−	0.240093i	$-0.0771780\pi$
$44$	12017.5	0.935795
$45$	0	0
$46$	−27484.4	−1.91510
$47$	− 8502.70i	− 0.561452i	−0.959788	−	0.280726i	$-0.909425\pi$
$47$	− 8502.70i	− 0.561452i	0.959788	−	0.280726i	$-0.0905751\pi$
$48$	20996.7i	1.31537i
$49$	−5529.90	−0.329023
$50$	0	0
$51$	31660.0	1.70445
$52$	− 4119.67i	− 0.211278i
$53$	− 7049.74i	− 0.344733i	−0.985033	−	0.172367i	$-0.944859\pi$
$53$	− 7049.74i	− 0.344733i	0.985033	−	0.172367i	$-0.0551414\pi$
$54$	24081.3	1.12382
$55$	0	0
$56$	8554.65	0.364529
$57$	7359.17i	0.300014i
$58$	61522.2i	2.40139i
$59$	21509.2	0.804440	0.402220	−	0.915543i	$-0.368239\pi$
$59$	21509.2	0.804440	0.402220	+	0.915543i	$0.368239\pi$
$60$	0	0
$61$	35743.1	1.22989	0.614947	−	0.788568i	$-0.289177\pi$
$61$	35743.1	1.22989	0.614947	+	0.788568i	$0.289177\pi$
$62$	60917.8i	2.01263i
$63$	8698.30i	0.276110i
$64$	15738.9	0.480314
$65$	0	0
$66$	−64241.3	−1.81533
$67$	15239.3i	0.414742i	0.978262	+	0.207371i	$0.0664907\pi$
$67$	15239.3i	0.414742i	−0.978262	+	0.207371i	$0.933509\pi$
$68$	− 44469.1i	− 1.16624i
$69$	63527.7	1.60635
$70$	0	0
$71$	69104.1	1.62689	0.813444	−	0.581643i	$-0.197590\pi$
$71$	69104.1	1.62689	0.813444	+	0.581643i	$0.197590\pi$
$72$	− 3331.30i	− 0.0757325i
$73$	4134.60i	0.0908085i	0.998969	+	0.0454043i	$0.0144576\pi$
$73$	4134.60i	0.0908085i	−0.998969	+	0.0454043i	$0.985542\pi$
$74$	47285.7	1.00381
$75$	0	0
$76$	10336.6	0.205278
$77$	73679.8i	1.41619i
$78$	22022.4i	0.409853i
$79$	16065.3	0.289615	0.144808	−	0.989460i	$-0.453744\pi$
$79$	16065.3	0.289615	0.144808	+	0.989460i	$0.453744\pi$
$80$	0	0
$81$	−69804.4	−1.18214
$82$	32274.0i	0.530052i
$83$	115739.i	1.84409i	0.387079	+	0.922047i	$0.373484\pi$
$83$	115739.i	1.84409i	−0.387079	+	0.922047i	$0.626516\pi$
$84$	63228.7	0.977724
$85$	0	0
$86$	43715.0	0.637360
$87$	− 142203.i	− 2.01424i
$88$	− 28218.1i	− 0.388438i
$89$	35250.6	0.471728	0.235864	−	0.971786i	$-0.424208\pi$
$89$	35250.6	0.471728	0.235864	+	0.971786i	$0.424208\pi$
$90$	0	0
$91$	25258.0	0.319738
$92$	− 89230.1i	− 1.09911i
$93$	− 140806.i	− 1.68816i
$94$	63842.1	0.745225
$95$	0	0
$96$	−125864.	−1.39388
$97$	32169.5i	0.347148i	0.984821	+	0.173574i	$0.0555316\pi$
$97$	32169.5i	0.347148i	−0.984821	+	0.173574i	$0.944468\pi$
$98$	− 41520.9i	− 0.436719i
$99$	28691.9	0.294220
$100$	0	0
$101$	−38266.9	−0.373267	−0.186634	−	0.982430i	$-0.559758\pi$
$101$	−38266.9	−0.373267	−0.186634	+	0.982430i	$0.559758\pi$
$102$	237717.i	2.26235i
$103$	− 52058.1i	− 0.483499i	−0.970339	−	0.241750i	$-0.922279\pi$
$103$	− 52058.1i	− 0.483499i	0.970339	−	0.241750i	$-0.0777212\pi$
$104$	−9673.37	−0.0876989
$105$	0	0
$106$	52932.6	0.457571
$107$	121233.i	1.02367i	0.859083	+	0.511836i	$0.171034\pi$
$107$	121233.i	1.02367i	−0.859083	+	0.511836i	$0.828966\pi$
$108$	78181.7i	0.644979i
$109$	−140412.	−1.13198	−0.565989	−	0.824413i	$-0.691506\pi$
$109$	−140412.	−1.13198	−0.565989	+	0.824413i	$0.691506\pi$
$110$	0	0
$111$	−109297.	−0.841977
$112$	180816.i	1.36204i
$113$	− 8725.69i	− 0.0642841i	−0.999483	−	0.0321421i	$-0.989767\pi$
$113$	− 8725.69i	− 0.0642841i	0.999483	−	0.0321421i	$-0.0102329\pi$
$114$	−55255.9	−0.398214
$115$	0	0
$116$	−199737.	−1.37820
$117$	− 9835.80i	− 0.0664270i
$118$	161500.i	1.06775i
$119$	272643.	1.76493
$120$	0	0
$121$	81986.8	0.509074
$122$	268375.i	1.63246i
$123$	− 74598.6i	− 0.444598i
$124$	−197774.	−1.15509
$125$	0	0
$126$	−65310.7	−0.366486
$127$	− 125413.i	− 0.689974i	−0.938607	−	0.344987i	$-0.887883\pi$
$127$	− 125413.i	− 0.689974i	0.938607	−	0.344987i	$-0.112117\pi$
$128$	− 113899.i	− 0.614461i
$129$	−101044.	−0.534606
$130$	0	0
$131$	−392187.	−1.99671	−0.998354	−	0.0573546i	$-0.981733\pi$
$131$	−392187.	−1.99671	−0.998354	+	0.0573546i	$0.981733\pi$
$132$	− 208564.i	− 1.04185i
$133$	63374.3i	0.310659i
$134$	−114423.	−0.550494
$135$	0	0
$136$	−104418.	−0.484091
$137$	10040.7i	0.0457051i	0.999739	+	0.0228525i	$0.00727482\pi$
$137$	10040.7i	0.0457051i	−0.999739	+	0.0228525i	$0.992725\pi$
$138$	476994.i	2.13214i
$139$	413226.	1.81406	0.907028	−	0.421071i	$-0.138346\pi$
$139$	413226.	1.81406	0.907028	+	0.421071i	$0.138346\pi$
$140$	0	0
$141$	−147565.	−0.625081
$142$	518864.i	2.15940i
$143$	− 83315.1i	− 0.340709i
$144$	70412.1	0.282970
$145$	0	0
$146$	−31044.4	−0.120532
$147$	95972.0i	0.366312i
$148$	153517.i	0.576105i
$149$	−111303.	−0.410716	−0.205358	−	0.978687i	$-0.565836\pi$
$149$	−111303.	−0.410716	−0.205358	+	0.978687i	$0.565836\pi$
$150$	0	0
$151$	329006.	1.17425	0.587127	−	0.809495i	$-0.300259\pi$
$151$	329006.	1.17425	0.587127	+	0.809495i	$0.300259\pi$
$152$	− 24271.3i	− 0.0852086i
$153$	− 106171.i	− 0.366672i
$154$	−553221.	−1.87973
$155$	0	0
$156$	−71497.3	−0.235222
$157$	324164.i	1.04958i	0.851232	+	0.524790i	$0.175856\pi$
$157$	324164.i	1.04958i	−0.851232	+	0.524790i	$0.824144\pi$
$158$	120626.i	0.384412i
$159$	−122349.	−0.383803
