LMFDB - Embedded newform 1150.4.a.q.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(1,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1150.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.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:	$67.8521965066$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 107x^{3} - 3x^{2} + 2151x - 2916 $ x^5 - 107x^3 - 3x^2 + 2151*x - 2916
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.53483$ of defining polynomial
Character	$\chi$	$=$	1150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +0.534830 q^{3} +4.00000 q^{4} -1.06966 q^{6} +28.9380 q^{7} -8.00000 q^{8} -26.7140 q^{9} +33.5234 q^{11} +2.13932 q^{12} -10.2917 q^{13} -57.8760 q^{14} +16.0000 q^{16} -56.5854 q^{17} +53.4279 q^{18} -5.26405 q^{19} +15.4769 q^{21} -67.0468 q^{22} -23.0000 q^{23} -4.27864 q^{24} +20.5834 q^{26} -28.7278 q^{27} +115.752 q^{28} -162.334 q^{29} -160.556 q^{31} -32.0000 q^{32} +17.9293 q^{33} +113.171 q^{34} -106.856 q^{36} +16.9178 q^{37} +10.5281 q^{38} -5.50431 q^{39} +22.7758 q^{41} -30.9538 q^{42} -333.620 q^{43} +134.094 q^{44} +46.0000 q^{46} -130.383 q^{47} +8.55728 q^{48} +494.408 q^{49} -30.2636 q^{51} -41.1668 q^{52} -673.206 q^{53} +57.4557 q^{54} -231.504 q^{56} -2.81537 q^{57} +324.669 q^{58} +291.958 q^{59} +454.587 q^{61} +321.113 q^{62} -773.048 q^{63} +64.0000 q^{64} -35.8586 q^{66} -132.389 q^{67} -226.342 q^{68} -12.3011 q^{69} +121.326 q^{71} +213.712 q^{72} -176.482 q^{73} -33.8356 q^{74} -21.0562 q^{76} +970.100 q^{77} +11.0086 q^{78} +563.082 q^{79} +705.912 q^{81} -45.5516 q^{82} -809.068 q^{83} +61.9076 q^{84} +667.241 q^{86} -86.8213 q^{87} -268.187 q^{88} -702.417 q^{89} -297.821 q^{91} -92.0000 q^{92} -85.8703 q^{93} +260.766 q^{94} -17.1146 q^{96} -342.945 q^{97} -988.815 q^{98} -895.543 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} - 5 q^{3} + 20 q^{4} + 10 q^{6} + 3 q^{7} - 40 q^{8} + 84 q^{9} - 26 q^{11} - 20 q^{12} + 61 q^{13} - 6 q^{14} + 80 q^{16} - 231 q^{17} - 168 q^{18} + 74 q^{19} - 88 q^{21} + 52 q^{22} - 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) $ 5 * q - 10 * q^2 - 5 * q^3 + 20 * q^4 + 10 * q^6 + 3 * q^7 - 40 * q^8 + 84 * q^9 - 26 * q^11 - 20 * q^12 + 61 * q^13 - 6 * q^14 + 80 * q^16 - 231 * q^17 - 168 * q^18 + 74 * q^19 - 88 * q^21 + 52 * q^22 - 115 * q^23 + 40 * q^24 - 122 * q^26 - 368 * q^27 + 12 * q^28 + 37 * q^29 + 220 * q^31 - 160 * q^32 - 391 * q^33 + 462 * q^34 + 336 * q^36 - 220 * q^37 - 148 * q^38 + 195 * q^39 + 112 * q^41 + 176 * q^42 - 97 * q^43 - 104 * q^44 + 230 * q^46 - 370 * q^47 - 80 * q^48 + 182 * q^49 + 563 * q^51 + 244 * q^52 - 1224 * q^53 + 736 * q^54 - 24 * q^56 + 1009 * q^57 - 74 * q^58 - 181 * q^59 + 1574 * q^61 - 440 * q^62 - 1829 * q^63 + 320 * q^64 + 782 * q^66 + 435 * q^67 - 924 * q^68 + 115 * q^69 - 30 * q^71 - 672 * q^72 - 2180 * q^73 + 440 * q^74 + 296 * q^76 + 1401 * q^77 - 390 * q^78 + 609 * q^79 + 1453 * q^81 - 224 * q^82 - 2018 * q^83 - 352 * q^84 + 194 * q^86 + 1485 * q^87 + 208 * q^88 - 1663 * q^89 + 361 * q^91 - 460 * q^92 - 2741 * q^93 + 740 * q^94 + 160 * q^96 + 1274 * q^97 - 364 * q^98 + 2397 * q^99

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0.534830	0.102928	0.0514640	−	0.998675i	$-0.483611\pi$
$3$	0.534830	0.102928	0.0514640	+	0.998675i	$0.483611\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.06966	−0.0727812
$7$	28.9380	1.56250	0.781252	−	0.624215i	$-0.214581\pi$
$7$	28.9380	1.56250	0.781252	+	0.624215i	$0.214581\pi$
$8$	−8.00000	−0.353553
$9$	−26.7140	−0.989406
$10$	0	0
$11$	33.5234	0.918880	0.459440	−	0.888209i	$-0.348050\pi$
$11$	33.5234	0.918880	0.459440	+	0.888209i	$0.348050\pi$
$12$	2.13932	0.0514640
$13$	−10.2917	−0.219570	−0.109785	−	0.993955i	$-0.535016\pi$
$13$	−10.2917	−0.219570	−0.109785	+	0.993955i	$0.535016\pi$
$14$	−57.8760	−1.10486
$15$	0	0
$16$	16.0000	0.250000
$17$	−56.5854	−0.807293	−0.403646	−	0.914915i	$-0.632257\pi$
$17$	−56.5854	−0.807293	−0.403646	+	0.914915i	$0.632257\pi$
$18$	53.4279	0.699616
$19$	−5.26405	−0.0635608	−0.0317804	−	0.999495i	$-0.510118\pi$
$19$	−5.26405	−0.0635608	−0.0317804	+	0.999495i	$0.510118\pi$
$20$	0	0
$21$	15.4769	0.160826
$22$	−67.0468	−0.649747
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−4.27864	−0.0363906
$25$	0	0
$26$	20.5834	0.155259
$27$	−28.7278	−0.204766
$28$	115.752	0.781252
$29$	−162.334	−1.03947	−0.519737	−	0.854327i	$-0.673970\pi$
$29$	−162.334	−1.03947	−0.519737	+	0.854327i	$0.673970\pi$
$30$	0	0
$31$	−160.556	−0.930218	−0.465109	−	0.885253i	$-0.653985\pi$
$31$	−160.556	−0.930218	−0.465109	+	0.885253i	$0.653985\pi$
$32$	−32.0000	−0.176777
$33$	17.9293	0.0945786
$34$	113.171	0.570842
$35$	0	0
$36$	−106.856	−0.494703
$37$	16.9178	0.0751694	0.0375847	−	0.999293i	$-0.488034\pi$
$37$	16.9178	0.0751694	0.0375847	+	0.999293i	$0.488034\pi$
$38$	10.5281	0.0449443
$39$	−5.50431	−0.0225999
$40$	0	0
$41$	22.7758	0.0867557	0.0433779	−	0.999059i	$-0.486188\pi$
$41$	22.7758	0.0867557	0.0433779	+	0.999059i	$0.486188\pi$
$42$	−30.9538	−0.113721
$43$	−333.620	−1.18318	−0.591589	−	0.806240i	$-0.701499\pi$
$43$	−333.620	−1.18318	−0.591589	+	0.806240i	$0.701499\pi$
$44$	134.094	0.459440
$45$	0	0
$46$	46.0000	0.147442
$47$	−130.383	−0.404645	−0.202322	−	0.979319i	$-0.564849\pi$
$47$	−130.383	−0.404645	−0.202322	+	0.979319i	$0.564849\pi$
$48$	8.55728	0.0257320
$49$	494.408	1.44142
$50$	0	0
$51$	−30.2636	−0.0830931
$52$	−41.1668	−0.109785
$53$	−673.206	−1.74475	−0.872376	−	0.488835i	$-0.837422\pi$
