LMFDB - Embedded newform 675.5.c.p.26.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,5,Mod(26,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("675.26");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	675.c (of order $2$, degree $1$, 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:	$69.7747250816$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 73x^{4} + 1456x^{2} + 4500 $ x^6 + 73x^4 + 1456x^2 + 4500
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.4
Root			$1.94025i$ of defining polynomial
Character	$\chi$	$=$	675.26
Dual form			675.5.c.p.26.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.94025i q^{2} +12.2354 q^{4} +50.3522 q^{7} +54.7837i q^{8} -202.956i q^{11} -183.564 q^{13} +97.6958i q^{14} +89.4731 q^{16} +174.739i q^{17} +306.208 q^{19} +393.785 q^{22} +121.013i q^{23} -356.160i q^{26} +616.082 q^{28} -1431.82i q^{29} +1115.55 q^{31} +1050.14i q^{32} -339.038 q^{34} +1645.59 q^{37} +594.120i q^{38} -510.825i q^{41} +544.361 q^{43} -2483.26i q^{44} -234.794 q^{46} -4157.23i q^{47} +134.346 q^{49} -2245.99 q^{52} +1814.46i q^{53} +2758.48i q^{56} +2778.09 q^{58} +4837.65i q^{59} -3605.33 q^{61} +2164.44i q^{62} -605.961 q^{64} -1367.51 q^{67} +2138.01i q^{68} -2401.14i q^{71} -2473.24 q^{73} +3192.85i q^{74} +3746.59 q^{76} -10219.3i q^{77} +11646.0 q^{79} +991.127 q^{82} -10062.9i q^{83} +1056.20i q^{86} +11118.7 q^{88} -8373.34i q^{89} -9242.87 q^{91} +1480.64i q^{92} +8066.04 q^{94} -1573.80 q^{97} +260.665i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 50 q^{4} - 4 q^{7} - 566 q^{13} - 638 q^{16} + 1124 q^{19} - 1402 q^{22} + 3584 q^{28} - 2582 q^{31} + 1558 q^{34} - 2040 q^{37} - 1922 q^{43} - 5862 q^{46} + 6782 q^{49} + 834 q^{52} + 7234 q^{58}+ \cdots + 27792 q^{97}+O(q^{100}) $ 6 * q - 50 * q^4 - 4 * q^7 - 566 * q^13 - 638 * q^16 + 1124 * q^19 - 1402 * q^22 + 3584 * q^28 - 2582 * q^31 + 1558 * q^34 - 2040 * q^37 - 1922 * q^43 - 5862 * q^46 + 6782 * q^49 + 834 * q^52 + 7234 * q^58 - 11656 * q^61 + 19734 * q^64 - 1288 * q^67 + 25828 * q^73 - 2520 * q^76 + 25690 * q^79 + 11692 * q^82 + 32994 * q^88 + 40676 * q^91 + 35266 * q^94 + 27792 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.94025i	0.485062i	0.970144	+	0.242531i	$0.0779776\pi$
$2$	1.94025i	0.485062i	−0.970144	+	0.242531i	$0.922022\pi$
$3$	0	0
$3$	0	0
$4$	12.2354	0.764715
$5$	0	0
$5$	0	0
$6$	0	0
$7$	50.3522	1.02760	0.513798	−	0.857911i	$-0.328238\pi$
$7$	50.3522	1.02760	0.513798	+	0.857911i	$0.328238\pi$
$8$	54.7837i	0.855996i
$9$	0	0
$10$	0	0
$11$	− 202.956i	− 1.67732i	−0.544653	−	0.838662i	$-0.683338\pi$
$11$	− 202.956i	− 1.67732i	0.544653	−	0.838662i	$-0.316662\pi$
$12$	0	0
$13$	−183.564	−1.08618	−0.543090	−	0.839675i	$-0.682746\pi$
$13$	−183.564	−1.08618	−0.543090	+	0.839675i	$0.682746\pi$
$14$	97.6958i	0.498448i
$15$	0	0
$16$	89.4731	0.349504
$17$	174.739i	0.604635i	0.953207	+	0.302317i	$0.0977603\pi$
$17$	174.739i	0.604635i	−0.953207	+	0.302317i	$0.902240\pi$
$18$	0	0
$19$	306.208	0.848223	0.424111	−	0.905610i	$-0.360587\pi$
$19$	306.208	0.848223	0.424111	+	0.905610i	$0.360587\pi$
$20$	0	0
$21$	0	0
$22$	393.785	0.813606
$23$	121.013i	0.228757i	0.993437	+	0.114379i	$0.0364877\pi$
$23$	121.013i	0.228757i	−0.993437	+	0.114379i	$0.963512\pi$
$24$	0	0
$25$	0	0
$26$	− 356.160i	− 0.526864i
$27$	0	0
$28$	616.082	0.785818
$29$	− 1431.82i	− 1.70252i	−0.524743	−	0.851261i	$-0.675838\pi$
$29$	− 1431.82i	− 1.70252i	0.524743	−	0.851261i	$-0.324162\pi$
$30$	0	0
$31$	1115.55	1.16082	0.580411	−	0.814324i	$-0.302892\pi$
$31$	1115.55	1.16082	0.580411	+	0.814324i	$0.302892\pi$
$32$	1050.14i	1.02553i
$33$	0	0
$34$	−339.038	−0.293285
$35$	0	0
$36$	0	0
$37$	1645.59	1.20204	0.601019	−	0.799235i	$-0.294762\pi$
$37$	1645.59	1.20204	0.601019	+	0.799235i	$0.294762\pi$
$38$	594.120i	0.411440i
$39$	0	0
$40$	0	0
$41$	− 510.825i	− 0.303882i	−0.988390	−	0.151941i	$-0.951448\pi$
$41$	− 510.825i	− 0.303882i	0.988390	−	0.151941i	$-0.0485523\pi$
$42$	0	0
$43$	544.361	0.294408	0.147204	−	0.989106i	$-0.452973\pi$
$43$	544.361	0.294408	0.147204	+	0.989106i	$0.452973\pi$
$44$	− 2483.26i	− 1.28267i
$45$	0	0
$46$	−234.794	−0.110961
$47$	− 4157.23i	− 1.88195i	−0.338478	−	0.940974i	$-0.609912\pi$
$47$	− 4157.23i	− 1.88195i	0.338478	−	0.940974i	$-0.390088\pi$
$48$	0	0
$49$	134.346	0.0559544
$50$	0	0
$51$	0	0
$52$	−2245.99	−0.830618
$53$	1814.46i	0.645946i	0.946408	+	0.322973i	$0.104682\pi$
$53$	1814.46i	0.645946i	−0.946408	+	0.322973i	$0.895318\pi$
$54$	0	0
$55$	0	0
$56$	2758.48i	0.879618i
$57$	0	0
$58$	2778.09	0.825828
$59$	4837.65i	1.38973i	0.719140	+	0.694866i	$0.244536\pi$
$59$	4837.65i	1.38973i	−0.719140	+	0.694866i	$0.755464\pi$
$60$	0	0
$61$	−3605.33	−0.968915	−0.484457	−	0.874815i	$-0.660983\pi$
$61$	−3605.33	−0.968915	−0.484457	+	0.874815i	$0.660983\pi$
$62$	2164.44i	0.563070i
$63$	0	0
$64$	−605.961	−0.147940