$160$	0	0
$161$	547076.	1.66335
$162$	− 524122.i	− 1.56908i
$163$	471803.i	1.39089i	0.718581	+	0.695443i	$0.244792\pi$
$163$	471803.i	1.39089i	−0.718581	+	0.695443i	$0.755208\pi$
$164$	−104780.	−0.304207
$165$	0	0
$166$	−869017.	−2.44770
$167$	318786.i	0.884522i	0.896886	+	0.442261i	$0.145823\pi$
$167$	318786.i	0.884522i	−0.896886	+	0.442261i	$0.854177\pi$
$168$	− 148467.i	− 0.405842i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	24678.8	0.0645407
$172$	141924.i	0.365793i
$173$	340132.i	0.864036i	0.901865	+	0.432018i	$0.142198\pi$
$173$	340132.i	0.864036i	−0.901865	+	0.432018i	$0.857802\pi$
$174$	1.06773e6	2.67354
$175$	0	0
$176$	596433.	1.45138
$177$	− 373294.i	− 0.895608i
$178$	264677.i	0.626134i
$179$	749176.	1.74764	0.873819	−	0.486251i	$-0.161636\pi$
$179$	749176.	1.74764	0.873819	+	0.486251i	$0.161636\pi$
$180$	0	0
$181$	−496170.	−1.12573	−0.562865	−	0.826549i	$-0.690301\pi$
$181$	−496170.	−1.12573	−0.562865	+	0.826549i	$0.690301\pi$
$182$	189648.i	0.424394i
$183$	− 620326.i	− 1.36928i
$184$	−209521.	−0.456228
$185$	0	0
$186$	1.05724e6	2.24073
$187$	− 899332.i	− 1.88068i
$188$	207268.i	0.427699i
$189$	−479337.	−0.976082
$190$	0	0
$191$	784516.	1.55603	0.778015	−	0.628245i	$-0.216227\pi$
$191$	784516.	1.55603	0.778015	+	0.628245i	$0.216227\pi$
$192$	− 273151.i	− 0.534748i
$193$	− 394155.i	− 0.761683i	−0.924640	−	0.380842i	$-0.875634\pi$
$193$	− 394155.i	− 0.761683i	0.924640	−	0.380842i	$-0.124366\pi$
$194$	−241543.	−0.460776
$195$	0	0
$196$	134801.	0.250641
$197$	− 913374.i	− 1.67681i	−0.545051	−	0.838403i	$-0.683490\pi$
$197$	− 913374.i	− 1.67681i	0.545051	−	0.838403i	$-0.316510\pi$
$198$	215432.i	0.390523i
$199$	340138.	0.608866	0.304433	−	0.952534i	$-0.401533\pi$
$199$	340138.	0.608866	0.304433	+	0.952534i	$0.401533\pi$
$200$	0	0
$201$	264480.	0.461745
$202$	− 287325.i	− 0.495444i
$203$	− 1.22460e6i	− 2.08571i
$204$	−771767.	−1.29841
$205$	0	0
$206$	390876.	0.641757
$207$	− 213039.i	− 0.345567i
$208$	− 204461.i	− 0.327682i
$209$	209044.	0.331034
$210$	0	0
$211$	−582499.	−0.900719	−0.450360	−	0.892847i	$-0.648704\pi$
$211$	−582499.	−0.900719	−0.450360	+	0.892847i	$0.648704\pi$
$212$	171850.i	0.262609i
$213$	− 1.19931e6i	− 1.81127i
$214$	−910269.	−1.35874
$215$	0	0
$216$	183578.	0.267723
$217$	− 1.21257e6i	− 1.74806i
$218$	− 1.05428e6i	− 1.50249i
$219$	71756.5	0.101100
$220$	0	0
$221$	−308297.	−0.424609
$222$	− 820649.i	− 1.11757i
$223$	− 580068.i	− 0.781119i	−0.920578	−	0.390559i	$-0.872282\pi$
$223$	− 580068.i	− 0.781119i	0.920578	−	0.390559i	$-0.127718\pi$
$224$	−1.08389e6	−1.44333
$225$	0	0
$226$	65516.4	0.0853255
$227$	833624.i	1.07376i	0.843660	+	0.536878i	$0.180396\pi$
$227$	833624.i	1.07376i	−0.843660	+	0.536878i	$0.819604\pi$
$228$	− 179393.i	− 0.228543i
$229$	10222.2	0.0128811	0.00644057	−	0.999979i	$-0.497950\pi$
$229$	10222.2	0.0128811	0.00644057	+	0.999979i	$0.497950\pi$
$230$	0	0
$231$	1.27872e6	1.57669
$232$	469000.i	0.572075i
$233$	932235.i	1.12496i	0.826812	+	0.562478i	$0.190152\pi$
$233$	932235.i	1.12496i	−0.826812	+	0.562478i	$0.809848\pi$
$234$	73851.5	0.0881698
$235$	0	0
$236$	−524324.	−0.612801
$237$	− 278815.i	− 0.322438i
$238$	2.04713e6i	2.34262i
$239$	1.35733e6	1.53706	0.768530	−	0.639814i	$-0.220988\pi$
$239$	1.35733e6	1.53706	0.768530	+	0.639814i	$0.220988\pi$
$240$	0	0
$241$	59546.1	0.0660406	0.0330203	−	0.999455i	$-0.489487\pi$
$241$	59546.1	0.0660406	0.0330203	+	0.999455i	$0.489487\pi$
$242$	615594.i	0.675703i
$243$	432107.i	0.469435i
$244$	−871301.	−0.936901
$245$	0	0
$246$	560119.	0.590123
$247$	− 71661.9i	− 0.0747387i
$248$	464392.i	0.479464i
$249$	2.00866e6	2.05309
$250$	0	0
$251$	−1.20903e6	−1.21130	−0.605651	−	0.795731i	$-0.707087\pi$
$251$	−1.20903e6	−1.21130	−0.605651	+	0.795731i	$0.707087\pi$
$252$	− 212036.i	− 0.210333i
$253$	− 1.80457e6i	− 1.77244i
$254$	941656.	0.915815
$255$	0	0
$256$	1.35885e6	1.29590
$257$	1.91933e6i	1.81266i	0.422573	+	0.906329i	$0.361127\pi$
$257$	1.91933e6i	1.81266i	−0.422573	+	0.906329i	$0.638873\pi$
$258$	− 758679.i	− 0.709593i
$259$	−941221.	−0.871851
$260$	0	0
$261$	−476875.	−0.433315
$262$	− 2.94471e6i	− 2.65027i
$263$	− 637187.i	− 0.568038i	−0.958819	−	0.284019i	$-0.908332\pi$
$263$	− 637187.i	− 0.568038i	0.958819	−	0.284019i	$-0.0916679\pi$
$264$	−489728.	−0.432460
$265$	0	0
$266$	−475842.	−0.412343
$267$	− 611779.i	− 0.525190i
$268$	− 371484.i	− 0.315939i
$269$	−250073.	−0.210711	−0.105355	−	0.994435i	$-0.533598\pi$
$269$	−250073.	−0.210711	−0.105355	+	0.994435i	$0.533598\pi$
$270$	0	0
$271$	994132.	0.822282	0.411141	−	0.911572i	$-0.365130\pi$
$271$	994132.	0.822282	0.411141	+	0.911572i	$0.365130\pi$
$272$	− 2.20703e6i	− 1.80878i
$273$	− 438355.i	− 0.355975i
$274$	−75390.4	−0.0606652
$275$	0	0
$276$	−1.54860e6	−1.22368
$277$	1.80516e6i	1.41356i	0.707432	+	0.706782i	$0.249854\pi$