$53$	−673.206	−1.74475	−0.872376	+	0.488835i	$0.837422\pi$
$54$	57.4557	0.144791
$55$	0	0
$56$	−231.504	−0.552429
$57$	−2.81537	−0.00654219
$58$	324.669	0.735019
$59$	291.958	0.644233	0.322116	−	0.946700i	$-0.395606\pi$
$59$	291.958	0.644233	0.322116	+	0.946700i	$0.395606\pi$
$60$	0	0
$61$	454.587	0.954162	0.477081	−	0.878859i	$-0.341695\pi$
$61$	454.587	0.954162	0.477081	+	0.878859i	$0.341695\pi$
$62$	321.113	0.657764
$63$	−773.048	−1.54595
$64$	64.0000	0.125000
$65$	0	0
$66$	−35.8586	−0.0668772
$67$	−132.389	−0.241401	−0.120701	−	0.992689i	$-0.538514\pi$
$67$	−132.389	−0.241401	−0.120701	+	0.992689i	$0.538514\pi$
$68$	−226.342	−0.403646
$69$	−12.3011	−0.0214620
$70$	0	0
$71$	121.326	0.202799	0.101399	−	0.994846i	$-0.467668\pi$
$71$	121.326	0.202799	0.101399	+	0.994846i	$0.467668\pi$
$72$	213.712	0.349808
$73$	−176.482	−0.282955	−0.141477	−	0.989942i	$-0.545185\pi$
$73$	−176.482	−0.282955	−0.141477	+	0.989942i	$0.545185\pi$
$74$	−33.8356	−0.0531528
$75$	0	0
$76$	−21.0562	−0.0317804
$77$	970.100	1.43576
$78$	11.0086	0.0159805
$79$	563.082	0.801920	0.400960	−	0.916096i	$-0.368677\pi$
$79$	563.082	0.801920	0.400960	+	0.916096i	$0.368677\pi$
$80$	0	0
$81$	705.912	0.968330
$82$	−45.5516	−0.0613456
$83$	−809.068	−1.06996	−0.534981	−	0.844864i	$-0.679681\pi$
$83$	−809.068	−1.06996	−0.534981	+	0.844864i	$0.679681\pi$
$84$	61.9076	0.0804128
$85$	0	0
$86$	667.241	0.836633
$87$	−86.8213	−0.106991
$88$	−268.187	−0.324873
$89$	−702.417	−0.836585	−0.418293	−	0.908312i	$-0.637371\pi$
$89$	−702.417	−0.836585	−0.418293	+	0.908312i	$0.637371\pi$
$90$	0	0
$91$	−297.821	−0.343079
$92$	−92.0000	−0.104257
$93$	−85.8703	−0.0957456
$94$	260.766	0.286127
$95$	0	0
$96$	−17.1146	−0.0181953
$97$	−342.945	−0.358977	−0.179489	−	0.983760i	$-0.557444\pi$
$97$	−342.945	−0.358977	−0.179489	+	0.983760i	$0.557444\pi$
$98$	−988.815	−1.01924
$99$	−895.543	−0.909146
$100$	0	0
$101$	1719.68	1.69420	0.847102	−	0.531431i	$-0.178345\pi$
$101$	1719.68	1.69420	0.847102	+	0.531431i	$0.178345\pi$
$102$	60.5272	0.0587557
$103$	−387.872	−0.371050	−0.185525	−	0.982640i	$-0.559399\pi$
$103$	−387.872	−0.371050	−0.185525	+	0.982640i	$0.559399\pi$
$104$	82.3337	0.0776296
$105$	0	0
$106$	1346.41	1.23373
$107$	−31.3904	−0.0283610	−0.0141805	−	0.999899i	$-0.504514\pi$
$107$	−31.3904	−0.0283610	−0.0141805	+	0.999899i	$0.504514\pi$
$108$	−114.911	−0.102383
$109$	686.159	0.602955	0.301478	−	0.953473i	$-0.402520\pi$
$109$	686.159	0.602955	0.301478	+	0.953473i	$0.402520\pi$
$110$	0	0
$111$	9.04814	0.00773704
$112$	463.008	0.390626
$113$	42.2350	0.0351605	0.0175802	−	0.999845i	$-0.494404\pi$
$113$	42.2350	0.0351605	0.0175802	+	0.999845i	$0.494404\pi$
$114$	5.63074	0.00462603
$115$	0	0
$116$	−649.337	−0.519737
$117$	274.932	0.217244
$118$	−583.917	−0.455541
$119$	−1637.47	−1.26140
$120$	0	0
$121$	−207.182	−0.155659
$122$	−909.173	−0.674694
$123$	12.1812	0.00892960
$124$	−642.225	−0.465109
$125$	0	0
$126$	1546.10	1.09315
$127$	−1234.48	−0.862538	−0.431269	−	0.902223i	$-0.641934\pi$
$127$	−1234.48	−0.862538	−0.431269	+	0.902223i	$0.641934\pi$
$128$	−128.000	−0.0883883
$129$	−178.430	−0.121782
$130$	0	0
$131$	−2216.74	−1.47846	−0.739228	−	0.673455i	$-0.764809\pi$
$131$	−2216.74	−1.47846	−0.739228	+	0.673455i	$0.764809\pi$
$132$	71.7173	0.0472893
$133$	−152.331	−0.0993140
$134$	264.778	0.170697
$135$	0	0
$136$	452.683	0.285421
$137$	1179.48	0.735545	0.367773	−	0.929916i	$-0.380120\pi$
$137$	1179.48	0.735545	0.367773	+	0.929916i	$0.380120\pi$
$138$	24.6022	0.0151759
$139$	241.224	0.147197	0.0735983	−	0.997288i	$-0.476552\pi$
$139$	241.224	0.147197	0.0735983	+	0.997288i	$0.476552\pi$
$140$	0	0
$141$	−69.7327	−0.0416493
$142$	−242.651	−0.143400
$143$	−345.013	−0.201758
$144$	−427.423	−0.247351
$145$	0	0
$146$	352.964	0.200079
$147$	264.424	0.148363
$148$	67.6711	0.0375847
$149$	−1028.14	−0.565294	−0.282647	−	0.959224i	$-0.591212\pi$
$149$	−1028.14	−0.565294	−0.282647	+	0.959224i	$0.591212\pi$
$150$	0	0
$151$	1511.24	0.814456	0.407228	−	0.913327i	$-0.366495\pi$
$151$	1511.24	0.814456	0.407228	+	0.913327i	$0.366495\pi$
$152$	42.1124	0.0224721
$153$	1511.62	0.798740
$154$	−1940.20	−1.01523
$155$	0	0
$156$	−22.0173	−0.0112999
$157$	2097.04	1.06600	0.532999	−	0.846116i	$-0.321065\pi$
$157$	2097.04	1.06600	0.532999	+	0.846116i	$0.321065\pi$
$158$	−1126.16	−0.567043
$159$	−360.051	−0.179584
$160$	0	0
$161$	−665.574	−0.325805
$162$	−1411.82	−0.684712
$163$	−703.046	−0.337833	−0.168917	−	0.985630i	$-0.554027\pi$
$163$	−703.046	−0.337833	−0.168917	+	0.985630i	$0.554027\pi$
$164$	91.1033	0.0433779
$165$	0	0
$166$	1618.14	0.756577
$167$	−3063.04	−1.41931	−0.709656	−	0.704549i	$-0.751150\pi$
$167$	−3063.04	−1.41931	−0.709656	+	0.704549i	$0.751150\pi$
$168$	−123.815	−0.0568605
$169$	−2091.08	−0.951789
$170$	0	0
$171$	140.624	0.0628874
$172$	−1334.48	−0.591589
$173$	−2810.34	−1.23506	−0.617531	−	0.786546i	$-0.711867\pi$
$173$	−2810.34	−1.23506	−0.617531	+	0.786546i	$0.711867\pi$
$174$	173.643	0.0756541
$175$	0	0
$176$	536.374	0.229720
$177$	156.148	0.0663096
$178$	1404.83	0.591555
$179$	−2516.53	−1.05081	−0.525403	−	0.850853i	$-0.676086\pi$
$179$	−2516.53	−1.05081	−0.525403	+	0.850853i	$0.676086\pi$
$180$	0	0
$181$	1085.01	0.445570	0.222785	−	0.974868i	$-0.428485\pi$
$181$	1085.01	0.445570	0.222785	+	0.974868i	$0.428485\pi$
$182$	595.643	0.242593
$183$	243.127	0.0982100
$184$	184.000	0.0737210
$185$	0	0
$186$	171.741	0.0677024
$187$	−1896.94	−0.741806
$188$	−521.532	−0.202322