$65$	0	0
$66$	0	0
$67$	−1367.51	−0.304635	−0.152317	−	0.988332i	$-0.548674\pi$
$67$	−1367.51	−0.304635	−0.152317	+	0.988332i	$0.548674\pi$
$68$	2138.01i	0.462373i
$69$	0	0
$70$	0	0
$71$	− 2401.14i	− 0.476322i	−0.971226	−	0.238161i	$-0.923455\pi$
$71$	− 2401.14i	− 0.476322i	0.971226	−	0.238161i	$-0.0765446\pi$
$72$	0	0
$73$	−2473.24	−0.464109	−0.232055	−	0.972703i	$-0.574545\pi$
$73$	−2473.24	−0.464109	−0.232055	+	0.972703i	$0.574545\pi$
$74$	3192.85i	0.583062i
$75$	0	0
$76$	3746.59	0.648649
$77$	− 10219.3i	− 1.72361i
$78$	0	0
$79$	11646.0	1.86604	0.933022	−	0.359820i	$-0.117162\pi$
$79$	11646.0	1.86604	0.933022	+	0.359820i	$0.117162\pi$
$80$	0	0
$81$	0	0
$82$	991.127	0.147401
$83$	− 10062.9i	− 1.46071i	−0.683066	−	0.730357i	$-0.739354\pi$
$83$	− 10062.9i	− 1.46071i	0.683066	−	0.730357i	$-0.260646\pi$
$84$	0	0
$85$	0	0
$86$	1056.20i	0.142806i
$87$	0	0
$88$	11118.7	1.43578
$89$	− 8373.34i	− 1.05711i	−0.848900	−	0.528553i	$-0.822735\pi$
$89$	− 8373.34i	− 1.05711i	0.848900	−	0.528553i	$-0.177265\pi$
$90$	0	0
$91$	−9242.87	−1.11615
$92$	1480.64i	0.174934i
$93$	0	0
$94$	8066.04	0.912862
$95$	0	0
$96$	0	0
$97$	−1573.80	−0.167266	−0.0836328	−	0.996497i	$-0.526652\pi$
$97$	−1573.80	−0.167266	−0.0836328	+	0.996497i	$0.526652\pi$
$98$	260.665i	0.0271413i
$99$	0	0
$100$	0	0
$101$	11677.7i	1.14476i	0.819988	+	0.572380i	$0.193980\pi$
$101$	11677.7i	1.14476i	−0.819988	+	0.572380i	$0.806020\pi$
$102$	0	0
$103$	13464.4	1.26915	0.634576	−	0.772861i	$-0.281175\pi$
$103$	13464.4	1.26915	0.634576	+	0.772861i	$0.281175\pi$
$104$	− 10056.3i	− 0.929765i
$105$	0	0
$106$	−3520.50	−0.313324
$107$	− 4631.07i	− 0.404495i	−0.979334	−	0.202248i	$-0.935175\pi$
$107$	− 4631.07i	− 0.404495i	0.979334	−	0.202248i	$-0.0648246\pi$
$108$	0	0
$109$	18127.8	1.52578	0.762890	−	0.646528i	$-0.223780\pi$
$109$	18127.8	1.52578	0.762890	+	0.646528i	$0.223780\pi$
$110$	0	0
$111$	0	0
$112$	4505.17	0.359149
$113$	− 12304.8i	− 0.963647i	−0.876268	−	0.481823i	$-0.839975\pi$
$113$	− 12304.8i	− 0.963647i	0.876268	−	0.481823i	$-0.160025\pi$
$114$	0	0
$115$	0	0
$116$	− 17519.0i	− 1.30194i
$117$	0	0
$118$	−9386.25	−0.674106
$119$	8798.52i	0.621321i
$120$	0	0
$121$	−26550.2	−1.81341
$122$	− 6995.23i	− 0.469984i
$123$	0	0
$124$	13649.2	0.887697
$125$	0	0
$126$	0	0
$127$	9955.91	0.617267	0.308634	−	0.951181i	$-0.400128\pi$
$127$	9955.91	0.617267	0.308634	+	0.951181i	$0.400128\pi$
$128$	15626.5i	0.953767i
$129$	0	0
$130$	0	0
$131$	155.782i	0.00907767i	0.999990	+	0.00453884i	$0.00144476\pi$
$131$	155.782i	0.00907767i	−0.999990	+	0.00453884i	$0.998555\pi$
$132$	0	0
$133$	15418.3	0.871630
$134$	− 2653.30i	− 0.147767i
$135$	0	0
$136$	−9572.88	−0.517565
$137$	23957.9i	1.27646i	0.769844	+	0.638232i	$0.220334\pi$
$137$	23957.9i	1.27646i	−0.769844	+	0.638232i	$0.779666\pi$
$138$	0	0
$139$	8399.21	0.434719	0.217360	−	0.976092i	$-0.430256\pi$
$139$	8399.21	0.434719	0.217360	+	0.976092i	$0.430256\pi$
$140$	0	0
$141$	0	0
$142$	4658.81	0.231046
$143$	37255.5i	1.82187i
$144$	0	0
$145$	0	0
$146$	− 4798.69i	− 0.225122i
$147$	0	0
$148$	20134.5	0.919216
$149$	− 388.709i	− 0.0175086i	−0.999962	−	0.00875432i	$-0.997213\pi$
$149$	− 388.709i	− 0.0175086i	0.999962	−	0.00875432i	$-0.00278662\pi$
$150$	0	0
$151$	27587.9	1.20994	0.604972	−	0.796247i	$-0.293184\pi$
$151$	27587.9	1.20994	0.604972	+	0.796247i	$0.293184\pi$
$152$	16775.2i	0.726075i
$153$	0	0
$154$	19828.0	0.836058
$155$	0	0
$156$	0	0
$157$	−34802.8	−1.41194	−0.705968	−	0.708244i	$-0.749488\pi$
$157$	−34802.8	−1.41194	−0.705968	+	0.708244i	$0.749488\pi$
$158$	22596.1i	0.905146i
$159$	0	0
$160$	0	0
$161$	6093.25i	0.235070i
$162$	0	0
$163$	37147.8	1.39816	0.699081	−	0.715043i	$-0.253593\pi$
$163$	37147.8	1.39816	0.699081	+	0.715043i	$0.253593\pi$
$164$	− 6250.17i	− 0.232383i
$165$	0	0
$166$	19524.4	0.708536
$167$	17254.1i	0.618672i	0.950953	+	0.309336i	$0.100107\pi$
$167$	17254.1i	0.618672i	−0.950953	+	0.309336i	$0.899893\pi$
$168$	0	0
$169$	5134.86	0.179786
$170$	0	0
$171$	0	0
$172$	6660.50	0.225139
$173$	− 16749.1i	− 0.559629i	−0.960054	−	0.279815i	$-0.909727\pi$
$173$	− 16749.1i	− 0.559629i	0.960054	−	0.279815i	$-0.0902730\pi$
$174$	0	0
$175$	0	0
$176$	− 18159.1i	− 0.586232i
$177$	0	0
$178$	16246.3	0.512762
$179$	37065.5i	1.15681i	0.815748	+	0.578407i	$0.196325\pi$
$179$	37065.5i	1.15681i	−0.815748	+	0.578407i	$0.803675\pi$
$180$	0	0
$181$	−25651.5	−0.782990	−0.391495	−	0.920180i	$-0.628042\pi$
$181$	−25651.5	−0.782990	−0.391495	+	0.920180i	$0.628042\pi$
$182$	− 17933.5i	− 0.541404i
$183$	0	0
$184$	−6629.52	−0.195815
$185$	0	0
$186$	0	0
$187$	35464.5	1.01417
$188$	− 50865.5i	− 1.43915i
$189$	0	0
$190$	0	0
$191$	11801.5i	0.323498i	0.986832	+	0.161749i	$0.0517135\pi$
$191$	11801.5i	0.323498i	−0.986832	+	0.161749i	$0.948286\pi$
$192$	0	0
$193$	30954.0	0.831002	0.415501	−	0.909593i	$-0.363606\pi$