$277$	1.80516e6i	1.41356i	−0.707432	+	0.706782i	$0.750146\pi$
$278$	3.10268e6i	2.40783i
$279$	−472190.	−0.363167
$280$	0	0
$281$	−349545.	−0.264081	−0.132040	−	0.991244i	$-0.542153\pi$
$281$	−349545.	−0.264081	−0.132040	+	0.991244i	$0.542153\pi$
$282$	− 1.10799e6i	− 0.829682i
$283$	204139.i	0.151516i	0.997126	+	0.0757582i	$0.0241377\pi$
$283$	204139.i	0.151516i	−0.997126	+	0.0757582i	$0.975862\pi$
$284$	−1.68453e6	−1.23932
$285$	0	0
$286$	625567.	0.452229
$287$	− 642413.i	− 0.460373i
$288$	422084.i	0.299859i
$289$	−1.90801e6	−1.34381
$290$	0	0
$291$	558305.	0.386491
$292$	− 100788.i	− 0.0691755i
$293$	− 78795.6i	− 0.0536207i	−0.999641	−	0.0268104i	$-0.991465\pi$
$293$	− 78795.6i	− 0.0536207i	0.999641	−	0.0268104i	$-0.00853503\pi$
$294$	−720600.	−0.486212
$295$	0	0
$296$	360471.	0.239134
$297$	1.58113e6i	1.04010i
$298$	− 835713.i	− 0.545150i
$299$	−618618.	−0.400170
$300$	0	0
$301$	−870146.	−0.553575
$302$	2.47033e6i	1.55861i
$303$	664126.i	0.415570i
$304$	513010.	0.318377
$305$	0	0
$306$	797179.	0.486690
$307$	− 2.05653e6i	− 1.24534i	−0.782483	−	0.622672i	$-0.786047\pi$
$307$	− 2.05653e6i	− 1.24534i	0.782483	−	0.622672i	$-0.213953\pi$
$308$	− 1.79607e6i	− 1.07882i
$309$	−903475.	−0.538295
$310$	0	0
$311$	−3119.84	−0.00182908	−0.000914538	−	1.00000i	$-0.500291\pi$
$311$	−3119.84	−0.00182908	−0.000914538		1.00000i	$0.500291\pi$
$312$	167882.i	0.0976379i
$313$	984315.i	0.567902i	0.958839	+	0.283951i	$0.0916453\pi$
$313$	984315.i	0.567902i	−0.958839	+	0.283951i	$0.908355\pi$
$314$	−2.43397e6	−1.39313
$315$	0	0
$316$	−391620.	−0.220621
$317$	1.90110e6i	1.06257i	0.847194	+	0.531283i	$0.178290\pi$
$317$	1.90110e6i	1.06257i	−0.847194	+	0.531283i	$0.821710\pi$
$318$	− 918651.i	− 0.509428i
$319$	−4.03942e6	−2.22250
$320$	0	0
$321$	2.10401e6	1.13969
$322$	4.10769e6i	2.20779i
$323$	− 773543.i	− 0.412552i
$324$	1.70160e6	0.900525
$325$	0	0
$326$	−3.54251e6	−1.84615
$327$	2.43687e6i	1.26027i
$328$	246033.i	0.126273i
$329$	−1.27077e6	−0.647260
$330$	0	0
$331$	−1.74888e6	−0.877386	−0.438693	−	0.898637i	$-0.644558\pi$
$331$	−1.74888e6	−0.877386	−0.438693	+	0.898637i	$0.644558\pi$
$332$	− 2.82133e6i	− 1.40478i
$333$	366524.i	0.181131i
$334$	−2.39359e6	−1.17404
$335$	0	0
$336$	3.13808e6	1.51640
$337$	1.09525e6i	0.525338i	0.964886	+	0.262669i	$0.0846027\pi$
$337$	1.09525e6i	0.525338i	−0.964886	+	0.262669i	$0.915397\pi$
$338$	− 214449.i	− 0.102101i
$339$	−151435.	−0.0715695
$340$	0	0
$341$	−3.99973e6	−1.86271
$342$	185300.i	0.0856661i
$343$	− 1.68542e6i	− 0.773524i
$344$	333251.	0.151836
$345$	0	0
$346$	−2.55386e6	−1.14685
$347$	1.99055e6i	0.887461i	0.896160	+	0.443730i	$0.146345\pi$
$347$	1.99055e6i	0.887461i	−0.896160	+	0.443730i	$0.853655\pi$
$348$	3.46645e6i	1.53439i
$349$	846086.	0.371835	0.185918	−	0.982565i	$-0.440474\pi$
$349$	846086.	0.371835	0.185918	+	0.982565i	$0.440474\pi$
$350$	0	0
$351$	542021.	0.234827
$352$	3.57530e6i	1.53800i
$353$	− 2.61806e6i	− 1.11826i	−0.829079	−	0.559131i	$-0.811135\pi$
$353$	− 2.61806e6i	− 1.11826i	0.829079	−	0.559131i	$-0.188865\pi$
$354$	2.80286e6	1.18876
$355$	0	0
$356$	−859296.	−0.359350
$357$	− 4.73175e6i	− 1.96495i
$358$	5.62515e6i	2.31967i
$359$	1.63224e6	0.668418	0.334209	−	0.942499i	$-0.391531\pi$
$359$	1.63224e6	0.668418	0.334209	+	0.942499i	$0.391531\pi$
$360$	0	0
$361$	−2.29629e6	−0.927384
$362$	− 3.72546e6i	− 1.49420i
$363$	− 1.42289e6i	− 0.566768i
$364$	−615706.	−0.243568
$365$	0	0
$366$	4.65768e6	1.81747
$367$	− 3.09644e6i	− 1.20005i	−0.799983	−	0.600023i	$-0.795158\pi$
$367$	− 3.09644e6i	− 1.20005i	0.799983	−	0.600023i	$-0.204842\pi$
$368$	− 4.42854e6i	− 1.70467i
$369$	−250165.	−0.0956445
$370$	0	0
$371$	−1.05362e6	−0.397420
$372$	3.43239e6i	1.28600i
$373$	2.44663e6i	0.910533i	0.890355	+	0.455266i	$0.150456\pi$
$373$	2.44663e6i	0.910533i	−0.890355	+	0.455266i	$0.849544\pi$
$374$	6.75258e6	2.49627
$375$	0	0
$376$	486685.	0.177533
$377$	1.38474e6i	0.501782i
$378$	− 3.59907e6i	− 1.29557i
$379$	4.31307e6	1.54237	0.771184	−	0.636612i	$-0.219665\pi$
$379$	4.31307e6	1.54237	0.771184	+	0.636612i	$0.219665\pi$
$380$	0	0
$381$	−2.17656e6	−0.768170
$382$	5.89049e6i	2.06535i
$383$	− 4.18340e6i	− 1.45724i	−0.684916	−	0.728622i	$-0.740161\pi$
$383$	− 4.18340e6i	− 1.45724i	0.684916	−	0.728622i	$-0.259839\pi$
$384$	−1.97673e6	−0.684098
$385$	0	0
$386$	2.95949e6	1.01100
$387$	338847.i	0.115008i
$388$	− 784187.i	− 0.264448i
$389$	3.01512e6	1.01025	0.505126	−	0.863045i	$-0.331446\pi$
$389$	3.01512e6	1.01025	0.505126	+	0.863045i	$0.331446\pi$
$390$	0	0
$391$	−6.67758e6	−2.20891
$392$	− 316525.i	− 0.104038i
$393$	6.80644e6i	2.22300i
$394$	6.85801e6	2.22566
$395$	0	0
$396$	−699416.	−0.224129
$397$	− 183544.i	− 0.0584471i	−0.999573	−	0.0292236i	$-0.990697\pi$
$397$	− 183544.i	− 0.0584471i	0.999573	−	0.0292236i	$-0.00930347\pi$
$398$	2.55390e6i	0.808159i