$189$	−831.326	−0.319947
$190$	0	0
$191$	−2081.87	−0.788683	−0.394342	−	0.918964i	$-0.629027\pi$
$191$	−2081.87	−0.788683	−0.394342	+	0.918964i	$0.629027\pi$
$192$	34.2291	0.0128660
$193$	−648.493	−0.241863	−0.120931	−	0.992661i	$-0.538588\pi$
$193$	−648.493	−0.241863	−0.120931	+	0.992661i	$0.538588\pi$
$194$	685.890	0.253835
$195$	0	0
$196$	1977.63	0.720711
$197$	1250.92	0.452409	0.226205	−	0.974080i	$-0.427368\pi$
$197$	1250.92	0.452409	0.226205	+	0.974080i	$0.427368\pi$
$198$	1791.09	0.642863
$199$	3135.54	1.11695	0.558473	−	0.829523i	$-0.311387\pi$
$199$	3135.54	1.11695	0.558473	+	0.829523i	$0.311387\pi$
$200$	0	0
$201$	−70.8056	−0.0248470
$202$	−3439.36	−1.19798
$203$	−4697.63	−1.62418
$204$	−121.054	−0.0415466
$205$	0	0
$206$	775.743	0.262372
$207$	614.421	0.206305
$208$	−164.667	−0.0548924
$209$	−176.469	−0.0584048
$210$	0	0
$211$	−844.189	−0.275433	−0.137716	−	0.990472i	$-0.543976\pi$
$211$	−844.189	−0.275433	−0.137716	+	0.990472i	$0.543976\pi$
$212$	−2692.82	−0.872376
$213$	64.8886	0.0208737
$214$	62.7808	0.0200542
$215$	0	0
$216$	229.823	0.0723956
$217$	−4646.18	−1.45347
$218$	−1372.32	−0.426354
$219$	−94.3880	−0.0291240
$220$	0	0
$221$	582.360	0.177257
$222$	−18.0963	−0.00547091
$223$	−5648.57	−1.69622	−0.848108	−	0.529824i	$-0.822258\pi$
$223$	−5648.57	−1.69622	−0.848108	+	0.529824i	$0.822258\pi$
$224$	−926.016	−0.276214
$225$	0	0
$226$	−84.4700	−0.0248622
$227$	3914.91	1.14468	0.572338	−	0.820018i	$-0.306036\pi$
$227$	3914.91	1.14468	0.572338	+	0.820018i	$0.306036\pi$
$228$	−11.2615	−0.00327110
$229$	−550.992	−0.158998	−0.0794990	−	0.996835i	$-0.525332\pi$
$229$	−550.992	−0.158998	−0.0794990	+	0.996835i	$0.525332\pi$
$230$	0	0
$231$	518.839	0.147780
$232$	1298.67	0.367509
$233$	−5785.67	−1.62675	−0.813373	−	0.581742i	$-0.802371\pi$
$233$	−5785.67	−1.62675	−0.813373	+	0.581742i	$0.802371\pi$
$234$	−549.864	−0.153614
$235$	0	0
$236$	1167.83	0.322116
$237$	301.153	0.0825400
$238$	3274.94	0.891944
$239$	−4525.71	−1.22487	−0.612435	−	0.790521i	$-0.709810\pi$
$239$	−4525.71	−1.22487	−0.612435	+	0.790521i	$0.709810\pi$
$240$	0	0
$241$	517.292	0.138264	0.0691322	−	0.997608i	$-0.477977\pi$
$241$	517.292	0.138264	0.0691322	+	0.997608i	$0.477977\pi$
$242$	414.363	0.110067
$243$	1153.19	0.304434
$244$	1818.35	0.477081
$245$	0	0
$246$	−24.3624	−0.00631418
$247$	54.1760	0.0139560
$248$	1284.45	0.328882
$249$	−432.714	−0.110129
$250$	0	0
$251$	−7039.00	−1.77011	−0.885055	−	0.465486i	$-0.845879\pi$
$251$	−7039.00	−1.77011	−0.885055	+	0.465486i	$0.845879\pi$
$252$	−3092.19	−0.772976
$253$	−771.038	−0.191600
$254$	2468.96	0.609906
$255$	0	0
$256$	256.000	0.0625000
$257$	4250.94	1.03177	0.515887	−	0.856656i	$-0.327462\pi$
$257$	4250.94	1.03177	0.515887	+	0.856656i	$0.327462\pi$
$258$	356.860	0.0861130
$259$	489.567	0.117453
$260$	0	0
$261$	4336.59	1.02846
$262$	4433.49	1.04543
$263$	−3598.83	−0.843778	−0.421889	−	0.906648i	$-0.638633\pi$
$263$	−3598.83	−0.843778	−0.421889	+	0.906648i	$0.638633\pi$
$264$	−143.435	−0.0334386
$265$	0	0
$266$	304.662	0.0702256
$267$	−375.674	−0.0861081
$268$	−529.556	−0.120701
$269$	−5062.52	−1.14746	−0.573731	−	0.819044i	$-0.694505\pi$
$269$	−5062.52	−1.14746	−0.573731	+	0.819044i	$0.694505\pi$
$270$	0	0
$271$	1854.58	0.415712	0.207856	−	0.978159i	$-0.433352\pi$
$271$	1854.58	0.415712	0.207856	+	0.978159i	$0.433352\pi$
$272$	−905.366	−0.201823
$273$	−159.284	−0.0353124
$274$	−2358.96	−0.520109
$275$	0	0
$276$	−49.2044	−0.0107310
$277$	−8747.91	−1.89751	−0.948756	−	0.316009i	$-0.897657\pi$
$277$	−8747.91	−1.89751	−0.948756	+	0.316009i	$0.897657\pi$
$278$	−482.447	−0.104084
$279$	4289.09	0.920363
$280$	0	0
$281$	4046.88	0.859134	0.429567	−	0.903035i	$-0.358666\pi$
$281$	4046.88	0.859134	0.429567	+	0.903035i	$0.358666\pi$
$282$	139.465	0.0294505
$283$	−1598.52	−0.335766	−0.167883	−	0.985807i	$-0.553693\pi$
$283$	−1598.52	−0.335766	−0.167883	+	0.985807i	$0.553693\pi$
$284$	485.303	0.101399
$285$	0	0
$286$	690.026	0.142665
$287$	659.086	0.135556
$288$	854.847	0.174904
$289$	−1711.09	−0.348278
$290$	0	0
$291$	−183.417	−0.0369488
$292$	−705.929	−0.141477
$293$	4542.62	0.905742	0.452871	−	0.891576i	$-0.350400\pi$
$293$	4542.62	0.905742	0.452871	+	0.891576i	$0.350400\pi$
$294$	−528.848	−0.104908
$295$	0	0
$296$	−135.342	−0.0265764
$297$	−963.055	−0.188155
$298$	2056.29	0.399723
$299$	236.709	0.0457834
$300$	0	0
$301$	−9654.31	−1.84872
$302$	−3022.48	−0.575907
$303$	919.736	0.174381
$304$	−84.2247	−0.0158902
$305$	0	0
$306$	−3023.24	−0.564795
$307$	2366.39	0.439925	0.219963	−	0.975508i	$-0.429406\pi$
$307$	2366.39	0.439925	0.219963	+	0.975508i	$0.429406\pi$
$308$	3880.40	0.717878
$309$	−207.445	−0.0381914
$310$	0	0
$311$	830.158	0.151363	0.0756816	−	0.997132i	$-0.475887\pi$
$311$	830.158	0.151363	0.0756816	+	0.997132i	$0.475887\pi$
$312$	44.0345	0.00799027
$313$	7182.03	1.29697	0.648486	−	0.761226i	$-0.275402\pi$
$313$	7182.03	1.29697	0.648486	+	0.761226i	$0.275402\pi$
$314$	−4194.07	−0.753775
$315$	0	0
$316$	2252.33	0.400960
$317$	−1910.06	−0.338422	−0.169211	−	0.985580i	$-0.554122\pi$
$317$	−1910.06	−0.338422	−0.169211	+	0.985580i	$0.554122\pi$
$318$	720.101	0.126985
$319$	−5442.00	−0.955152
$320$	0	0
$321$	−16.7885	−0.00291914
$322$	1331.15	0.230379
$323$	297.868	0.0513122
$324$	2823.65	0.484165
$325$	0	0
$326$	1406.09	0.238884
$327$	366.979	0.0620610
$328$	−182.207	−0.0306728
$329$	−3773.02	−0.632260
$330$	0	0
$331$	7540.40	1.25214	0.626069	−	0.779767i	$-0.284663\pi$