$193$	30954.0	0.831002	0.415501	+	0.909593i	$0.363606\pi$
$194$	− 3053.56i	− 0.0811341i
$195$	0	0
$196$	1643.79	0.0427892
$197$	− 12462.4i	− 0.321120i	−0.987026	−	0.160560i	$-0.948670\pi$
$197$	− 12462.4i	− 0.321120i	0.987026	−	0.160560i	$-0.0513301\pi$
$198$	0	0
$199$	68252.0	1.72349	0.861746	−	0.507340i	$-0.169371\pi$
$199$	68252.0	1.72349	0.861746	+	0.507340i	$0.169371\pi$
$200$	0	0
$201$	0	0
$202$	−22657.6	−0.555280
$203$	− 72095.4i	− 1.74951i
$204$	0	0
$205$	0	0
$206$	26124.3i	0.615617i
$207$	0	0
$208$	−16424.1	−0.379624
$209$	− 62146.9i	− 1.42274i
$210$	0	0
$211$	−21887.4	−0.491621	−0.245810	−	0.969318i	$-0.579054\pi$
$211$	−21887.4	−0.491621	−0.245810	+	0.969318i	$0.579054\pi$
$212$	22200.7i	0.493964i
$213$	0	0
$214$	8985.42	0.196205
$215$	0	0
$216$	0	0
$217$	56170.4	1.19286
$218$	35172.4i	0.740098i
$219$	0	0
$220$	0	0
$221$	− 32075.9i	− 0.656742i
$222$	0	0
$223$	−5919.40	−0.119033	−0.0595166	−	0.998227i	$-0.518956\pi$
$223$	−5919.40	−0.119033	−0.0595166	+	0.998227i	$0.518956\pi$
$224$	52876.9i	1.05383i
$225$	0	0
$226$	23874.4	0.467428
$227$	55687.9i	1.08071i	0.841437	+	0.540355i	$0.181710\pi$
$227$	55687.9i	1.08071i	−0.841437	+	0.540355i	$0.818290\pi$
$228$	0	0
$229$	6312.99	0.120383	0.0601914	−	0.998187i	$-0.480829\pi$
$229$	6312.99	0.120383	0.0601914	+	0.998187i	$0.480829\pi$
$230$	0	0
$231$	0	0
$232$	78440.5	1.45735
$233$	− 84453.4i	− 1.55563i	−0.628495	−	0.777814i	$-0.716329\pi$
$233$	− 84453.4i	− 1.55563i	0.628495	−	0.777814i	$-0.283671\pi$
$234$	0	0
$235$	0	0
$236$	59190.8i	1.06275i
$237$	0	0
$238$	−17071.3	−0.301379
$239$	106001.i	1.85572i	0.372930	+	0.927860i	$0.378353\pi$
$239$	106001.i	1.85572i	−0.372930	+	0.927860i	$0.621647\pi$
$240$	0	0
$241$	−22592.3	−0.388979	−0.194490	−	0.980905i	$-0.562305\pi$
$241$	−22592.3	−0.388979	−0.194490	+	0.980905i	$0.562305\pi$
$242$	− 51513.9i	− 0.879618i
$243$	0	0
$244$	−44112.8	−0.740944
$245$	0	0
$246$	0	0
$247$	−56208.9	−0.921322
$248$	61113.9i	0.993658i
$249$	0	0
$250$	0	0
$251$	6641.52i	0.105419i	0.998610	+	0.0527096i	$0.0167858\pi$
$251$	6641.52i	0.105419i	−0.998610	+	0.0527096i	$0.983214\pi$
$252$	0	0
$253$	24560.2	0.383700
$254$	19316.9i	0.299413i
$255$	0	0
$256$	−40014.7	−0.610576
$257$	− 63640.5i	− 0.963534i	−0.876299	−	0.481767i	$-0.839995\pi$
$257$	− 63640.5i	− 0.963534i	0.876299	−	0.481767i	$-0.160005\pi$
$258$	0	0
$259$	82859.0	1.23521
$260$	0	0
$261$	0	0
$262$	−302.256	−0.00440323
$263$	63237.1i	0.914241i	0.889405	+	0.457120i	$0.151119\pi$
$263$	63237.1i	0.914241i	−0.889405	+	0.457120i	$0.848881\pi$
$264$	0	0
$265$	0	0
$266$	29915.3i	0.422795i
$267$	0	0
$268$	−16732.0	−0.232959
$269$	− 3204.84i	− 0.0442896i	−0.999755	−	0.0221448i	$-0.992951\pi$
$269$	− 3204.84i	− 0.0442896i	0.999755	−	0.0221448i	$-0.00704949\pi$
$270$	0	0
$271$	38026.4	0.517781	0.258891	−	0.965907i	$-0.416643\pi$
$271$	38026.4	0.517781	0.258891	+	0.965907i	$0.416643\pi$
$272$	15634.5i	0.211322i
$273$	0	0
$274$	−46484.3	−0.619164
$275$	0	0
$276$	0	0
$277$	−118697.	−1.54696	−0.773481	−	0.633819i	$-0.781486\pi$
$277$	−118697.	−1.54696	−0.773481	+	0.633819i	$0.781486\pi$
$278$	16296.5i	0.210866i
$279$	0	0
$280$	0	0
$281$	− 79375.0i	− 1.00524i	−0.864506	−	0.502622i	$-0.832369\pi$
$281$	− 79375.0i	− 1.00524i	0.864506	−	0.502622i	$-0.167631\pi$
$282$	0	0
$283$	10423.0	0.130143	0.0650716	−	0.997881i	$-0.479272\pi$
$283$	10423.0	0.130143	0.0650716	+	0.997881i	$0.479272\pi$
$284$	− 29379.0i	− 0.364251i
$285$	0	0
$286$	−72284.9	−0.883722
$287$	− 25721.2i	− 0.312268i
$288$	0	0
$289$	52987.1	0.634417
$290$	0	0
$291$	0	0
$292$	−30261.2	−0.354911
$293$	108970.i	1.26932i	0.772792	+	0.634659i	$0.218859\pi$
$293$	108970.i	1.26932i	−0.772792	+	0.634659i	$0.781141\pi$
$294$	0	0
$295$	0	0
$296$	90151.5i	1.02894i
$297$	0	0
$298$	754.192	0.00849277
$299$	− 22213.6i	− 0.248471i
$300$	0	0
$301$	27409.8	0.302533
$302$	53527.4i	0.586897i
$303$	0	0
$304$	27397.4	0.296457
$305$	0	0
$306$	0	0
$307$	−29250.9	−0.310357	−0.155179	−	0.987886i	$-0.549595\pi$
$307$	−29250.9	−0.310357	−0.155179	+	0.987886i	$0.549595\pi$
$308$	− 125038.i	− 1.31807i
$309$	0	0
$310$	0	0
$311$	− 93436.3i	− 0.966039i	−0.875609	−	0.483020i	$-0.839540\pi$
$311$	− 93436.3i	− 0.966039i	0.875609	−	0.483020i	$-0.160460\pi$
$312$	0	0
$313$	−135052.	−1.37852	−0.689258	−	0.724516i	$-0.742063\pi$
$313$	−135052.	−1.37852	−0.689258	+	0.724516i	$0.742063\pi$
$314$	− 67526.0i	− 0.684876i
$315$	0	0
$316$	142494.	1.42699
$317$	167909.i	1.67092i	0.549549	+	0.835462i	$0.314800\pi$
$317$	167909.i	1.67092i	−0.549549	+	0.835462i	$0.685200\pi$
$318$	0	0
$319$	−290597.	−2.85568
$320$	0	0
$321$	0	0
$322$	−11822.4	−0.114024
$323$	53506.7i	0.512865i
$324$	0	0
$325$	0	0
$326$	72075.8i	0.678195i
$327$	0	0
$328$	27984.9	0.260122
$329$	− 209326.i	− 1.93388i
$330$	0	0
$331$	−159004.	−1.45129	−0.725644	−	0.688071i	$-0.758458\pi$
$331$	−159004.	−1.45129	−0.725644	+	0.688071i	$0.758458\pi$