$399$	1.09987e6	0.345866
$400$	0	0
$401$	−3.10417e6	−0.964015	−0.482008	−	0.876167i	$-0.660092\pi$
$401$	−3.10417e6	−0.964015	−0.482008	+	0.876167i	$0.660092\pi$
$402$	1.98583e6i	0.612882i
$403$	1.37114e6i	0.420551i
$404$	932822.	0.284345
$405$	0	0
$406$	9.19483e6	2.76840
$407$	3.10468e6i	0.929033i
$408$	1.81218e6i	0.538953i
$409$	−2.45114e6	−0.724535	−0.362268	−	0.932074i	$-0.617997\pi$
$409$	−2.45114e6	−0.724535	−0.362268	+	0.932074i	$0.617997\pi$
$410$	0	0
$411$	174258.	0.0508849
$412$	1.26901e6i	0.368317i
$413$	− 3.21466e6i	− 0.927385i
$414$	1.59959e6	0.458678
$415$	0	0
$416$	1.22564e6	0.347240
$417$	− 7.17158e6i	− 2.01964i
$418$	1.56960e6i	0.439387i
$419$	1.38177e6	0.384503	0.192251	−	0.981346i	$-0.438421\pi$
$419$	1.38177e6	0.384503	0.192251	+	0.981346i	$0.438421\pi$
$420$	0	0
$421$	−4.38636e6	−1.20614	−0.603071	−	0.797687i	$-0.706057\pi$
$421$	−4.38636e6	−1.20614	−0.603071	+	0.797687i	$0.706057\pi$
$422$	− 4.37366e6i	− 1.19554i
$423$	494857.i	0.134471i
$424$	403519.	0.109006
$425$	0	0
$426$	9.00495e6	2.40413
$427$	− 5.34200e6i	− 1.41786i
$428$	− 2.95526e6i	− 0.779805i
$429$	−1.44594e6	−0.379322
$430$	0	0
$431$	1.36275e6	0.353365	0.176682	−	0.984268i	$-0.443463\pi$
$431$	1.36275e6	0.353365	0.176682	+	0.984268i	$0.443463\pi$
$432$	3.88020e6i	1.00033i
$433$	5.59344e6i	1.43370i	0.697226	+	0.716851i	$0.254417\pi$
$433$	5.59344e6i	1.43370i	−0.697226	+	0.716851i	$0.745583\pi$
$434$	9.10449e6	2.32023
$435$	0	0
$436$	3.42279e6	0.862310
$437$	− 1.55216e6i	− 0.388807i
$438$	538780.i	0.134192i
$439$	4.86469e6	1.20474	0.602371	−	0.798216i	$-0.294223\pi$
$439$	4.86469e6	1.20474	0.602371	+	0.798216i	$0.294223\pi$
$440$	0	0
$441$	321840.	0.0788031
$442$	− 2.31483e6i	− 0.563591i
$443$	4.59200e6i	1.11171i	0.831278	+	0.555857i	$0.187610\pi$
$443$	4.59200e6i	1.11171i	−0.831278	+	0.555857i	$0.812390\pi$
$444$	2.66430e6	0.641395
$445$	0	0
$446$	4.35541e6	1.03679
$447$	1.93168e6i	0.457263i
$448$	− 2.35227e6i	− 0.553722i
$449$	351998.	0.0823994	0.0411997	−	0.999151i	$-0.486882\pi$
$449$	351998.	0.0823994	0.0411997	+	0.999151i	$0.486882\pi$
$450$	0	0
$451$	−2.11904e6	−0.490567
$452$	212704.i	0.0489699i
$453$	− 5.70994e6i	− 1.30733i
$454$	−6.25922e6	−1.42521
$455$	0	0
$456$	−421231.	−0.0948654
$457$	546819.i	0.122477i	0.998123	+	0.0612383i	$0.0195050\pi$
$457$	546819.i	0.122477i	−0.998123	+	0.0612383i	$0.980495\pi$
$458$	76752.6i	0.0170974i
$459$	5.85076e6	1.29623
$460$	0	0
$461$	−6.35474e6	−1.39266	−0.696331	−	0.717721i	$-0.745185\pi$
$461$	−6.35474e6	−1.39266	−0.696331	+	0.717721i	$0.745185\pi$
$462$	9.60121e6i	2.09277i
$463$	− 2.90818e6i	− 0.630476i	−0.949013	−	0.315238i	$-0.897916\pi$
$463$	− 2.90818e6i	− 0.630476i	0.949013	−	0.315238i	$-0.102084\pi$
$464$	−9.91303e6	−2.13753
$465$	0	0
$466$	−6.99963e6	−1.49317
$467$	− 3.33677e6i	− 0.708002i	−0.935245	−	0.354001i	$-0.884821\pi$
$467$	− 3.33677e6i	− 0.708002i	0.935245	−	0.354001i	$-0.115179\pi$
$468$	239765.i	0.0506023i
$469$	2.27759e6	0.478128
$470$	0	0
$471$	5.62590e6	1.16853
$472$	1.23116e6i	0.254366i
$473$	2.87024e6i	0.589882i
$474$	2.09347e6	0.427978
$475$	0	0
$476$	−6.64615e6	−1.34447
$477$	410295.i	0.0825658i
$478$	1.01914e7i	2.04017i
$479$	−1.93937e6	−0.386208	−0.193104	−	0.981178i	$-0.561856\pi$
$479$	−1.93937e6	−0.386208	−0.193104	+	0.981178i	$0.561856\pi$
$480$	0	0
$481$	1.06431e6	0.209751
$482$	447099.i	0.0876568i
$483$	− 9.49456e6i	− 1.85185i
$484$	−1.99857e6	−0.387799
$485$	0	0
$486$	−3.24445e6	−0.623089
$487$	− 5.88271e6i	− 1.12397i	−0.827147	−	0.561985i	$-0.810038\pi$
$487$	− 5.88271e6i	− 1.12397i	0.827147	−	0.561985i	$-0.189962\pi$
$488$	2.04590e6i	0.388896i
$489$	8.18820e6	1.54852
$490$	0	0
$491$	8.39552e6	1.57161	0.785804	−	0.618476i	$-0.212250\pi$
$491$	8.39552e6	1.57161	0.785804	+	0.618476i	$0.212250\pi$
$492$	1.81847e6i	0.338683i
$493$	1.49474e7i	2.76980i
$494$	538069.	0.0992021
$495$	0	0
$496$	−9.81564e6	−1.79149
$497$	− 1.03280e7i	− 1.87553i
$498$	1.50819e7i	2.72510i
$499$	−8.24513e6	−1.48234	−0.741168	−	0.671320i	$-0.765728\pi$
$499$	−8.24513e6	−1.48234	−0.741168	+	0.671320i	$0.765728\pi$
$500$	0	0
$501$	5.53257e6	0.984766
$502$	− 9.07792e6i	− 1.60778i
$503$	− 8.99706e6i	− 1.58555i	−0.609513	−	0.792776i	$-0.708635\pi$
$503$	− 8.99706e6i	− 1.58555i	0.609513	−	0.792776i	$-0.291365\pi$
$504$	−497880.	−0.0873069
$505$	0	0
$506$	1.35495e7	2.35259
$507$	495679.i	0.0856409i
$508$	3.05716e6i	0.525604i
$509$	−6.27444e6	−1.07345	−0.536723	−	0.843758i	$-0.680338\pi$
$509$	−6.27444e6	−1.07345	−0.536723	+	0.843758i	$0.680338\pi$
$510$	0	0
$511$	617938.	0.104687
$512$	6.55808e6i	1.10561i
$513$	1.35998e6i	0.228159i
$514$	−1.44111e7	−2.40597
$515$	0	0
$516$	2.46311e6	0.407249
$517$	4.19174e6i	0.689712i
$518$	− 7.06710e6i	− 1.15722i
$519$	5.90302e6	0.961958
$520$	0	0
$521$	−8.31964e6	−1.34280	−0.671398	−	0.741097i	$-0.734306\pi$