$331$	7540.40	1.25214	0.626069	+	0.779767i	$0.284663\pi$
$332$	−3236.27	−0.534981
$333$	−451.941	−0.0743730
$334$	6126.08	1.00360
$335$	0	0
$336$	247.631	0.0402064
$337$	10069.4	1.62764	0.813822	−	0.581114i	$-0.197383\pi$
$337$	10069.4	1.62764	0.813822	+	0.581114i	$0.197383\pi$
$338$	4182.16	0.673017
$339$	22.5885	0.00361900
$340$	0	0
$341$	−5382.39	−0.854759
$342$	−281.247	−0.0444681
$343$	4381.43	0.689723
$344$	2668.96	0.418316
$345$	0	0
$346$	5620.67	0.873321
$347$	−6031.06	−0.933039	−0.466519	−	0.884511i	$-0.654492\pi$
$347$	−6031.06	−0.933039	−0.466519	+	0.884511i	$0.654492\pi$
$348$	−347.285	−0.0534955
$349$	−11352.0	−1.74114	−0.870569	−	0.492047i	$-0.836249\pi$
$349$	−11352.0	−1.74114	−0.870569	+	0.492047i	$0.836249\pi$
$350$	0	0
$351$	295.659	0.0449604
$352$	−1072.75	−0.162437
$353$	4864.81	0.733506	0.366753	−	0.930318i	$-0.380469\pi$
$353$	4864.81	0.733506	0.366753	+	0.930318i	$0.380469\pi$
$354$	−312.296	−0.0468880
$355$	0	0
$356$	−2809.67	−0.418293
$357$	−875.767	−0.129833
$358$	5033.06	0.743032
$359$	−5809.49	−0.854076	−0.427038	−	0.904234i	$-0.640443\pi$
$359$	−5809.49	−0.854076	−0.427038	+	0.904234i	$0.640443\pi$
$360$	0	0
$361$	−6831.29	−0.995960
$362$	−2170.02	−0.315066
$363$	−110.807	−0.0160216
$364$	−1191.29	−0.171539
$365$	0	0
$366$	−486.253	−0.0694450
$367$	−6984.38	−0.993410	−0.496705	−	0.867919i	$-0.665457\pi$
$367$	−6984.38	−0.993410	−0.496705	+	0.867919i	$0.665457\pi$
$368$	−368.000	−0.0521286
$369$	−608.432	−0.0858366
$370$	0	0
$371$	−19481.2	−2.72618
$372$	−343.481	−0.0478728
$373$	−8551.53	−1.18708	−0.593541	−	0.804804i	$-0.702270\pi$
$373$	−8551.53	−1.18708	−0.593541	+	0.804804i	$0.702270\pi$
$374$	3793.87	0.524536
$375$	0	0
$376$	1043.06	0.143064
$377$	1670.70	0.228237
$378$	1662.65	0.226237
$379$	9933.40	1.34629	0.673145	−	0.739510i	$-0.264943\pi$
$379$	9933.40	1.34629	0.673145	+	0.739510i	$0.264943\pi$
$380$	0	0
$381$	−660.236	−0.0887794
$382$	4163.73	0.557683
$383$	−9748.52	−1.30059	−0.650295	−	0.759682i	$-0.725355\pi$
$383$	−9748.52	−1.30059	−0.650295	+	0.759682i	$0.725355\pi$
$384$	−68.4582	−0.00909764
$385$	0	0
$386$	1296.99	0.171023
$387$	8912.32	1.17064
$388$	−1371.78	−0.179489
$389$	−10517.0	−1.37078	−0.685391	−	0.728175i	$-0.740369\pi$
$389$	−10517.0	−1.37078	−0.685391	+	0.728175i	$0.740369\pi$
$390$	0	0
$391$	1301.46	0.168332
$392$	−3955.26	−0.509619
$393$	−1185.58	−0.152175
$394$	−2501.85	−0.319902
$395$	0	0
$396$	−3582.17	−0.454573
$397$	−4117.94	−0.520588	−0.260294	−	0.965529i	$-0.583819\pi$
$397$	−4117.94	−0.520588	−0.260294	+	0.965529i	$0.583819\pi$
$398$	−6271.07	−0.789800
$399$	−81.4712	−0.0102222
$400$	0	0
$401$	−7374.29	−0.918340	−0.459170	−	0.888348i	$-0.651853\pi$
$401$	−7374.29	−0.918340	−0.459170	+	0.888348i	$0.651853\pi$
$402$	141.611	0.0175695
$403$	1652.40	0.204248
$404$	6878.72	0.847102
$405$	0	0
$406$	9395.26	1.14847
$407$	567.142	0.0690717
$408$	242.109	0.0293778
$409$	12941.5	1.56459	0.782296	−	0.622907i	$-0.214049\pi$
$409$	12941.5	1.56459	0.782296	+	0.622907i	$0.214049\pi$
$410$	0	0
$411$	630.821	0.0757083
$412$	−1551.49	−0.185525
$413$	8448.69	1.00662
$414$	−1228.84	−0.145880
$415$	0	0
$416$	329.335	0.0388148
$417$	129.014	0.0151507
$418$	352.937	0.0412984
$419$	5633.46	0.656832	0.328416	−	0.944533i	$-0.393485\pi$
$419$	5633.46	0.656832	0.328416	+	0.944533i	$0.393485\pi$
$420$	0	0
$421$	−4205.49	−0.486848	−0.243424	−	0.969920i	$-0.578271\pi$
$421$	−4205.49	−0.486848	−0.243424	+	0.969920i	$0.578271\pi$
$422$	1688.38	0.194761
$423$	3483.04	0.400358
$424$	5385.64	0.616863
$425$	0	0
$426$	−129.777	−0.0147599
$427$	13154.8	1.49088
$428$	−125.562	−0.0141805
$429$	−184.523	−0.0207666
$430$	0	0
$431$	15075.7	1.68486	0.842428	−	0.538809i	$-0.181125\pi$
$431$	15075.7	1.68486	0.842428	+	0.538809i	$0.181125\pi$
$432$	−459.645	−0.0511914
$433$	−3098.05	−0.343840	−0.171920	−	0.985111i	$-0.554997\pi$
$433$	−3098.05	−0.343840	−0.171920	+	0.985111i	$0.554997\pi$
$434$	9292.36	1.02776
$435$	0	0
$436$	2744.64	0.301478
$437$	121.073	0.0132533
$438$	188.776	0.0205938
$439$	6867.74	0.746649	0.373325	−	0.927701i	$-0.378218\pi$
$439$	6867.74	0.746649	0.373325	+	0.927701i	$0.378218\pi$
$440$	0	0
$441$	−13207.6	−1.42615
$442$	−1164.72	−0.125340
$443$	5878.29	0.630442	0.315221	−	0.949018i	$-0.397921\pi$
$443$	5878.29	0.630442	0.315221	+	0.949018i	$0.397921\pi$
$444$	36.1926	0.00386852
$445$	0	0
$446$	11297.1	1.19941
$447$	−549.882	−0.0581846
$448$	1852.03	0.195313
$449$	13670.7	1.43689	0.718443	−	0.695586i	$-0.244855\pi$
$449$	13670.7	1.43689	0.718443	+	0.695586i	$0.244855\pi$
$450$	0	0
$451$	763.523	0.0797182
$452$	168.940	0.0175802
$453$	808.256	0.0838304
$454$	−7829.81	−0.809408
$455$	0	0
$456$	22.5230	0.00231301
$457$	16947.2	1.73470	0.867349	−	0.497700i	$-0.165822\pi$
$457$	16947.2	1.73470	0.867349	+	0.497700i	$0.165822\pi$
$458$	1101.98	0.112429
$459$	1625.58	0.165306
$460$	0	0
$461$	−5583.33	−0.564081	−0.282041	−	0.959402i	$-0.591011\pi$
$461$	−5583.33	−0.564081	−0.282041	+	0.959402i	$0.591011\pi$
$462$	−1037.68	−0.104496
$463$	5858.02	0.588003	0.294001	−	0.955805i	$-0.405013\pi$
$463$	5858.02	0.588003	0.294001	+	0.955805i	$0.405013\pi$
$464$	−2597.35	−0.259868
$465$	0	0
$466$	11571.3	1.15028
$467$	13949.8	1.38227	0.691136	−	0.722725i	$-0.257111\pi$
$467$	13949.8	1.38227	0.691136	+	0.722725i	$0.257111\pi$
$468$	1099.73	0.108622
$469$	−3831.07	−0.377191
$470$	0	0
$471$	1121.56	0.109721
$472$	−2335.67	−0.227771
$473$	−11184.1	−1.08720
$474$	−602.306	−0.0583646
$475$	0	0