$332$	− 123123.i	− 1.11703i
$333$	0	0
$334$	−33477.3	−0.300094
$335$	0	0
$336$	0	0
$337$	−195916.	−1.72508	−0.862541	−	0.505987i	$-0.831128\pi$
$337$	−195916.	−1.72508	−0.862541	+	0.505987i	$0.831128\pi$
$338$	9962.90i	0.0872072i
$339$	0	0
$340$	0	0
$341$	− 226408.i	− 1.94707i
$342$	0	0
$343$	−114131.	−0.970098
$344$	29822.1i	0.252012i
$345$	0	0
$346$	32497.5	0.271455
$347$	− 145963.i	− 1.21223i	−0.795378	−	0.606113i	$-0.792728\pi$
$347$	− 145963.i	− 1.21223i	0.795378	−	0.606113i	$-0.207272\pi$
$348$	0	0
$349$	−217782.	−1.78801	−0.894006	−	0.448055i	$-0.852117\pi$
$349$	−217782.	−1.78801	−0.894006	+	0.448055i	$0.852117\pi$
$350$	0	0
$351$	0	0
$352$	213132.	1.72014
$353$	410.870i	0.00329727i	0.999999	+	0.00164864i	$0.000524777\pi$
$353$	410.870i	0.00329727i	−0.999999	+	0.00164864i	$0.999475\pi$
$354$	0	0
$355$	0	0
$356$	− 102451.i	− 0.808385i
$357$	0	0
$358$	−71916.2	−0.561126
$359$	− 147600.i	− 1.14524i	−0.819821	−	0.572620i	$-0.805927\pi$
$359$	− 147600.i	− 1.14524i	0.819821	−	0.572620i	$-0.194073\pi$
$360$	0	0
$361$	−36557.4	−0.280518
$362$	− 49770.3i	− 0.379799i
$363$	0	0
$364$	−113091.	−0.853540
$365$	0	0
$366$	0	0
$367$	103326.	0.767145	0.383572	−	0.923511i	$-0.374694\pi$
$367$	103326.	0.767145	0.383572	+	0.923511i	$0.374694\pi$
$368$	10827.4i	0.0799516i
$369$	0	0
$370$	0	0
$371$	91362.2i	0.663771i
$372$	0	0
$373$	−48216.2	−0.346558	−0.173279	−	0.984873i	$-0.555436\pi$
$373$	−48216.2	−0.346558	−0.173279	+	0.984873i	$0.555436\pi$
$374$	68809.8i	0.491934i
$375$	0	0
$376$	227748.	1.61094
$377$	262831.i	1.84924i
$378$	0	0
$379$	173643.	1.20887	0.604433	−	0.796656i	$-0.293400\pi$
$379$	173643.	1.20887	0.604433	+	0.796656i	$0.293400\pi$
$380$	0	0
$381$	0	0
$382$	−22897.9	−0.156917
$383$	− 195296.i	− 1.33136i	−0.746238	−	0.665679i	$-0.768142\pi$
$383$	− 195296.i	− 1.33136i	0.746238	−	0.665679i	$-0.231858\pi$
$384$	0	0
$385$	0	0
$386$	60058.4i	0.403087i
$387$	0	0
$388$	−19256.2	−0.127910
$389$	− 39786.7i	− 0.262929i	−0.991321	−	0.131465i	$-0.958032\pi$
$389$	− 39786.7i	− 0.262929i	0.991321	−	0.131465i	$-0.0419679\pi$
$390$	0	0
$391$	−21145.7	−0.138315
$392$	7360.00i	0.0478967i
$393$	0	0
$394$	24180.1	0.155763
$395$	0	0
$396$	0	0
$397$	59732.1	0.378989	0.189495	−	0.981882i	$-0.439315\pi$
$397$	59732.1	0.378989	0.189495	+	0.981882i	$0.439315\pi$
$398$	132426.i	0.836000i
$399$	0	0
$400$	0	0
$401$	233569.i	1.45253i	0.687414	+	0.726266i	$0.258746\pi$
$401$	233569.i	1.45253i	−0.687414	+	0.726266i	$0.741254\pi$
$402$	0	0
$403$	−204775.	−1.26086
$404$	142882.i	0.875416i
$405$	0	0
$406$	139883.	0.848618
$407$	− 333982.i	− 2.01620i
$408$	0	0
$409$	−331803.	−1.98350	−0.991752	−	0.128168i	$-0.959090\pi$
$409$	−331803.	−1.98350	−0.991752	+	0.128168i	$0.959090\pi$
$410$	0	0
$411$	0	0
$412$	164743.	0.970539
$413$	243587.i	1.42808i
$414$	0	0
$415$	0	0
$416$	− 192768.i	− 1.11391i
$417$	0	0
$418$	120580.	0.690119
$419$	207488.i	1.18186i	0.806723	+	0.590930i	$0.201239\pi$
$419$	207488.i	1.18186i	−0.806723	+	0.590930i	$0.798761\pi$
$420$	0	0
$421$	190580.	1.07526	0.537631	−	0.843181i	$-0.319320\pi$
$421$	190580.	1.07526	0.537631	+	0.843181i	$0.319320\pi$
$422$	− 42467.1i	− 0.238466i
$423$	0	0
$424$	−99403.0	−0.552927
$425$	0	0
$426$	0	0
$427$	−181536.	−0.995653
$428$	− 56663.2i	− 0.309324i
$429$	0	0
$430$	0	0
$431$	− 43553.5i	− 0.234460i	−0.993105	−	0.117230i	$-0.962599\pi$
$431$	− 43553.5i	− 0.234460i	0.993105	−	0.117230i	$-0.0374015\pi$
$432$	0	0
$433$	−123115.	−0.656650	−0.328325	−	0.944565i	$-0.606484\pi$
$433$	−123115.	−0.656650	−0.328325	+	0.944565i	$0.606484\pi$
$434$	108984.i	0.578609i
$435$	0	0
$436$	221802.	1.16679
$437$	37055.1i	0.194037i
$438$	0	0
$439$	125357.	0.650459	0.325229	−	0.945635i	$-0.394558\pi$
$439$	125357.	0.650459	0.325229	+	0.945635i	$0.394558\pi$
$440$	0	0
$441$	0	0
$442$	62235.2	0.318560
$443$	− 67781.5i	− 0.345385i	−0.984976	−	0.172693i	$-0.944753\pi$
$443$	− 67781.5i	− 0.345385i	0.984976	−	0.172693i	$-0.0552468\pi$
$444$	0	0
$445$	0	0
$446$	− 11485.1i	− 0.0577385i
$447$	0	0
$448$	−30511.5	−0.152022
$449$	11399.8i	0.0565462i	0.999600	+	0.0282731i	$0.00900081\pi$
$449$	11399.8i	0.0565462i	−0.999600	+	0.0282731i	$0.990999\pi$
$450$	0	0
$451$	−103675.	−0.509708
$452$	− 150555.i	− 0.736915i
$453$	0	0
$454$	−108048.	−0.524211
$455$	0	0
$456$	0	0
$457$	173541.	0.830939	0.415470	−	0.909607i	$-0.363617\pi$
$457$	173541.	0.830939	0.415470	+	0.909607i	$0.363617\pi$
$458$	12248.8i	0.0583931i
$459$	0	0
$460$	0	0
$461$	207824.i	0.977900i	0.872312	+	0.488950i	$0.162620\pi$
$461$	207824.i	0.977900i	−0.872312	+	0.488950i	$0.837380\pi$
$462$	0	0
$463$	−295939.	−1.38051	−0.690256	−	0.723565i	$-0.742502\pi$
$463$	−295939.	−1.38051	−0.690256	+	0.723565i	$0.742502\pi$
$464$	− 128109.i	− 0.595038i
$465$	0	0
$466$	163861.	0.754575
$467$	38962.8i	0.178655i	0.996002	+	0.0893277i	$0.0284718\pi$