$521$	−8.31964e6	−1.34280	−0.671398	+	0.741097i	$0.734306\pi$
$522$	− 3.58059e6i	− 0.575147i
$523$	8.50917e6i	1.36029i	0.733076	+	0.680147i	$0.238084\pi$
$523$	8.50917e6i	1.36029i	−0.733076	+	0.680147i	$0.761916\pi$
$524$	9.56023e6	1.52104
$525$	0	0
$526$	4.78428e6	0.753967
$527$	1.48005e7i	2.32140i
$528$	− 1.03512e7i	− 1.61586i
$529$	−6.96264e6	−1.08177
$530$	0	0
$531$	−1.25183e6	−0.192668
$532$	− 1.54486e6i	− 0.236652i
$533$	726423.i	0.110757i
$534$	4.59351e6	0.697094
$535$	0	0
$536$	−872280.	−0.131143
$537$	− 1.30020e7i	− 1.94570i
$538$	− 1.87766e6i	− 0.279680i
$539$	2.72618e6	0.404187
$540$	0	0
$541$	−1.15207e6	−0.169233	−0.0846164	−	0.996414i	$-0.526966\pi$
$541$	−1.15207e6	−0.169233	−0.0846164	+	0.996414i	$0.526966\pi$
$542$	7.46438e6i	1.09143i
$543$	8.61109e6i	1.25331i
$544$	1.32300e7	1.91673
$545$	0	0
$546$	3.29136e6	0.472491
$547$	− 943671.i	− 0.134850i	−0.997724	−	0.0674252i	$-0.978522\pi$
$547$	− 943671.i	− 0.134850i	0.997724	−	0.0674252i	$-0.0214784\pi$
$548$	− 244761.i	− 0.0348169i
$549$	−2.08025e6	−0.294567
$550$	0	0
$551$	−3.47443e6	−0.487534
$552$	3.63625e6i	0.507933i
$553$	− 2.40105e6i	− 0.333878i
$554$	−1.35539e7	−1.87625
$555$	0	0
$556$	−1.00731e7	−1.38190
$557$	− 1.19731e7i	− 1.63519i	−0.575791	−	0.817597i	$-0.695306\pi$
$557$	− 1.19731e7i	− 1.63519i	0.575791	−	0.817597i	$-0.304694\pi$
$558$	− 3.54541e6i	− 0.482038i
$559$	983938.	0.133180
$560$	0	0
$561$	−1.56080e7	−2.09382
$562$	− 2.62454e6i	− 0.350519i
$563$	3.19566e6i	0.424903i	0.977172	+	0.212452i	$0.0681448\pi$
$563$	3.19566e6i	0.424903i	−0.977172	+	0.212452i	$0.931855\pi$
$564$	3.59716e6	0.476170
$565$	0	0
$566$	−1.53276e6	−0.201110
$567$	1.04326e7i	1.36281i
$568$	3.95544e6i	0.514427i
$569$	−3.18417e6	−0.412303	−0.206151	−	0.978520i	$-0.566094\pi$
$569$	−3.18417e6	−0.412303	−0.206151	+	0.978520i	$0.566094\pi$
$570$	0	0
$571$	−6.71439e6	−0.861820	−0.430910	−	0.902395i	$-0.641807\pi$
$571$	−6.71439e6	−0.861820	−0.430910	+	0.902395i	$0.641807\pi$
$572$	2.03095e6i	0.259543i
$573$	− 1.36154e7i	− 1.73238i
$574$	4.82352e6	0.611061
$575$	0	0
$576$	−916005.	−0.115038
$577$	4.75161e6i	0.594157i	0.954853	+	0.297078i	$0.0960122\pi$
$577$	4.75161e6i	0.594157i	−0.954853	+	0.297078i	$0.903988\pi$
$578$	− 1.43262e7i	− 1.78366i
$579$	−6.84061e6	−0.848005
$580$	0	0
$581$	1.72977e7	2.12593
$582$	4.19200e6i	0.512996i
$583$	3.47544e6i	0.423486i
$584$	−236660.	−0.0287139
$585$	0	0
$586$	591632.	0.0711718
$587$	− 639190.i	− 0.0765658i	−0.999267	−	0.0382829i	$-0.987811\pi$
$587$	− 639190.i	− 0.0765658i	0.999267	−	0.0382829i	$-0.0121888\pi$
$588$	− 2.33948e6i	− 0.279047i
$589$	−3.44030e6	−0.408609
$590$	0	0
$591$	−1.58517e7	−1.86684
$592$	7.61911e6i	0.893512i
$593$	− 1.37403e6i	− 0.160458i	−0.996776	−	0.0802288i	$-0.974435\pi$
$593$	− 1.37403e6i	− 0.160458i	0.996776	−	0.0802288i	$-0.0255651\pi$
$594$	−1.18718e7	−1.38054
$595$	0	0
$596$	2.71320e6	0.312872
$597$	− 5.90313e6i	− 0.677870i
$598$	− 4.64486e6i	− 0.531153i
$599$	1.28982e7	1.46880	0.734401	−	0.678716i	$-0.237463\pi$
$599$	1.28982e7	1.46880	0.734401	+	0.678716i	$0.237463\pi$
$600$	0	0
$601$	6.50916e6	0.735087	0.367543	−	0.930006i	$-0.380199\pi$
$601$	6.50916e6	0.735087	0.367543	+	0.930006i	$0.380199\pi$
$602$	− 6.53345e6i	− 0.734769i
$603$	− 886926.i	− 0.0993332i
$604$	−8.02010e6	−0.894515
$605$	0	0
$606$	−4.98655e6	−0.551593
$607$	6.58784e6i	0.725724i	0.931843	+	0.362862i	$0.118200\pi$
$607$	6.58784e6i	0.725724i	−0.931843	+	0.362862i	$0.881800\pi$
$608$	3.07523e6i	0.337379i
$609$	−2.12530e7	−2.32208
$610$	0	0
$611$	1.43696e6	0.155719
$612$	2.58810e6i	0.279321i
$613$	6.83056e6i	0.734184i	0.930185	+	0.367092i	$0.119647\pi$
$613$	6.83056e6i	0.734184i	−0.930185	+	0.367092i	$0.880353\pi$
$614$	1.54413e7	1.65297
$615$	0	0
$616$	−4.21735e6	−0.447803
$617$	− 5.07265e6i	− 0.536441i	−0.963357	−	0.268221i	$-0.913564\pi$
$617$	− 5.07265e6i	− 0.536441i	0.963357	−	0.268221i	$-0.0864356\pi$
$618$	− 6.78369e6i	− 0.714488i
$619$	−2.32710e6	−0.244112	−0.122056	−	0.992523i	$-0.538949\pi$
$619$	−2.32710e6	−0.244112	−0.122056	+	0.992523i	$0.538949\pi$
$620$	0	0
$621$	1.17399e7	1.22162
$622$	− 23425.2i	− 0.00242776i
$623$	− 5.26840e6i	− 0.543824i
$624$	−3.54845e6	−0.364819
$625$	0	0
$626$	−7.39067e6	−0.753786
$627$	− 3.62799e6i	− 0.368551i
$628$	− 7.90206e6i	− 0.799542i
$629$	1.14885e7	1.15781
$630$	0	0
$631$	1.62914e7	1.62886	0.814432	−	0.580259i	$-0.197049\pi$
$631$	1.62914e7	1.62886	0.814432	+	0.580259i	$0.197049\pi$
$632$	919560.i	0.0915772i
$633$	1.01093e7i	1.00280i
$634$	−1.42743e7	−1.41036
$635$	0	0
$636$	2.98247e6	0.292370
$637$	− 934552.i	− 0.0912547i
$638$	− 3.03298e7i	− 2.94997i
$639$	−4.02186e6	−0.389650
$640$	0	0
$641$	9.20916e6	0.885268	0.442634	−	0.896702i	$-0.354044\pi$
$641$	9.20916e6	0.885268	0.442634	+	0.896702i	$0.354044\pi$
$642$	1.57978e7i	1.51272i