$476$	−6549.87	−0.630699
$477$	17984.0	1.72627
$478$	9051.43	0.866114
$479$	7154.84	0.682491	0.341245	−	0.939974i	$-0.389151\pi$
$479$	7154.84	0.682491	0.341245	+	0.939974i	$0.389151\pi$
$480$	0	0
$481$	−174.113	−0.0165049
$482$	−1034.58	−0.0977677
$483$	−355.969	−0.0335345
$484$	−828.727	−0.0778293
$485$	0	0
$486$	−2306.39	−0.215267
$487$	14811.9	1.37821	0.689106	−	0.724661i	$-0.258004\pi$
$487$	14811.9	1.37821	0.689106	+	0.724661i	$0.258004\pi$
$488$	−3636.69	−0.337347
$489$	−376.010	−0.0347725
$490$	0	0
$491$	3092.94	0.284282	0.142141	−	0.989846i	$-0.454601\pi$
$491$	3092.94	0.284282	0.142141	+	0.989846i	$0.454601\pi$
$492$	48.7248	0.00446480
$493$	9185.75	0.839159
$494$	−108.352	−0.00986840
$495$	0	0
$496$	−2568.90	−0.232555
$497$	3510.92	0.316874
$498$	865.428	0.0778730
$499$	17250.6	1.54758	0.773790	−	0.633443i	$-0.218359\pi$
$499$	17250.6	1.54758	0.773790	+	0.633443i	$0.218359\pi$
$500$	0	0
$501$	−1638.21	−0.146087
$502$	14078.0	1.25166
$503$	5534.20	0.490572	0.245286	−	0.969451i	$-0.421118\pi$
$503$	5534.20	0.490572	0.245286	+	0.969451i	$0.421118\pi$
$504$	6184.39	0.546576
$505$	0	0
$506$	1542.08	0.135482
$507$	−1118.37	−0.0979658
$508$	−4937.91	−0.431269
$509$	21995.5	1.91539	0.957693	−	0.287791i	$-0.0929208\pi$
$509$	21995.5	1.91539	0.957693	+	0.287791i	$0.0929208\pi$
$510$	0	0
$511$	−5107.04	−0.442118
$512$	−512.000	−0.0441942
$513$	151.225	0.0130151
$514$	−8501.87	−0.729575
$515$	0	0
$516$	−713.721	−0.0608911
$517$	−4370.88	−0.371820
$518$	−979.134	−0.0830515
$519$	−1503.05	−0.127123
$520$	0	0
$521$	150.733	0.0126751	0.00633756	−	0.999980i	$-0.497983\pi$
$521$	150.733	0.0126751	0.00633756	+	0.999980i	$0.497983\pi$
$522$	−8673.18	−0.727232
$523$	21831.2	1.82526	0.912630	−	0.408787i	$-0.134048\pi$
$523$	21831.2	1.82526	0.912630	+	0.408787i	$0.134048\pi$
$524$	−8866.97	−0.739228
$525$	0	0
$526$	7197.67	0.596641
$527$	9085.14	0.750958
$528$	286.869	0.0236447
$529$	529.000	0.0434783
$530$	0	0
$531$	−7799.36	−0.637408
$532$	−609.324	−0.0496570
$533$	−234.402	−0.0190489
$534$	751.348	0.0608877
$535$	0	0
$536$	1059.11	0.0853483
$537$	−1345.92	−0.108158
$538$	10125.0	0.811379
$539$	16574.2	1.32449
$540$	0	0
$541$	5796.22	0.460627	0.230313	−	0.973117i	$-0.426025\pi$
$541$	5796.22	0.460627	0.230313	+	0.973117i	$0.426025\pi$
$542$	−3709.16	−0.293952
$543$	580.297	0.0458617
$544$	1810.73	0.142711
$545$	0	0
$546$	318.568	0.0249697
$547$	−9736.32	−0.761051	−0.380526	−	0.924770i	$-0.624257\pi$
$547$	−9736.32	−0.761051	−0.380526	+	0.924770i	$0.624257\pi$
$548$	4717.92	0.367773
$549$	−12143.8	−0.944053
$550$	0	0
$551$	854.535	0.0660698
$552$	98.4087	0.00758796
$553$	16294.5	1.25300
$554$	17495.8	1.34174
$555$	0	0
$556$	964.895	0.0735983
$557$	−5064.36	−0.385249	−0.192625	−	0.981273i	$-0.561700\pi$
$557$	−5064.36	−0.385249	−0.192625	+	0.981273i	$0.561700\pi$
$558$	−8578.19	−0.650795
$559$	3433.52	0.259790
$560$	0	0
$561$	−1014.54	−0.0763526
$562$	−8093.76	−0.607499
$563$	−3091.61	−0.231431	−0.115716	−	0.993282i	$-0.536916\pi$
$563$	−3091.61	−0.231431	−0.115716	+	0.993282i	$0.536916\pi$
$564$	−278.931	−0.0208247
$565$	0	0
$566$	3197.03	0.237423
$567$	20427.7	1.51302
$568$	−970.605	−0.0717002
$569$	8120.51	0.598294	0.299147	−	0.954207i	$-0.403298\pi$
$569$	8120.51	0.598294	0.299147	+	0.954207i	$0.403298\pi$
$570$	0	0
$571$	−5547.16	−0.406552	−0.203276	−	0.979121i	$-0.565159\pi$
$571$	−5547.16	−0.406552	−0.203276	+	0.979121i	$0.565159\pi$
$572$	−1380.05	−0.100879
$573$	−1113.44	−0.0811777
$574$	−1318.17	−0.0958527
$575$	0	0
$576$	−1709.69	−0.123676
$577$	−23746.0	−1.71327	−0.856636	−	0.515922i	$-0.827450\pi$
$577$	−23746.0	−1.71327	−0.856636	+	0.515922i	$0.827450\pi$
$578$	3422.18	0.246270
$579$	−346.833	−0.0248945
$580$	0	0
$581$	−23412.8	−1.67182
$582$	366.835	0.0261268
$583$	−22568.1	−1.60322
$584$	1411.86	0.100040
$585$	0	0
$586$	−9085.23	−0.640456
$587$	−17023.6	−1.19700	−0.598500	−	0.801123i	$-0.704237\pi$
$587$	−17023.6	−1.19700	−0.598500	+	0.801123i	$0.704237\pi$
$588$	1057.70	0.0741814
$589$	845.176	0.0591254
$590$	0	0
$591$	669.032	0.0465656
$592$	270.685	0.0187923
$593$	−2675.31	−0.185264	−0.0926321	−	0.995700i	$-0.529528\pi$
$593$	−2675.31	−0.185264	−0.0926321	+	0.995700i	$0.529528\pi$
$594$	1926.11	0.133046
$595$	0	0
$596$	−4112.57	−0.282647
$597$	1676.98	0.114965
$598$	−473.419	−0.0323738
$599$	15350.4	1.04708	0.523538	−	0.852002i	$-0.324612\pi$
$599$	15350.4	1.04708	0.523538	+	0.852002i	$0.324612\pi$
$600$	0	0
$601$	−15429.3	−1.04721	−0.523604	−	0.851962i	$-0.675413\pi$
$601$	−15429.3	−1.04721	−0.523604	+	0.851962i	$0.675413\pi$
$602$	19308.6	1.30724
$603$	3536.63	0.238844
$604$	6044.95	0.407228
$605$	0	0
$606$	−1839.47	−0.123306
$607$	6027.74	0.403061	0.201531	−	0.979482i	$-0.435408\pi$
$607$	6027.74	0.403061	0.201531	+	0.979482i	$0.435408\pi$
$608$	168.449	0.0112361
$609$	−2512.43	−0.167174
$610$	0	0
$611$	1341.86	0.0888478
$612$	6046.48	0.399370
$613$	21260.9	1.40085	0.700423	−	0.713728i	$-0.252995\pi$
$613$	21260.9	1.40085	0.700423	+	0.713728i	$0.252995\pi$
$614$	−4732.78	−0.311074
$615$	0	0
$616$	−7760.80	−0.507616
$617$	−20739.7	−1.35324	−0.676621	−	0.736332i	$-0.736556\pi$
$617$	−20739.7	−1.35324	−0.676621	+	0.736332i	$0.736556\pi$
$618$	414.891	0.0270054
$619$	−17994.4	−1.16843	−0.584214	−	0.811600i	$-0.698597\pi$
$619$	−17994.4	−1.16843	−0.584214	+	0.811600i	$0.698597\pi$
$620$	0	0
$621$	660.740	0.0426966
$622$	−1660.32	−0.107030
$623$	−20326.6	−1.30717
$624$	−88.0690	−0.00564997
$625$	0	0