$467$	38962.8i	0.178655i	−0.996002	+	0.0893277i	$0.971528\pi$
$468$	0	0
$469$	−68856.9	−0.313042
$470$	0	0
$471$	0	0
$472$	−265025.	−1.18960
$473$	− 110481.i	− 0.493818i
$474$	0	0
$475$	0	0
$476$	107654.i	0.475133i
$477$	0	0
$478$	−205667.	−0.900138
$479$	− 42930.4i	− 0.187109i	−0.995614	−	0.0935543i	$-0.970177\pi$
$479$	− 42930.4i	− 0.187109i	0.995614	−	0.0935543i	$-0.0298229\pi$
$480$	0	0
$481$	−302071.	−1.30563
$482$	− 43834.7i	− 0.188679i
$483$	0	0
$484$	−324853.	−1.38675
$485$	0	0
$486$	0	0
$487$	242477.	1.02238	0.511191	−	0.859467i	$-0.329204\pi$
$487$	242477.	1.02238	0.511191	+	0.859467i	$0.329204\pi$
$488$	− 197514.i	− 0.829387i
$489$	0	0
$490$	0	0
$491$	423092.i	1.75498i	0.479598	+	0.877488i	$0.340783\pi$
$491$	423092.i	1.75498i	−0.479598	+	0.877488i	$0.659217\pi$
$492$	0	0
$493$	250196.	1.02940
$494$	− 109059.i	− 0.446898i
$495$	0	0
$496$	99811.6	0.405712
$497$	− 120903.i	− 0.489467i
$498$	0	0
$499$	−152654.	−0.613065	−0.306533	−	0.951860i	$-0.599169\pi$
$499$	−152654.	−0.613065	−0.306533	+	0.951860i	$0.599169\pi$
$500$	0	0
$501$	0	0
$502$	−12886.2	−0.0511348
$503$	− 334692.i	− 1.32285i	−0.750013	−	0.661423i	$-0.769953\pi$
$503$	− 334692.i	− 1.32285i	0.750013	−	0.661423i	$-0.230047\pi$
$504$	0	0
$505$	0	0
$506$	47652.9i	0.186118i
$507$	0	0
$508$	121815.	0.472034
$509$	63575.3i	0.245388i	0.992445	+	0.122694i	$0.0391533\pi$
$509$	63575.3i	0.245388i	−0.992445	+	0.122694i	$0.960847\pi$
$510$	0	0
$511$	−124533.	−0.476917
$512$	172386.i	0.657600i
$513$	0	0
$514$	123478.	0.467374
$515$	0	0
$516$	0	0
$517$	−843734.	−3.15664
$518$	160767.i	0.599153i
$519$	0	0
$520$	0	0
$521$	− 412565.i	− 1.51991i	−0.649977	−	0.759954i	$-0.725222\pi$
$521$	− 412565.i	− 1.51991i	0.649977	−	0.759954i	$-0.274778\pi$
$522$	0	0
$523$	−436558.	−1.59602	−0.798011	−	0.602643i	$-0.794114\pi$
$523$	−436558.	−1.59602	−0.798011	+	0.602643i	$0.794114\pi$
$524$	1906.06i	0.00694183i
$525$	0	0
$526$	−122696.	−0.443463
$527$	194930.i	0.701873i
$528$	0	0
$529$	265197.	0.947670
$530$	0	0
$531$	0	0
$532$	188649.	0.666549
$533$	93769.3i	0.330070i
$534$	0	0
$535$	0	0
$536$	− 74917.1i	− 0.260766i
$537$	0	0
$538$	6218.18	0.0214832
$539$	− 27266.4i	− 0.0938536i
$540$	0	0
$541$	−386165.	−1.31941	−0.659703	−	0.751526i	$-0.729318\pi$
$541$	−386165.	−1.31941	−0.659703	+	0.751526i	$0.729318\pi$
$542$	73780.6i	0.251156i
$543$	0	0
$544$	−183501.	−0.620069
$545$	0	0
$546$	0	0
$547$	−99445.7	−0.332362	−0.166181	−	0.986095i	$-0.553144\pi$
$547$	−99445.7	−0.332362	−0.166181	+	0.986095i	$0.553144\pi$
$548$	293136.i	0.976131i
$549$	0	0
$550$	0	0
$551$	− 438436.i	− 1.44412i
$552$	0	0
$553$	586401.	1.91754
$554$	− 230301.i	− 0.750372i
$555$	0	0
$556$	102768.	0.332436
$557$	220057.i	0.709292i	0.935001	+	0.354646i	$0.115399\pi$
$557$	220057.i	0.709292i	−0.935001	+	0.354646i	$0.884601\pi$
$558$	0	0
$559$	−99925.3	−0.319780
$560$	0	0
$561$	0	0
$562$	154007.	0.487605
$563$	− 73858.7i	− 0.233016i	−0.993190	−	0.116508i	$-0.962830\pi$
$563$	− 73858.7i	− 0.233016i	0.993190	−	0.116508i	$-0.0371700\pi$
$564$	0	0
$565$	0	0
$566$	20223.3i	0.0631275i
$567$	0	0
$568$	131543.	0.407730
$569$	517855.i	1.59950i	0.600335	+	0.799749i	$0.295034\pi$
$569$	517855.i	1.59950i	−0.600335	+	0.799749i	$0.704966\pi$
$570$	0	0
$571$	−30998.5	−0.0950754	−0.0475377	−	0.998869i	$-0.515137\pi$
$571$	−30998.5	−0.0950754	−0.0475377	+	0.998869i	$0.515137\pi$
$572$	455838.i	1.39321i
$573$	0	0
$574$	49905.5	0.151469
$575$	0	0
$576$	0	0
$577$	−42846.7	−0.128696	−0.0643481	−	0.997928i	$-0.520497\pi$
$577$	−42846.7	−0.128696	−0.0643481	+	0.997928i	$0.520497\pi$
$578$	102808.i	0.307731i
$579$	0	0
$580$	0	0
$581$	− 506687.i	− 1.50102i
$582$	0	0
$583$	368256.	1.08346
$584$	− 135493.i	− 0.397276i
$585$	0	0
$586$	−211428.	−0.615698
$587$	329934.i	0.957528i	0.877944	+	0.478764i	$0.158915\pi$
$587$	329934.i	0.957528i	−0.877944	+	0.478764i	$0.841085\pi$
$588$	0	0
$589$	341590.	0.984635
$590$	0	0
$591$	0	0
$592$	147236.	0.420117
$593$	167618.i	0.476664i	0.971184	+	0.238332i	$0.0766006\pi$
$593$	167618.i	0.476664i	−0.971184	+	0.238332i	$0.923399\pi$
$594$	0	0
$595$	0	0
$596$	− 4756.03i	− 0.0133891i
$597$	0	0
$598$	43099.9	0.120524
$599$	227547.i	0.634188i	0.948394	+	0.317094i	$0.102707\pi$
$599$	227547.i	0.634188i	−0.948394	+	0.317094i	$0.897293\pi$
$600$	0	0
$601$	46077.2	0.127567	0.0637833	−	0.997964i	$-0.479683\pi$
$601$	46077.2	0.127567	0.0637833	+	0.997964i	$0.479683\pi$
$602$	53181.8i	0.146747i
$603$	0	0
$604$	337550.	0.925262
$605$	0	0
$606$	0	0
$607$	−277108.	−0.752095	−0.376047	−	0.926600i	$-0.622717\pi$
$607$	−277108.	−0.752095	−0.376047	+	0.926600i	$0.622717\pi$
$608$	321562.i	0.869875i
$609$	0	0
$610$	0	0
$611$	763118.i	2.04413i
$612$	0	0
$613$	−315023.	−0.838343	−0.419171	−	0.907907i	$-0.637679\pi$
$613$	−315023.	−0.838343	−0.419171	+	0.907907i	$0.637679\pi$
$614$	− 56753.9i	− 0.150543i
$615$	0	0
$616$	559851.	1.47540