$643$	− 1.30502e7i	− 1.24477i	−0.782710	−	0.622387i	$-0.786163\pi$
$643$	− 1.30502e7i	− 1.24477i	0.782710	−	0.622387i	$-0.213837\pi$
$644$	−1.33359e7	−1.26709
$645$	0	0
$646$	5.80811e6	0.547587
$647$	− 4.14869e6i	− 0.389628i	−0.980840	−	0.194814i	$-0.937590\pi$
$647$	− 4.14869e6i	− 0.389628i	0.980840	−	0.194814i	$-0.0624103\pi$
$648$	− 3.99552e6i	− 0.373797i
$649$	−1.06038e7	−0.988210
$650$	0	0
$651$	−2.10442e7	−1.94617
$652$	− 1.15010e7i	− 1.05954i
$653$	− 1.41931e7i	− 1.30255i	−0.758844	−	0.651273i	$-0.774235\pi$
$653$	− 1.41931e7i	− 1.30255i	0.758844	−	0.651273i	$-0.225765\pi$
$654$	−1.82971e7	−1.67277
$655$	0	0
$656$	−5.20029e6	−0.471811
$657$	− 240634.i	− 0.0217492i
$658$	− 9.54154e6i	− 0.859120i
$659$	−7.37184e6	−0.661245	−0.330622	−	0.943763i	$-0.607259\pi$
$659$	−7.37184e6	−0.661245	−0.330622	+	0.943763i	$0.607259\pi$
$660$	0	0
$661$	1.77538e7	1.58047	0.790235	−	0.612803i	$-0.209958\pi$
$661$	1.77538e7	1.58047	0.790235	+	0.612803i	$0.209958\pi$
$662$	− 1.31314e7i	− 1.16457i
$663$	5.35054e6i	0.472730i
$664$	−6.62474e6	−0.583108
$665$	0	0
$666$	−2.75203e6	−0.240418
$667$	2.99929e7i	2.61038i
$668$	− 7.77097e6i	− 0.673805i
$669$	−1.00672e7	−0.869644
$670$	0	0
$671$	−1.76210e7	−1.51086
$672$	1.88111e7i	1.60691i
$673$	274222.i	0.0233381i	0.999932	+	0.0116690i	$0.00371446\pi$
$673$	274222.i	0.0233381i	−0.999932	+	0.0116690i	$0.996286\pi$
$674$	−8.22363e6	−0.697290
$675$	0	0
$676$	696224.	0.0585979
$677$	− 3.03927e6i	− 0.254857i	−0.991848	−	0.127429i	$-0.959328\pi$
$677$	− 3.03927e6i	− 0.254857i	0.991848	−	0.127429i	$-0.0406724\pi$
$678$	− 1.13704e6i	− 0.0949955i
$679$	4.80790e6	0.400204
$680$	0	0
$681$	1.44676e7	1.19545
$682$	− 3.00318e7i	− 2.47241i
$683$	− 1.26346e7i	− 1.03636i	−0.855273	−	0.518178i	$-0.826610\pi$
$683$	− 1.26346e7i	− 1.03636i	0.855273	−	0.518178i	$-0.173390\pi$
$684$	−601589.	−0.0491654
$685$	0	0
$686$	1.26549e7	1.02671
$687$	− 177407.i	− 0.0143410i
$688$	7.04377e6i	0.567328i
$689$	1.19141e6	0.0956119
$690$	0	0
$691$	1.53558e7	1.22343	0.611713	−	0.791080i	$-0.290481\pi$
$691$	1.53558e7	1.22343	0.611713	+	0.791080i	$0.290481\pi$
$692$	− 8.29130e6i	− 0.658199i
$693$	− 4.28816e6i	− 0.339186i
$694$	−1.49459e7	−1.17794
$695$	0	0
$696$	8.13955e6	0.636909
$697$	7.84127e6i	0.611370i
$698$	6.35279e6i	0.493544i
$699$	1.61790e7	1.25245
$700$	0	0
$701$	−2.13816e7	−1.64341	−0.821703	−	0.569916i	$-0.806976\pi$
$701$	−2.13816e7	−1.64341	−0.821703	+	0.569916i	$0.806976\pi$
$702$	4.06974e6i	0.311690i
$703$	2.67043e6i	0.203795i
$704$	−7.75911e6	−0.590038
$705$	0	0
$706$	1.96576e7	1.48429
$707$	5.71919e6i	0.430315i
$708$	9.09970e6i	0.682250i
$709$	2.39380e7	1.78843	0.894214	−	0.447639i	$-0.147735\pi$
$709$	2.39380e7	1.78843	0.894214	+	0.447639i	$0.147735\pi$
$710$	0	0
$711$	−935001.	−0.0693646
$712$	2.01771e6i	0.149162i
$713$	2.96982e7i	2.18779i
$714$	3.55281e7	2.60811
$715$	0	0
$716$	−1.82625e7	−1.33130
$717$	− 2.35566e7i	− 1.71126i
$718$	1.22556e7i	0.887203i
$719$	2.50209e6	0.180501	0.0902507	−	0.995919i	$-0.471233\pi$
$719$	2.50209e6	0.180501	0.0902507	+	0.995919i	$0.471233\pi$
$720$	0	0
$721$	−7.78037e6	−0.557394
$722$	− 1.72416e7i	− 1.23093i
$723$	− 1.03343e6i	− 0.0735250i
$724$	1.20950e7	0.857550
$725$	0	0
$726$	1.06837e7	0.752281
$727$	− 2.78011e7i	− 1.95086i	−0.220315	−	0.975429i	$-0.570709\pi$
$727$	− 2.78011e7i	− 1.95086i	0.220315	−	0.975429i	$-0.429291\pi$
$728$	1.44574e6i	0.101102i
$729$	−9.46320e6	−0.659506
$730$	0	0
$731$	1.06210e7	0.735141
$732$	1.51215e7i	1.04308i
$733$	4.46741e6i	0.307111i	0.988140	+	0.153556i	$0.0490724\pi$
$733$	4.46741e6i	0.307111i	−0.988140	+	0.153556i	$0.950928\pi$
$734$	2.32495e7	1.59284
$735$	0	0
$736$	2.65468e7	1.80641
$737$	− 7.51280e6i	− 0.509487i
$738$	− 1.87835e6i	− 0.126951i
$739$	1.23320e7	0.830656	0.415328	−	0.909672i	$-0.363667\pi$
$739$	1.23320e7	0.830656	0.415328	+	0.909672i	$0.363667\pi$
$740$	0	0
$741$	−1.24370e6	−0.0832090
$742$	− 7.91106e6i	− 0.527503i
$743$	− 9.81721e6i	− 0.652403i	−0.945300	−	0.326202i	$-0.894231\pi$
$743$	− 9.81721e6i	− 0.652403i	0.945300	−	0.326202i	$-0.105769\pi$
$744$	8.05958e6	0.533802
$745$	0	0
$746$	−1.83704e7	−1.20857
$747$	− 6.73598e6i	− 0.441672i
$748$	2.19228e7i	1.43265i
$749$	1.81189e7	1.18012
$750$	0	0
$751$	4.20655e6	0.272161	0.136081	−	0.990698i	$-0.456549\pi$
$751$	4.20655e6	0.272161	0.136081	+	0.990698i	$0.456549\pi$
$752$	1.02868e7i	0.663341i
$753$	2.09828e7i	1.34858i
$754$	−1.03973e7	−0.666025
$755$	0	0
$756$	1.16847e7	0.743553
$757$	2.89772e7i	1.83788i	0.394400	+	0.918939i	$0.370952\pi$
$757$	2.89772e7i	1.83788i	−0.394400	+	0.918939i	$0.629048\pi$
$758$	3.23844e7i	2.04721i
$759$	−3.13185e7	−1.97331
$760$	0	0
$761$	−1.57354e6	−0.0984957	−0.0492478	−	0.998787i	$-0.515682\pi$
$761$	−1.57354e6	−0.0984957	−0.0492478	+	0.998787i	$0.515682\pi$
$762$	− 1.63425e7i	− 1.01961i