$626$	−14364.1	−0.917098
$627$	−94.3808	−0.00601149
$628$	8388.15	0.532999
$629$	−957.300	−0.0606837
$630$	0	0
$631$	22242.4	1.40326	0.701630	−	0.712542i	$-0.252456\pi$
$631$	22242.4	1.40326	0.701630	+	0.712542i	$0.252456\pi$
$632$	−4504.65	−0.283521
$633$	−451.498	−0.0283498
$634$	3820.13	0.239301
$635$	0	0
$636$	−1440.20	−0.0897920
$637$	−5088.30	−0.316492
$638$	10884.0	0.675394
$639$	−3241.09	−0.200650
$640$	0	0
$641$	12480.7	0.769046	0.384523	−	0.923115i	$-0.374366\pi$
$641$	12480.7	0.769046	0.384523	+	0.923115i	$0.374366\pi$
$642$	33.5771	0.00206415
$643$	3063.19	0.187870	0.0939352	−	0.995578i	$-0.470055\pi$
$643$	3063.19	0.187870	0.0939352	+	0.995578i	$0.470055\pi$
$644$	−2662.30	−0.162902
$645$	0	0
$646$	−595.736	−0.0362832
$647$	−20052.9	−1.21849	−0.609243	−	0.792984i	$-0.708526\pi$
$647$	−20052.9	−1.21849	−0.609243	+	0.792984i	$0.708526\pi$
$648$	−5647.30	−0.342356
$649$	9787.43	0.591973
$650$	0	0
$651$	−2484.92	−0.149603
$652$	−2812.18	−0.168917
$653$	16978.6	1.01749	0.508746	−	0.860917i	$-0.330109\pi$
$653$	16978.6	1.01749	0.508746	+	0.860917i	$0.330109\pi$
$654$	−733.957	−0.0438838
$655$	0	0
$656$	364.413	0.0216889
$657$	4714.54	0.279957
$658$	7546.04	0.447075
$659$	−15608.4	−0.922637	−0.461319	−	0.887235i	$-0.652623\pi$
$659$	−15608.4	−0.922637	−0.461319	+	0.887235i	$0.652623\pi$
$660$	0	0
$661$	3744.26	0.220325	0.110163	−	0.993914i	$-0.464863\pi$
$661$	3744.26	0.220325	0.110163	+	0.993914i	$0.464863\pi$
$662$	−15080.8	−0.885396
$663$	311.464	0.0182447
$664$	6472.55	0.378288
$665$	0	0
$666$	903.882	0.0525897
$667$	3733.69	0.216745
$668$	−12252.2	−0.709656
$669$	−3021.02	−0.174588
$670$	0	0
$671$	15239.3	0.876761
$672$	−495.261	−0.0284302
$673$	−8962.72	−0.513355	−0.256677	−	0.966497i	$-0.582628\pi$
$673$	−8962.72	−0.513355	−0.256677	+	0.966497i	$0.582628\pi$
$674$	−20138.8	−1.15092
$675$	0	0
$676$	−8364.32	−0.475895
$677$	−30864.5	−1.75217	−0.876086	−	0.482155i	$-0.839854\pi$
$677$	−30864.5	−1.75217	−0.876086	+	0.482155i	$0.839854\pi$
$678$	−45.1771	−0.00255902
$679$	−9924.14	−0.560904
$680$	0	0
$681$	2093.81	0.117819
$682$	10764.8	0.604406
$683$	−2135.02	−0.119611	−0.0598055	−	0.998210i	$-0.519048\pi$
$683$	−2135.02	−0.119611	−0.0598055	+	0.998210i	$0.519048\pi$
$684$	562.494	0.0314437
$685$	0	0
$686$	−8762.86	−0.487708
$687$	−294.687	−0.0163654
$688$	−5337.93	−0.295794
$689$	6928.44	0.383095
$690$	0	0
$691$	−14271.3	−0.785680	−0.392840	−	0.919607i	$-0.628507\pi$
$691$	−14271.3	−0.785680	−0.392840	+	0.919607i	$0.628507\pi$
$692$	−11241.3	−0.617531
$693$	−25915.2	−1.42054
$694$	12062.1	0.659758
$695$	0	0
$696$	694.570	0.0378270
$697$	−1288.78	−0.0700373
$698$	22703.9	1.23117
$699$	−3094.35	−0.167438
$700$	0	0
$701$	−13473.5	−0.725945	−0.362973	−	0.931800i	$-0.618238\pi$
$701$	−13473.5	−0.725945	−0.362973	+	0.931800i	$0.618238\pi$
$702$	−591.317	−0.0317918
$703$	−89.0560	−0.00477782
$704$	2145.50	0.114860
$705$	0	0
$706$	−9729.61	−0.518667
$707$	49764.1	2.64720
$708$	624.592	0.0331548
$709$	−8117.53	−0.429986	−0.214993	−	0.976616i	$-0.568973\pi$
$709$	−8117.53	−0.429986	−0.214993	+	0.976616i	$0.568973\pi$
$710$	0	0
$711$	−15042.1	−0.793424
$712$	5619.34	0.295778
$713$	3692.80	0.193964
$714$	1751.53	0.0918061
$715$	0	0
$716$	−10066.1	−0.525403
$717$	−2420.49	−0.126074
$718$	11619.0	0.603923
$719$	−34187.5	−1.77327	−0.886634	−	0.462471i	$-0.846963\pi$
$719$	−34187.5	−1.77327	−0.886634	+	0.462471i	$0.846963\pi$
$720$	0	0
$721$	−11224.2	−0.579767
$722$	13662.6	0.704250
$723$	276.663	0.0142313
$724$	4340.04	0.222785
$725$	0	0
$726$	221.614	0.0113290
$727$	−26638.2	−1.35895	−0.679476	−	0.733698i	$-0.737793\pi$
$727$	−26638.2	−1.35895	−0.679476	+	0.733698i	$0.737793\pi$
$728$	2382.57	0.121297
$729$	−18442.9	−0.936995
$730$	0	0
$731$	18878.0	0.955171
$732$	972.507	0.0491050
$733$	26875.8	1.35427	0.677135	−	0.735858i	$-0.263221\pi$
$733$	26875.8	1.35427	0.677135	+	0.735858i	$0.263221\pi$
$734$	13968.8	0.702447
$735$	0	0
$736$	736.000	0.0368605
$737$	−4438.13	−0.221819
$738$	1216.86	0.0606957
$739$	5873.86	0.292386	0.146193	−	0.989256i	$-0.453298\pi$
$739$	5873.86	0.292386	0.146193	+	0.989256i	$0.453298\pi$
$740$	0	0
$741$	28.9750	0.00143647
$742$	38962.4	1.92770
$743$	16863.9	0.832673	0.416337	−	0.909210i	$-0.363314\pi$
$743$	16863.9	0.832673	0.416337	+	0.909210i	$0.363314\pi$
$744$	686.963	0.0338512
$745$	0	0
$746$	17103.1	0.839394
$747$	21613.4	1.05863
$748$	−7587.74	−0.370903
$749$	−908.376	−0.0443142
$750$	0	0
$751$	24885.3	1.20916	0.604578	−	0.796546i	$-0.293342\pi$
$751$	24885.3	1.20916	0.604578	+	0.796546i	$0.293342\pi$
$752$	−2086.13	−0.101161
$753$	−3764.67	−0.182194
$754$	−3341.39	−0.161388
$755$	0	0
$756$	−3325.30	−0.159974
$757$	14692.8	0.705442	0.352721	−	0.935728i	$-0.385256\pi$
$757$	14692.8	0.705442	0.352721	+	0.935728i	$0.385256\pi$
$758$	−19866.8	−0.951971
$759$	−412.374	−0.0197210
$760$	0	0
$761$	−7531.01	−0.358737	−0.179369	−	0.983782i	$-0.557405\pi$
$761$	−7531.01	−0.358737	−0.179369	+	0.983782i	$0.557405\pi$
$762$	1320.47	0.0627765
$763$	19856.1	0.942121
$764$	−8327.46	−0.394342
$765$	0	0
$766$	19497.0	0.919656
$767$	−3004.75	−0.141454
$768$	136.916	0.00643301
$769$	20747.5	0.972916	0.486458	−	0.873704i	$-0.338289\pi$
$769$	20747.5	0.972916	0.486458	+	0.873704i	$0.338289\pi$
$770$	0	0
$771$	2273.53	0.106199
$772$	−2593.97	−0.120931
$773$	940.128	0.0437439	0.0218720	−	0.999761i	$-0.493037\pi$
$773$	940.128	0.0437439	0.0218720	+	0.999761i	$0.493037\pi$
$774$	−17824.6	−0.827769
$775$	0	0