$617$	− 282133.i	− 0.741112i	−0.928810	−	0.370556i	$-0.879167\pi$
$617$	− 282133.i	− 0.741112i	0.928810	−	0.370556i	$-0.120833\pi$
$618$	0	0
$619$	288014.	0.751679	0.375839	−	0.926685i	$-0.377354\pi$
$619$	288014.	0.751679	0.375839	+	0.926685i	$0.377354\pi$
$620$	0	0
$621$	0	0
$622$	181290.	0.468589
$623$	− 421616.i	− 1.08628i
$624$	0	0
$625$	0	0
$626$	− 262034.i	− 0.668665i
$627$	0	0
$628$	−425828.	−1.07973
$629$	287549.i	0.726793i
$630$	0	0
$631$	−77926.3	−0.195716	−0.0978578	−	0.995200i	$-0.531199\pi$
$631$	−77926.3	−0.195716	−0.0978578	+	0.995200i	$0.531199\pi$
$632$	638010.i	1.59733i
$633$	0	0
$634$	−325786.	−0.810501
$635$	0	0
$636$	0	0
$637$	−24661.2	−0.0607765
$638$	− 563830.i	− 1.38518i
$639$	0	0
$640$	0	0
$641$	− 339129.i	− 0.825372i	−0.910873	−	0.412686i	$-0.864591\pi$
$641$	− 339129.i	− 0.825372i	0.910873	−	0.412686i	$-0.135409\pi$
$642$	0	0
$643$	178034.	0.430606	0.215303	−	0.976547i	$-0.430926\pi$
$643$	178034.	0.430606	0.215303	+	0.976547i	$0.430926\pi$
$644$	74553.6i	0.179762i
$645$	0	0
$646$	−103816.	−0.248771
$647$	− 778198.i	− 1.85901i	−0.368809	−	0.929505i	$-0.620234\pi$
$647$	− 778198.i	− 1.85901i	0.368809	−	0.929505i	$-0.379766\pi$
$648$	0	0
$649$	981832.	2.33103
$650$	0	0
$651$	0	0
$652$	454519.	1.06920
$653$	570456.i	1.33782i	0.743346	+	0.668908i	$0.233238\pi$
$653$	570456.i	1.33782i	−0.743346	+	0.668908i	$0.766762\pi$
$654$	0	0
$655$	0	0
$656$	− 45705.1i	− 0.106208i
$657$	0	0
$658$	406143.	0.938053
$659$	− 493094.i	− 1.13543i	−0.823227	−	0.567713i	$-0.807828\pi$
$659$	− 493094.i	− 1.13543i	0.823227	−	0.567713i	$-0.192172\pi$
$660$	0	0
$661$	−654379.	−1.49771	−0.748853	−	0.662736i	$-0.769395\pi$
$661$	−654379.	−1.49771	−0.748853	+	0.662736i	$0.769395\pi$
$662$	− 308508.i	− 0.703964i
$663$	0	0
$664$	551281.	1.25036
$665$	0	0
$666$	0	0
$667$	173268.	0.389464
$668$	211112.i	0.473108i
$669$	0	0
$670$	0	0
$671$	731724.i	1.62518i
$672$	0	0
$673$	−1427.22	−0.00315109	−0.00157554	−	0.999999i	$-0.500502\pi$
$673$	−1427.22	−0.00315109	−0.00157554	+	0.999999i	$0.500502\pi$
$674$	− 380125.i	− 0.836771i
$675$	0	0
$676$	62827.3	0.137485
$677$	− 397819.i	− 0.867976i	−0.900919	−	0.433988i	$-0.857106\pi$
$677$	− 397819.i	− 0.867976i	0.900919	−	0.433988i	$-0.142894\pi$
$678$	0	0
$679$	−79244.4	−0.171881
$680$	0	0
$681$	0	0
$682$	439287.	0.944450
$683$	− 137477.i	− 0.294707i	−0.989084	−	0.147353i	$-0.952925\pi$
$683$	− 137477.i	− 0.294707i	0.989084	−	0.147353i	$-0.0470755\pi$
$684$	0	0
$685$	0	0
$686$	− 221442.i	− 0.470557i
$687$	0	0
$688$	48705.7	0.102897
$689$	− 333070.i	− 0.701613i
$690$	0	0
$691$	−307963.	−0.644975	−0.322488	−	0.946574i	$-0.604519\pi$
$691$	−307963.	−0.644975	−0.322488	+	0.946574i	$0.604519\pi$
$692$	− 204933.i	− 0.427957i
$693$	0	0
$694$	283204.	0.588005
$695$	0	0
$696$	0	0
$697$	89261.3	0.183737
$698$	− 422550.i	− 0.867296i
$699$	0	0
$700$	0	0
$701$	779311.i	1.58590i	0.609289	+	0.792948i	$0.291455\pi$
$701$	779311.i	1.58590i	−0.609289	+	0.792948i	$0.708545\pi$
$702$	0	0
$703$	503893.	1.01959
$704$	122984.i	0.248143i
$705$	0	0
$706$	−797.189	−0.00159938
$707$	587998.i	1.17635i
$708$	0	0
$709$	350809.	0.697876	0.348938	−	0.937146i	$-0.386542\pi$
$709$	350809.	0.697876	0.348938	+	0.937146i	$0.386542\pi$
$710$	0	0
$711$	0	0
$712$	458723.	0.904878
$713$	134995.i	0.265546i
$714$	0	0
$715$	0	0
$716$	453512.i	0.884633i
$717$	0	0
$718$	286380.	0.555512
$719$	840579.i	1.62600i	0.582263	+	0.813001i	$0.302167\pi$
$719$	840579.i	1.62600i	−0.582263	+	0.813001i	$0.697833\pi$
$720$	0	0
$721$	677964.	1.30418
$722$	− 70930.5i	− 0.136069i
$723$	0	0
$724$	−313858.	−0.598764
$725$	0	0
$726$	0	0
$727$	−1.01200e6	−1.91475	−0.957374	−	0.288851i	$-0.906727\pi$
$727$	−1.01200e6	−1.91475	−0.957374	+	0.288851i	$0.906727\pi$
$728$	− 506359.i	− 0.955423i
$729$	0	0
$730$	0	0
$731$	95121.4i	0.178010i
$732$	0	0
$733$	421087.	0.783725	0.391863	−	0.920024i	$-0.371831\pi$
$733$	421087.	0.783725	0.391863	+	0.920024i	$0.371831\pi$
$734$	200478.i	0.372113i
$735$	0	0
$736$	−127080.	−0.234597
$737$	277544.i	0.510971i
$738$	0	0
$739$	−282437.	−0.517170	−0.258585	−	0.965989i	$-0.583256\pi$
$739$	−282437.	−0.517170	−0.258585	+	0.965989i	$0.583256\pi$
$740$	0	0
$741$	0	0
$742$	−177265.	−0.321970
$743$	− 369786.i	− 0.669842i	−0.942246	−	0.334921i	$-0.891290\pi$
$743$	− 369786.i	− 0.669842i	0.942246	−	0.334921i	$-0.108710\pi$
$744$	0	0
$745$	0	0
$746$	− 93551.4i	− 0.168102i
$747$	0	0
$748$	433923.	0.775550
$749$	− 233185.i	− 0.415658i
$750$	0	0
$751$	86095.1	0.152651	0.0763253	−	0.997083i	$-0.475681\pi$
$751$	86095.1	0.152651	0.0763253	+	0.997083i	$0.475681\pi$
$752$	− 371960.i	− 0.657749i
$753$	0	0
$754$	−509958.	−0.896998
$755$	0	0
$756$	0	0
$757$	189172.	0.330115	0.165058	−	0.986284i	$-0.447219\pi$
$757$	189172.	0.330115	0.165058	+	0.986284i	$0.447219\pi$
$758$	336910.i	0.586374i
$759$	0	0
$760$	0	0