$763$	2.09853e7i	1.30498i
$764$	−1.91239e7	−1.18534
$765$	0	0
$766$	3.14108e7	1.93423
$767$	3.63505e6i	0.223112i
$768$	− 2.35830e7i	− 1.44276i
$769$	1.98322e7	1.20936	0.604680	−	0.796468i	$-0.293301\pi$
$769$	1.98322e7	1.20936	0.604680	+	0.796468i	$0.293301\pi$
$770$	0	0
$771$	3.33101e7	2.01809
$772$	9.60822e6i	0.580230i
$773$	− 2.42031e7i	− 1.45687i	−0.685112	−	0.728437i	$-0.740247\pi$
$773$	− 2.42031e7i	− 1.45687i	0.685112	−	0.728437i	$-0.259753\pi$
$774$	−2.54421e6	−0.152652
$775$	0	0
$776$	−1.84134e6	−0.109769
$777$	1.63350e7i	0.970658i
$778$	2.26388e7i	1.34093i
$779$	−1.82266e6	−0.107612
$780$	0	0
$781$	−3.40675e7	−1.99854
$782$	− 5.01382e7i	− 2.93192i
$783$	− 2.62792e7i	− 1.53182i
$784$	6.69024e6	0.388733
$785$	0	0
$786$	−5.11058e7	−2.95062
$787$	1.03509e6i	0.0595721i	0.999556	+	0.0297860i	$0.00948259\pi$
$787$	1.03509e6i	0.0595721i	−0.999556	+	0.0297860i	$0.990517\pi$
$788$	2.22651e7i	1.27735i
$789$	−1.10584e7	−0.632414
$790$	0	0
$791$	−1.30410e6	−0.0741089
$792$	1.64229e6i	0.0930331i
$793$	6.04059e6i	0.341111i
$794$	1.37813e6	0.0775779
$795$	0	0
$796$	−8.29144e6	−0.463818
$797$	− 2.57309e7i	− 1.43486i	−0.696632	−	0.717429i	$-0.745319\pi$
$797$	− 2.57309e7i	− 1.43486i	0.696632	−	0.717429i	$-0.254681\pi$
$798$	8.25829e6i	0.459074i
$799$	1.55110e7	0.859554
$800$	0	0
$801$	−2.05159e6	−0.112982
$802$	− 2.33075e7i	− 1.27955i
$803$	− 2.03831e6i	− 0.111553i
$804$	−6.44715e6	−0.351745
$805$	0	0
$806$	−1.02951e7	−0.558204
$807$	4.34005e6i	0.234591i
$808$	− 2.19035e6i	− 0.118028i
$809$	−3.62907e7	−1.94950	−0.974751	−	0.223297i	$-0.928318\pi$
$809$	−3.62907e7	−1.94950	−0.974751	+	0.223297i	$0.928318\pi$
$810$	0	0
$811$	−2.80415e7	−1.49709	−0.748546	−	0.663083i	$-0.769248\pi$
$811$	−2.80415e7	−1.49709	−0.748546	+	0.663083i	$0.769248\pi$
$812$	2.98517e7i	1.58884i
$813$	− 1.72533e7i	− 0.915472i
$814$	−2.33113e7	−1.23312
$815$	0	0
$816$	−3.83032e7	−2.01377
$817$	2.46878e6i	0.129398i
$818$	− 1.84042e7i	− 0.961688i
$819$	−1.47001e6	−0.0765793
$820$	0	0
$821$	−1.73781e7	−0.899795	−0.449897	−	0.893080i	$-0.648539\pi$
$821$	−1.73781e7	−0.899795	−0.449897	+	0.893080i	$0.648539\pi$
$822$	1.30841e6i	0.0675404i
$823$	− 8.26775e6i	− 0.425488i	−0.977108	−	0.212744i	$-0.931760\pi$
$823$	− 8.26775e6i	− 0.425488i	0.977108	−	0.212744i	$-0.0682401\pi$
$824$	2.97975e6	0.152884
$825$	0	0
$826$	2.41371e7	1.23094
$827$	− 9.80834e6i	− 0.498691i	−0.968415	−	0.249346i	$-0.919784\pi$
$827$	− 9.80834e6i	− 0.498691i	0.968415	−	0.249346i	$-0.0802155\pi$
$828$	5.19319e6i	0.263244i
$829$	−2.34318e7	−1.18419	−0.592093	−	0.805870i	$-0.701698\pi$
$829$	−2.34318e7	−1.18419	−0.592093	+	0.805870i	$0.701698\pi$
$830$	0	0
$831$	3.13287e7	1.57376
$832$	2.65988e6i	0.133215i
$833$	− 1.00879e7i	− 0.503718i
$834$	5.38474e7	2.68071
$835$	0	0
$836$	−5.09582e6	−0.252173
$837$	− 2.60210e7i	− 1.28384i
$838$	1.03749e7i	0.510357i
$839$	1.14704e7	0.562568	0.281284	−	0.959625i	$-0.409240\pi$
$839$	1.14704e7	0.562568	0.281284	+	0.959625i	$0.409240\pi$
$840$	0	0
$841$	4.66262e7	2.27321
$842$	− 3.29347e7i	− 1.60093i
$843$	6.06639e6i	0.294010i
$844$	1.41994e7	0.686144
$845$	0	0
$846$	−3.71561e6	−0.178486
$847$	− 1.22534e7i	− 0.586877i
$848$	8.52899e6i	0.407294i
$849$	3.54285e6	0.168688
$850$	0	0
$851$	2.30524e7	1.09117
$852$	2.92352e7i	1.37977i
$853$	− 2.31885e7i	− 1.09119i	−0.838049	−	0.545594i	$-0.816304\pi$
$853$	− 2.31885e7i	− 1.09119i	0.838049	−	0.545594i	$-0.183696\pi$
$854$	4.01101e7	1.88196
$855$	0	0
$856$	−6.93922e6	−0.323688
$857$	3.35616e7i	1.56096i	0.625184	+	0.780478i	$0.285024\pi$
$857$	3.35616e7i	1.56096i	−0.625184	+	0.780478i	$0.714976\pi$
$858$	− 1.08568e7i	− 0.503481i
$859$	2.84860e7	1.31719	0.658594	−	0.752498i	$-0.271151\pi$
$859$	2.84860e7	1.31719	0.658594	+	0.752498i	$0.271151\pi$
$860$	0	0
$861$	−1.11492e7	−0.512547
$862$	1.02321e7i	0.469028i
$863$	3.40787e7i	1.55760i	0.627273	+	0.778799i	$0.284171\pi$
$863$	3.40787e7i	1.55760i	−0.627273	+	0.778799i	$0.715829\pi$
$864$	−2.32598e7	−1.06004
$865$	0	0
$866$	−4.19980e7	−1.90298
$867$	3.31138e7i	1.49610i
$868$	2.95584e7i	1.33162i
$869$	−7.92002e6	−0.355776
$870$	0	0
$871$	−2.57544e6	−0.115029
$872$	− 8.03702e6i	− 0.357935i
$873$	− 1.87226e6i	− 0.0831441i
$874$	1.16543e7	0.516070
$875$	0	0
$876$	−1.74919e6	−0.0770152
$877$	1.29489e7i	0.568505i	0.958750	+	0.284252i	$0.0917453\pi$
$877$	1.29489e7i	0.568505i	−0.958750	+	0.284252i	$0.908255\pi$
$878$	3.65263e7i	1.59908i
$879$	−1.36751e6	−0.0596976
$880$	0	0
$881$	8.39894e6	0.364573	0.182287	−	0.983245i	$-0.441650\pi$
$881$	8.39894e6	0.364573	0.182287	+	0.983245i	$0.441650\pi$
$882$	2.41652e6i	0.104597i
$883$	− 2.38721e7i	− 1.03036i	−0.857083	−	0.515179i	$-0.827725\pi$
$883$	− 2.38721e7i	− 1.03036i	0.857083	−	0.515179i	$-0.172275\pi$
$884$	7.51528e6	0.323456
$885$	0	0
$886$	−3.44788e7	−1.47560
$887$	− 8.56803e6i	− 0.365655i	−0.983145	−	0.182828i	$-0.941475\pi$