$776$	2743.56	0.126918
$777$	261.835	0.0120892
$778$	21034.1	0.969290
$779$	−119.893	−0.00551426
$780$	0	0
$781$	4067.25	0.186348
$782$	−2602.93	−0.119029
$783$	4663.51	0.212849
$784$	7910.52	0.360355
$785$	0	0
$786$	2371.16	0.107604
$787$	6072.14	0.275030	0.137515	−	0.990500i	$-0.456088\pi$
$787$	6072.14	0.275030	0.137515	+	0.990500i	$0.456088\pi$
$788$	5003.70	0.226205
$789$	−1924.76	−0.0868484
$790$	0	0
$791$	1222.20	0.0549384
$792$	7164.34	0.321432
$793$	−4678.47	−0.209505
$794$	8235.88	0.368111
$795$	0	0
$796$	12542.1	0.558473
$797$	−5424.76	−0.241098	−0.120549	−	0.992707i	$-0.538465\pi$
$797$	−5424.76	−0.241098	−0.120549	+	0.992707i	$0.538465\pi$
$798$	162.942	0.00722819
$799$	7377.77	0.326667
$800$	0	0
$801$	18764.4	0.827723
$802$	14748.6	0.649365
$803$	−5916.28	−0.260001
$804$	−283.223	−0.0124235
$805$	0	0
$806$	−3304.80	−0.144425
$807$	−2707.59	−0.118106
$808$	−13757.4	−0.598991
$809$	−13646.3	−0.593051	−0.296526	−	0.955025i	$-0.595828\pi$
$809$	−13646.3	−0.593051	−0.296526	+	0.955025i	$0.595828\pi$
$810$	0	0
$811$	−10015.4	−0.433647	−0.216823	−	0.976211i	$-0.569570\pi$
$811$	−10015.4	−0.433647	−0.216823	+	0.976211i	$0.569570\pi$
$812$	−18790.5	−0.812091
$813$	991.886	0.0427884
$814$	−1134.28	−0.0488410
$815$	0	0
$816$	−484.217	−0.0207733
$817$	1756.19	0.0752037
$818$	−25883.1	−1.10633
$819$	7955.99	0.339444
$820$	0	0
$821$	6525.48	0.277394	0.138697	−	0.990335i	$-0.455709\pi$
$821$	6525.48	0.277394	0.138697	+	0.990335i	$0.455709\pi$
$822$	−1261.64	−0.0535338
$823$	−42874.0	−1.81591	−0.907954	−	0.419069i	$-0.862356\pi$
$823$	−42874.0	−1.81591	−0.907954	+	0.419069i	$0.862356\pi$
$824$	3102.97	0.131186
$825$	0	0
$826$	−16897.4	−0.711785
$827$	21654.4	0.910517	0.455259	−	0.890359i	$-0.349547\pi$
$827$	21654.4	0.910517	0.455259	+	0.890359i	$0.349547\pi$
$828$	2457.68	0.103153
$829$	−22392.7	−0.938153	−0.469077	−	0.883158i	$-0.655413\pi$
$829$	−22392.7	−0.938153	−0.469077	+	0.883158i	$0.655413\pi$
$830$	0	0
$831$	−4678.64	−0.195307
$832$	−658.669	−0.0274462
$833$	−27976.2	−1.16365
$834$	−258.027	−0.0107131
$835$	0	0
$836$	−705.875	−0.0292024
$837$	4612.44	0.190477
$838$	−11266.9	−0.464450
$839$	21335.4	0.877926	0.438963	−	0.898505i	$-0.355346\pi$
$839$	21335.4	0.877926	0.438963	+	0.898505i	$0.355346\pi$
$840$	0	0
$841$	1963.43	0.0805049
$842$	8410.97	0.344253
$843$	2164.39	0.0884290
$844$	−3376.76	−0.137716
$845$	0	0
$846$	−6966.09	−0.283096
$847$	−5995.42	−0.243217
$848$	−10771.3	−0.436188
$849$	−854.934	−0.0345598
$850$	0	0
$851$	−389.109	−0.0156739
$852$	259.554	0.0104368
$853$	11610.2	0.466034	0.233017	−	0.972473i	$-0.425140\pi$
$853$	11610.2	0.466034	0.233017	+	0.972473i	$0.425140\pi$
$854$	−26309.7	−1.05421
$855$	0	0
$856$	251.123	0.0100271
$857$	−27905.6	−1.11230	−0.556148	−	0.831084i	$-0.687721\pi$
$857$	−27905.6	−1.11230	−0.556148	+	0.831084i	$0.687721\pi$
$858$	369.047	0.0146842
$859$	4529.52	0.179913	0.0899565	−	0.995946i	$-0.471327\pi$
$859$	4529.52	0.179913	0.0899565	+	0.995946i	$0.471327\pi$
$860$	0	0
$861$	352.499	0.0139525
$862$	−30151.5	−1.19137
$863$	10511.6	0.414622	0.207311	−	0.978275i	$-0.433529\pi$
$863$	10511.6	0.414622	0.207311	+	0.978275i	$0.433529\pi$
$864$	919.291	0.0361978
$865$	0	0
$866$	6196.11	0.243132
$867$	−915.143	−0.0358476
$868$	−18584.7	−0.726735
$869$	18876.4	0.736868
$870$	0	0
$871$	1362.51	0.0530044
$872$	−5489.27	−0.213177
$873$	9161.42	0.355174
$874$	−242.146	−0.00937153
$875$	0	0
$876$	−377.552	−0.0145620
$877$	−20949.8	−0.806643	−0.403321	−	0.915058i	$-0.632144\pi$
$877$	−20949.8	−0.806643	−0.403321	+	0.915058i	$0.632144\pi$
$878$	−13735.5	−0.527961
$879$	2429.53	0.0932263
$880$	0	0
$881$	39907.1	1.52611	0.763057	−	0.646332i	$-0.223698\pi$
$881$	39907.1	1.52611	0.763057	+	0.646332i	$0.223698\pi$
$882$	26415.2	1.00844
$883$	16565.2	0.631329	0.315665	−	0.948871i	$-0.397773\pi$
$883$	16565.2	0.631329	0.315665	+	0.948871i	$0.397773\pi$
$884$	2329.44	0.0886285
$885$	0	0
$886$	−11756.6	−0.445790
$887$	−8217.44	−0.311065	−0.155532	−	0.987831i	$-0.549709\pi$
$887$	−8217.44	−0.311065	−0.155532	+	0.987831i	$0.549709\pi$
$888$	−72.3851	−0.00273546
$889$	−35723.3	−1.34772
$890$	0	0
$891$	23664.6	0.889779
$892$	−22594.3	−0.848108
$893$	686.342	0.0257195
$894$	1099.76	0.0411427
$895$	0	0
$896$	−3704.06	−0.138107
$897$	126.599	0.00471240
$898$	−27341.5	−1.01603
$899$	26063.8	0.966937
$900$	0	0
$901$	38093.6	1.40853
$902$	−1527.05	−0.0563692
$903$	−5163.41	−0.190285
$904$	−337.880	−0.0124311
$905$	0	0
$906$	−1616.51	−0.0592770
$907$	−25129.0	−0.919952	−0.459976	−	0.887931i	$-0.652142\pi$
$907$	−25129.0	−0.919952	−0.459976	+	0.887931i	$0.652142\pi$
$908$	15659.6	0.572338
$909$	−45939.5	−1.67625
$910$	0	0
$911$	−9275.75	−0.337343	−0.168671	−	0.985672i	$-0.553948\pi$
$911$	−9275.75	−0.337343	−0.168671	+	0.985672i	$0.553948\pi$
$912$	−45.0459	−0.00163555
$913$	−27122.7	−0.983166
$914$	−33894.4	−1.22662
$915$	0	0
$916$	−2203.97	−0.0794990
$917$	−64148.1	−2.31010
$918$	−3251.15	−0.116889
$919$	−47822.1	−1.71655	−0.858273	−	0.513194i	$-0.828462\pi$
$919$	−47822.1	−1.71655	−0.858273	+	0.513194i	$0.828462\pi$
$920$	0	0
$921$	1265.62	0.0452807
$922$	11166.7	0.398866
$923$	−1248.65	−0.0445285
$924$	2075.35	0.0738898
$925$	0	0
$926$	−11716.0	−0.415781
$927$	10361.6	0.367119
$928$	5194.70	0.183755
$929$	−447.878	−0.0158175	−0.00790873	−	0.999969i	$-0.502517\pi$
$929$	−447.878	−0.0158175	−0.00790873	+	0.999969i	$0.502517\pi$
$930$	0	0
$931$	−2602.58	−0.0916179
$932$	−23142.7	−0.813373