$761$	− 473747.i	− 0.818045i	−0.912524	−	0.409022i	$-0.865870\pi$
$761$	− 473747.i	− 0.818045i	0.912524	−	0.409022i	$-0.134130\pi$
$762$	0	0
$763$	912775.	1.56789
$764$	144397.i	0.247384i
$765$	0	0
$766$	378922.	0.645791
$767$	− 888021.i	− 1.50950i
$768$	0	0
$769$	783721.	1.32528	0.662642	−	0.748936i	$-0.269435\pi$
$769$	783721.	1.32528	0.662642	+	0.748936i	$0.269435\pi$
$770$	0	0
$771$	0	0
$772$	378736.	0.635480
$773$	393774.i	0.659004i	0.944155	+	0.329502i	$0.106881\pi$
$773$	393774.i	0.659004i	−0.944155	+	0.329502i	$0.893119\pi$
$774$	0	0
$775$	0	0
$776$	− 86218.7i	− 0.143179i
$777$	0	0
$778$	77196.0	0.127537
$779$	− 156419.i	− 0.257759i
$780$	0	0
$781$	−487326.	−0.798947
$782$	− 41027.8i	− 0.0670911i
$783$	0	0
$784$	12020.4	0.0195563
$785$	0	0
$786$	0	0
$787$	262539.	0.423881	0.211940	−	0.977283i	$-0.432022\pi$
$787$	262539.	0.423881	0.211940	+	0.977283i	$0.432022\pi$
$788$	− 152482.i	− 0.245566i
$789$	0	0
$790$	0	0
$791$	− 619574.i	− 0.990240i
$792$	0	0
$793$	661810.	1.05242
$794$	115895.i	0.183833i
$795$	0	0
$796$	835094.	1.31798
$797$	433451.i	0.682375i	0.939995	+	0.341187i	$0.110829\pi$
$797$	433451.i	0.682375i	−0.939995	+	0.341187i	$0.889171\pi$
$798$	0	0
$799$	726431.	1.13789
$800$	0	0
$801$	0	0
$802$	−453181.	−0.704568
$803$	501959.i	0.778461i
$804$	0	0
$805$	0	0
$806$	− 397314.i	− 0.611595i
$807$	0	0
$808$	−639748.	−0.979910
$809$	394934.i	0.603431i	0.953398	+	0.301716i	$0.0975593\pi$
$809$	394934.i	0.603431i	−0.953398	+	0.301716i	$0.902441\pi$
$810$	0	0
$811$	−795473.	−1.20944	−0.604719	−	0.796439i	$-0.706715\pi$
$811$	−795473.	−1.20944	−0.604719	+	0.796439i	$0.706715\pi$
$812$	− 882119.i	− 1.33787i
$813$	0	0
$814$	648008.	0.977984
$815$	0	0
$816$	0	0
$817$	166688.	0.249724
$818$	− 643779.i	− 0.962122i
$819$	0	0
$820$	0	0
$821$	288613.i	0.428183i	0.976814	+	0.214092i	$0.0686791\pi$
$821$	288613.i	0.428183i	−0.976814	+	0.214092i	$0.931321\pi$
$822$	0	0
$823$	−90831.5	−0.134103	−0.0670513	−	0.997750i	$-0.521359\pi$
$823$	−90831.5	−0.134103	−0.0670513	+	0.997750i	$0.521359\pi$
$824$	737632.i	1.08639i
$825$	0	0
$826$	−472618.	−0.692708
$827$	− 442580.i	− 0.647114i	−0.946209	−	0.323557i	$-0.895121\pi$
$827$	− 442580.i	− 0.647114i	0.946209	−	0.323557i	$-0.104879\pi$
$828$	0	0
$829$	−259965.	−0.378273	−0.189137	−	0.981951i	$-0.560569\pi$
$829$	−259965.	−0.378273	−0.189137	+	0.981951i	$0.560569\pi$
$830$	0	0
$831$	0	0
$832$	111233.	0.160689
$833$	23475.6i	0.0338320i
$834$	0	0
$835$	0	0
$836$	− 760394.i	− 1.08799i
$837$	0	0
$838$	−402579.	−0.573275
$839$	− 413062.i	− 0.586802i	−0.955989	−	0.293401i	$-0.905213\pi$
$839$	− 413062.i	− 0.586802i	0.955989	−	0.293401i	$-0.0947871\pi$
$840$	0	0
$841$	−1.34283e6	−1.89858
$842$	369773.i	0.521568i
$843$	0	0
$844$	−267803.	−0.375950
$845$	0	0
$846$	0	0
$847$	−1.33686e6	−1.86346
$848$	162345.i	0.225761i
$849$	0	0
$850$	0	0
$851$	199137.i	0.274975i
$852$	0	0
$853$	−48283.6	−0.0663593	−0.0331796	−	0.999449i	$-0.510563\pi$
$853$	−48283.6	−0.0663593	−0.0331796	+	0.999449i	$0.510563\pi$
$854$	− 352226.i	− 0.482953i
$855$	0	0
$856$	253707.	0.346246
$857$	639554.i	0.870794i	0.900238	+	0.435397i	$0.143392\pi$
$857$	639554.i	0.870794i	−0.900238	+	0.435397i	$0.856608\pi$
$858$	0	0
$859$	391575.	0.530675	0.265337	−	0.964156i	$-0.414517\pi$
$859$	391575.	0.530675	0.265337	+	0.964156i	$0.414517\pi$
$860$	0	0
$861$	0	0
$862$	84504.6	0.113728
$863$	1.20719e6i	1.62089i	0.585816	+	0.810444i	$0.300774\pi$
$863$	1.20719e6i	1.62089i	−0.585816	+	0.810444i	$0.699226\pi$
$864$	0	0
$865$	0	0
$866$	− 238873.i	− 0.318516i
$867$	0	0
$868$	687269.	0.912195
$869$	− 2.36362e6i	− 3.12996i
$870$	0	0
$871$	251025.	0.330888
$872$	993108.i	1.30606i
$873$	0	0
$874$	−71896.0	−0.0941200
$875$	0	0
$876$	0	0
$877$	413297.	0.537357	0.268678	−	0.963230i	$-0.413413\pi$
$877$	413297.	0.537357	0.268678	+	0.963230i	$0.413413\pi$
$878$	243224.i	0.315513i
$879$	0	0
$880$	0	0
$881$	789955.i	1.01777i	0.860834	+	0.508886i	$0.169943\pi$
$881$	789955.i	1.01777i	−0.860834	+	0.508886i	$0.830057\pi$
$882$	0	0
$883$	−558493.	−0.716302	−0.358151	−	0.933664i	$-0.616593\pi$
$883$	−558493.	−0.716302	−0.358151	+	0.933664i	$0.616593\pi$
$884$	− 392463.i	− 0.502220i
$885$	0	0
$886$	131513.	0.167533
$887$	− 124383.i	− 0.158093i	−0.996871	−	0.0790465i	$-0.974812\pi$
$887$	− 124383.i	− 0.158093i	0.996871	−	0.0790465i	$-0.0251876\pi$
$888$	0	0
$889$	501302.	0.634302
$890$	0	0
$891$	0	0
$892$	−72426.5	−0.0910265
$893$	− 1.27298e6i	− 1.59631i
$894$	0	0
$895$	0	0
$896$	786830.i	0.980088i
$897$	0	0
$898$	−22118.4	−0.0274284
$899$	− 1.59727e6i	− 1.97632i
$900$	0	0
$901$	−317058.	−0.390561
$902$	− 201155.i	− 0.247240i
$903$	0	0
$904$	674103.	0.824877
$905$	0	0
$906$	0	0
$907$	683001.	0.830246	0.415123	−	0.909765i	$-0.363739\pi$
$907$	683001.	0.830246	0.415123	+	0.909765i	$0.363739\pi$
$908$	681366.i	0.826435i
$909$	0	0
$910$	0	0