$887$	− 8.56803e6i	− 0.365655i	0.983145	−	0.182828i	$-0.0585250\pi$
$888$	− 6.25602e6i	− 0.266236i
$889$	−1.87436e7	−0.795425
$890$	0	0
$891$	3.44128e7	1.45220
$892$	1.41402e7i	0.595035i
$893$	3.60544e6i	0.151297i
$894$	−1.45039e7	−0.606933
$895$	0	0
$896$	−1.70228e7	−0.708370
$897$	1.07362e7i	0.445522i
$898$	2.64295e6i	0.109370i
$899$	6.64777e7	2.74332
$900$	0	0
$901$	1.28605e7	0.527769
$902$	− 1.59107e7i	− 0.651139i
$903$	1.51015e7i	0.616312i
$904$	499449.	0.0203268
$905$	0	0
$906$	4.28728e7	1.73525
$907$	2.84173e7i	1.14700i	0.819204	+	0.573502i	$0.194416\pi$
$907$	2.84173e7i	1.14700i	−0.819204	+	0.573502i	$0.805584\pi$
$908$	− 2.03210e7i	− 0.817958i
$909$	2.22713e6	0.0893997
$910$	0	0
$911$	−1.90880e7	−0.762015	−0.381008	−	0.924572i	$-0.624423\pi$
$911$	−1.90880e7	−0.762015	−0.381008	+	0.924572i	$0.624423\pi$
$912$	− 8.90335e6i	− 0.354459i
$913$	− 5.70578e7i	− 2.26536i
$914$	−4.10576e6	−0.162565
$915$	0	0
$916$	−249183.	−0.00981251
$917$	5.86144e7i	2.30187i
$918$	4.39301e7i	1.72050i
$919$	1.95252e7	0.762619	0.381310	−	0.924447i	$-0.375473\pi$
$919$	1.95252e7	0.762619	0.381310	+	0.924447i	$0.375473\pi$
$920$	0	0
$921$	−3.56913e7	−1.38648
$922$	− 4.77142e7i	− 1.84850i
$923$	1.16786e7i	0.451218i
$924$	−3.11710e7	−1.20108
$925$	0	0
$926$	2.18359e7	0.836843
$927$	3.02978e6i	0.115801i
$928$	− 5.94235e7i	− 2.26510i
$929$	1.62698e7	0.618504	0.309252	−	0.950980i	$-0.399921\pi$
$929$	1.62698e7	0.618504	0.309252	+	0.950980i	$0.399921\pi$
$930$	0	0
$931$	2.34487e6	0.0886634
$932$	− 2.27248e7i	− 0.856961i
$933$	54145.2i	0.00203637i
$934$	2.50540e7	0.939744
$935$	0	0
$936$	562990.	0.0210044
$937$	3.49494e7i	1.30044i	0.759745	+	0.650221i	$0.225324\pi$
$937$	3.49494e7i	1.30044i	−0.759745	+	0.650221i	$0.774676\pi$
$938$	1.71012e7i	0.634628i
$939$	1.70829e7	0.632263
$940$	0	0
$941$	2.04858e7	0.754187	0.377094	−	0.926175i	$-0.376923\pi$
$941$	2.04858e7	0.754187	0.377094	+	0.926175i	$0.376923\pi$
$942$	4.22418e7i	1.55101i
$943$	1.57340e7i	0.576182i
$944$	−2.60225e7	−0.950426
$945$	0	0
$946$	−2.15510e7	−0.782961
$947$	3.59599e7i	1.30300i	0.758650	+	0.651498i	$0.225859\pi$
$947$	3.59599e7i	1.30300i	−0.758650	+	0.651498i	$0.774141\pi$
$948$	6.79661e6i	0.245624i
$949$	−698748.	−0.0251858
$950$	0	0
$951$	3.29938e7	1.18299
$952$	1.56058e7i	0.558076i
$953$	2.09887e7i	0.748606i	0.927306	+	0.374303i	$0.122118\pi$
$953$	2.09887e7i	0.748606i	−0.927306	+	0.374303i	$0.877882\pi$
$954$	−3.08068e6	−0.109591
$955$	0	0
$956$	−3.30873e7	−1.17089
$957$	7.01046e7i	2.47438i
$958$	− 1.45616e7i	− 0.512621i
$959$	1.50064e6	0.0526903
$960$	0	0
$961$	3.71955e7	1.29922
$962$	7.99129e6i	0.278406i
$963$	− 7.05574e6i	− 0.245176i
$964$	−1.45154e6	−0.0503079
$965$	0	0
$966$	7.12894e7	2.45800
$967$	5.30179e6i	0.182329i	0.995836	+	0.0911646i	$0.0290589\pi$
$967$	5.30179e6i	0.182329i	−0.995836	+	0.0911646i	$0.970941\pi$
$968$	4.69283e6i	0.160971i
$969$	−1.34249e7	−0.459306
$970$	0	0
$971$	−5.19084e7	−1.76681	−0.883405	−	0.468611i	$-0.844755\pi$
$971$	−5.19084e7	−1.76681	−0.883405	+	0.468611i	$0.844755\pi$
$972$	− 1.05334e7i	− 0.357603i
$973$	− 6.17588e7i	− 2.09130i
$974$	4.41700e7	1.49187
$975$	0	0
$976$	−4.32431e7	−1.45309
$977$	− 5.09506e7i	− 1.70771i	−0.520515	−	0.853853i	$-0.674260\pi$
$977$	− 5.09506e7i	− 1.70771i	0.520515	−	0.853853i	$-0.325740\pi$
$978$	6.14806e7i	2.05537i
$979$	−1.73782e7	−0.579492
$980$	0	0
$981$	8.17198e6	0.271116
$982$	6.30373e7i	2.08602i
$983$	− 4.03520e7i	− 1.33193i	−0.745984	−	0.665964i	$-0.768020\pi$
$983$	− 4.03520e7i	− 1.33193i	0.745984	−	0.665964i	$-0.231980\pi$
$984$	4.26994e6	0.140583
$985$	0	0
$986$	−1.12232e8	−3.67640
$987$	2.20544e7i	0.720614i
$988$	1.74688e6i	0.0569340i
$989$	2.13116e7	0.692829
$990$	0	0
$991$	−2.26833e7	−0.733705	−0.366852	−	0.930279i	$-0.619565\pi$
$991$	−2.26833e7	−0.733705	−0.366852	+	0.930279i	$0.619565\pi$
$992$	− 5.88396e7i	− 1.89841i
$993$	3.03521e7i	0.976821i
$994$	7.75470e7	2.48943
$995$	0	0
$996$	−4.89645e7	−1.56399
$997$	840910.i	0.0267924i	0.999910	+	0.0133962i	$0.00426427\pi$
$997$	840910.i	0.0267924i	−0.999910	+	0.0133962i	$0.995736\pi$
$998$	− 6.19081e7i	− 1.96753i
$999$	−2.01981e7	−0.640318

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.h.274.15		18
5.2	odd	4		325.6.a.h.1.3	✓	9
5.3	odd	4		325.6.a.i.1.7	yes	9
5.4	even	2	inner	325.6.b.h.274.4		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.3	✓	9	5.2	odd	4
325.6.a.i.1.7	yes	9	5.3	odd	4
325.6.b.h.274.4		18	5.4	even	2	inner
325.6.b.h.274.15		18	1.1	even	1	trivial

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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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	325.6.b.h.274.15

Level	$325$
Weight	$6$
Character	325.274
Analytic conductor	$52.125$
Analytic rank	$0$
Dimension	$18$
Inner twists	$2$