$933$	443.994	0.0155795
$934$	−27899.6	−0.977414
$935$	0	0
$936$	−2199.46	−0.0768072
$937$	3402.08	0.118614	0.0593069	−	0.998240i	$-0.481111\pi$
$937$	3402.08	0.118614	0.0593069	+	0.998240i	$0.481111\pi$
$938$	7662.15	0.266714
$939$	3841.17	0.133495
$940$	0	0
$941$	14084.0	0.487912	0.243956	−	0.969786i	$-0.421555\pi$
$941$	14084.0	0.487912	0.243956	+	0.969786i	$0.421555\pi$
$942$	−2243.12	−0.0775846
$943$	−523.844	−0.0180898
$944$	4671.33	0.161058
$945$	0	0
$946$	22368.2	0.768766
$947$	10020.6	0.343850	0.171925	−	0.985110i	$-0.445001\pi$
$947$	10020.6	0.343850	0.171925	+	0.985110i	$0.445001\pi$
$948$	1204.61	0.0412700
$949$	1816.30	0.0621283
$950$	0	0
$951$	−1021.56	−0.0348331
$952$	13099.7	0.445972
$953$	45482.6	1.54599	0.772993	−	0.634415i	$-0.218759\pi$
$953$	45482.6	1.54599	0.772993	+	0.634415i	$0.218759\pi$
$954$	−35968.0	−1.22066
$955$	0	0
$956$	−18102.9	−0.612435
$957$	−2910.54	−0.0983120
$958$	−14309.7	−0.482594
$959$	34131.8	1.14929
$960$	0	0
$961$	−4012.67	−0.134694
$962$	348.226	0.0116707
$963$	838.562	0.0280605
$964$	2069.17	0.0691322
$965$	0	0
$966$	711.938	0.0237124
$967$	−50955.7	−1.69455	−0.847273	−	0.531158i	$-0.821757\pi$
$967$	−50955.7	−1.69455	−0.847273	+	0.531158i	$0.821757\pi$
$968$	1657.45	0.0550336
$969$	159.309	0.00528146
$970$	0	0
$971$	−38192.3	−1.26226	−0.631128	−	0.775679i	$-0.717408\pi$
$971$	−38192.3	−1.26226	−0.631128	+	0.775679i	$0.717408\pi$
$972$	4612.78	0.152217
$973$	6980.53	0.229995
$974$	−29623.7	−0.974543
$975$	0	0
$976$	7273.39	0.238540
$977$	28889.6	0.946018	0.473009	−	0.881058i	$-0.343168\pi$
$977$	28889.6	0.946018	0.473009	+	0.881058i	$0.343168\pi$
$978$	752.020	0.0245879
$979$	−23547.4	−0.768722
$980$	0	0
$981$	−18330.0	−0.596568
$982$	−6185.88	−0.201018
$983$	−51725.2	−1.67831	−0.839154	−	0.543894i	$-0.816949\pi$
$983$	−51725.2	−1.67831	−0.839154	+	0.543894i	$0.816949\pi$
$984$	−97.4495	−0.00315709
$985$	0	0
$986$	−18371.5	−0.593375
$987$	−2017.93	−0.0650773
$988$	216.704	0.00697801
$989$	7673.27	0.246710
$990$	0	0
$991$	24851.8	0.796613	0.398307	−	0.917252i	$-0.369598\pi$
$991$	24851.8	0.796613	0.398307	+	0.917252i	$0.369598\pi$
$992$	5137.80	0.164441
$993$	4032.83	0.128880
$994$	−7021.84	−0.224064
$995$	0	0
$996$	−1730.86	−0.0550645
$997$	−2537.99	−0.0806209	−0.0403104	−	0.999187i	$-0.512835\pi$
$997$	−2537.99	−0.0806209	−0.0403104	+	0.999187i	$0.512835\pi$
$998$	−34501.2	−1.09430
$999$	−486.011	−0.0153921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.q.1.3	✓	5
5.2	odd	4		1150.4.b.r.599.3		10
5.3	odd	4		1150.4.b.r.599.8		10
5.4	even	2		1150.4.a.v.1.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.q.1.3	✓	5	1.1	even	1	trivial
1150.4.a.v.1.3	yes	5	5.4	even	2
1150.4.b.r.599.3		10	5.2	odd	4
1150.4.b.r.599.8		10	5.3	odd	4

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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +0.534830 q^{3} +4.00000 q^{4} -1.06966 q^{6} +28.9380 q^{7} -8.00000 q^{8} -26.7140 q^{9} +33.5234 q^{11} +2.13932 q^{12} -10.2917 q^{13} -57.8760 q^{14} +16.0000 q^{16} -56.5854 q^{17} +53.4279 q^{18} -5.26405 q^{19} +15.4769 q^{21} -67.0468 q^{22} -23.0000 q^{23} -4.27864 q^{24} +20.5834 q^{26} -28.7278 q^{27} +115.752 q^{28} -162.334 q^{29} -160.556 q^{31} -32.0000 q^{32} +17.9293 q^{33} +113.171 q^{34} -106.856 q^{36} +16.9178 q^{37} +10.5281 q^{38} -5.50431 q^{39} +22.7758 q^{41} -30.9538 q^{42} -333.620 q^{43} +134.094 q^{44} +46.0000 q^{46} -130.383 q^{47} +8.55728 q^{48} +494.408 q^{49} -30.2636 q^{51} -41.1668 q^{52} -673.206 q^{53} +57.4557 q^{54} -231.504 q^{56} -2.81537 q^{57} +324.669 q^{58} +291.958 q^{59} +454.587 q^{61} +321.113 q^{62} -773.048 q^{63} +64.0000 q^{64} -35.8586 q^{66} -132.389 q^{67} -226.342 q^{68} -12.3011 q^{69} +121.326 q^{71} +213.712 q^{72} -176.482 q^{73} -33.8356 q^{74} -21.0562 q^{76} +970.100 q^{77} +11.0086 q^{78} +563.082 q^{79} +705.912 q^{81} -45.5516 q^{82} -809.068 q^{83} +61.9076 q^{84} +667.241 q^{86} -86.8213 q^{87} -268.187 q^{88} -702.417 q^{89} -297.821 q^{91} -92.0000 q^{92} -85.8703 q^{93} +260.766 q^{94} -17.1146 q^{96} -342.945 q^{97} -988.815 q^{98} -895.543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} - 5 q^{3} + 20 q^{4} + 10 q^{6} + 3 q^{7} - 40 q^{8} + 84 q^{9} - 26 q^{11} - 20 q^{12} + 61 q^{13} - 6 q^{14} + 80 q^{16} - 231 q^{17} - 168 q^{18} + 74 q^{19} - 88 q^{21} + 52 q^{22} - 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 - 5 * q^3 + 20 * q^4 + 10 * q^6 + 3 * q^7 - 40 * q^8 + 84 * q^9 - 26 * q^11 - 20 * q^12 + 61 * q^13 - 6 * q^14 + 80 * q^16 - 231 * q^17 - 168 * q^18 + 74 * q^19 - 88 * q^21 + 52 * q^22 - 115 * q^23 + 40 * q^24 - 122 * q^26 - 368 * q^27 + 12 * q^28 + 37 * q^29 + 220 * q^31 - 160 * q^32 - 391 * q^33 + 462 * q^34 + 336 * q^36 - 220 * q^37 - 148 * q^38 + 195 * q^39 + 112 * q^41 + 176 * q^42 - 97 * q^43 - 104 * q^44 + 230 * q^46 - 370 * q^47 - 80 * q^48 + 182 * q^49 + 563 * q^51 + 244 * q^52 - 1224 * q^53 + 736 * q^54 - 24 * q^56 + 1009 * q^57 - 74 * q^58 - 181 * q^59 + 1574 * q^61 - 440 * q^62 - 1829 * q^63 + 320 * q^64 + 782 * q^66 + 435 * q^67 - 924 * q^68 + 115 * q^69 - 30 * q^71 - 672 * q^72 - 2180 * q^73 + 440 * q^74 + 296 * q^76 + 1401 * q^77 - 390 * q^78 + 609 * q^79 + 1453 * q^81 - 224 * q^82 - 2018 * q^83 - 352 * q^84 + 194 * q^86 + 1485 * q^87 + 208 * q^88 - 1663 * q^89 + 361 * q^91 - 460 * q^92 - 2741 * q^93 + 740 * q^94 + 160 * q^96 + 1274 * q^97 - 364 * q^98 + 2397 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more