$911$	− 683114.i	− 0.823107i	−0.911386	−	0.411553i	$-0.864986\pi$
$911$	− 683114.i	− 0.823107i	0.911386	−	0.411553i	$-0.135014\pi$
$912$	0	0
$913$	−2.04232e6	−2.45009
$914$	336712.i	0.403057i
$915$	0	0
$916$	77242.2	0.0920585
$917$	7843.97i	0.00932819i
$918$	0	0
$919$	247081.	0.292556	0.146278	−	0.989244i	$-0.453271\pi$
$919$	247081.	0.292556	0.146278	+	0.989244i	$0.453271\pi$
$920$	0	0
$921$	0	0
$922$	−403230.	−0.474342
$923$	440764.i	0.517372i
$924$	0	0
$925$	0	0
$926$	− 574195.i	− 0.669633i
$927$	0	0
$928$	1.50361e6	1.74598
$929$	377632.i	0.437560i	0.975774	+	0.218780i	$0.0702077\pi$
$929$	377632.i	0.437560i	−0.975774	+	0.218780i	$0.929792\pi$
$930$	0	0
$931$	41138.0	0.0474618
$932$	− 1.03333e6i	− 1.18961i
$933$	0	0
$934$	−75597.4	−0.0866589
$935$	0	0
$936$	0	0
$937$	987404.	1.12465	0.562323	−	0.826918i	$-0.309908\pi$
$937$	987404.	1.12465	0.562323	+	0.826918i	$0.309908\pi$
$938$	− 133599.i	− 0.151845i
$939$	0	0
$940$	0	0
$941$	− 795104.i	− 0.897934i	−0.893548	−	0.448967i	$-0.851792\pi$
$941$	− 795104.i	− 0.897934i	0.893548	−	0.448967i	$-0.148208\pi$
$942$	0	0
$943$	61816.3	0.0695151
$944$	432840.i	0.485717i
$945$	0	0
$946$	214361.	0.239532
$947$	1.54212e6i	1.71956i	0.510665	+	0.859780i	$0.329399\pi$
$947$	1.54212e6i	1.71956i	−0.510665	+	0.859780i	$0.670601\pi$
$948$	0	0
$949$	453998.	0.504106
$950$	0	0
$951$	0	0
$952$	−482016.	−0.531848
$953$	706894.i	0.778339i	0.921166	+	0.389169i	$0.127238\pi$
$953$	706894.i	0.778339i	−0.921166	+	0.389169i	$0.872762\pi$
$954$	0	0
$955$	0	0
$956$	1.29696e6i	1.41910i
$957$	0	0
$958$	83295.6	0.0907593
$959$	1.20634e6i	1.31169i
$960$	0	0
$961$	320929.	0.347506
$962$	− 586093.i	− 0.633310i
$963$	0	0
$964$	−276427.	−0.297458
$965$	0	0
$966$	0	0
$967$	619740.	0.662761	0.331380	−	0.943497i	$-0.392486\pi$
$967$	619740.	0.662761	0.331380	+	0.943497i	$0.392486\pi$
$968$	− 1.45452e6i	− 1.55227i
$969$	0	0
$970$	0	0
$971$	− 777633.i	− 0.824776i	−0.911008	−	0.412388i	$-0.864695\pi$
$971$	− 777633.i	− 0.824776i	0.911008	−	0.412388i	$-0.135305\pi$
$972$	0	0
$973$	422919.	0.446716
$974$	470466.i	0.495919i
$975$	0	0
$976$	−322580.	−0.338640
$977$	− 1.35355e6i	− 1.41802i	−0.705196	−	0.709012i	$-0.749141\pi$
$977$	− 1.35355e6i	− 1.41802i	0.705196	−	0.709012i	$-0.250859\pi$
$978$	0	0
$979$	−1.69942e6	−1.77311
$980$	0	0
$981$	0	0
$982$	−820902.	−0.851272
$983$	1.41672e6i	1.46614i	0.680151	+	0.733072i	$0.261914\pi$
$983$	1.41672e6i	1.46614i	−0.680151	+	0.733072i	$0.738086\pi$
$984$	0	0
$985$	0	0
$986$	485441.i	0.499325i
$987$	0	0
$988$	−687741.	−0.704549
$989$	65874.6i	0.0673481i
$990$	0	0
$991$	−697784.	−0.710516	−0.355258	−	0.934768i	$-0.615607\pi$
$991$	−697784.	−0.710516	−0.355258	+	0.934768i	$0.615607\pi$
$992$	1.17148e6i	1.19045i
$993$	0	0
$994$	234581.	0.237422
$995$	0	0
$996$	0	0
$997$	−537542.	−0.540782	−0.270391	−	0.962751i	$-0.587153\pi$
$997$	−537542.	−0.540782	−0.270391	+	0.962751i	$0.587153\pi$
$998$	− 296186.i	− 0.297375i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.5.c.p.26.4		6
3.2	odd	2	inner	675.5.c.p.26.3		6
5.2	odd	4		675.5.d.l.674.6		12
5.3	odd	4		675.5.d.l.674.7		12
5.4	even	2		135.5.c.c.26.3	✓	6
15.2	even	4		675.5.d.l.674.8		12
15.8	even	4		675.5.d.l.674.5		12
15.14	odd	2		135.5.c.c.26.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.5.c.c.26.3	✓	6	5.4	even	2
135.5.c.c.26.4	yes	6	15.14	odd	2
675.5.c.p.26.3		6	3.2	odd	2	inner
675.5.c.p.26.4		6	1.1	even	1	trivial
675.5.d.l.674.5		12	15.8	even	4
675.5.d.l.674.6		12	5.2	odd	4
675.5.d.l.674.7		12	5.3	odd	4
675.5.d.l.674.8		12	15.2	even	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	675.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.94025i q^{2} +12.2354 q^{4} +50.3522 q^{7} +54.7837i q^{8} -202.956i q^{11} -183.564 q^{13} +97.6958i q^{14} +89.4731 q^{16} +174.739i q^{17} +306.208 q^{19} +393.785 q^{22} +121.013i q^{23} -356.160i q^{26} +616.082 q^{28} -1431.82i q^{29} +1115.55 q^{31} +1050.14i q^{32} -339.038 q^{34} +1645.59 q^{37} +594.120i q^{38} -510.825i q^{41} +544.361 q^{43} -2483.26i q^{44} -234.794 q^{46} -4157.23i q^{47} +134.346 q^{49} -2245.99 q^{52} +1814.46i q^{53} +2758.48i q^{56} +2778.09 q^{58} +4837.65i q^{59} -3605.33 q^{61} +2164.44i q^{62} -605.961 q^{64} -1367.51 q^{67} +2138.01i q^{68} -2401.14i q^{71} -2473.24 q^{73} +3192.85i q^{74} +3746.59 q^{76} -10219.3i q^{77} +11646.0 q^{79} +991.127 q^{82} -10062.9i q^{83} +1056.20i q^{86} +11118.7 q^{88} -8373.34i q^{89} -9242.87 q^{91} +1480.64i q^{92} +8066.04 q^{94} -1573.80 q^{97} +260.665i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 50 q^{4} - 4 q^{7} - 566 q^{13} - 638 q^{16} + 1124 q^{19} - 1402 q^{22} + 3584 q^{28} - 2582 q^{31} + 1558 q^{34} - 2040 q^{37} - 1922 q^{43} - 5862 q^{46} + 6782 q^{49} + 834 q^{52} + 7234 q^{58}+ \cdots + 27792 q^{97}+O(q^{100}) \) 6 * q - 50 * q^4 - 4 * q^7 - 566 * q^13 - 638 * q^16 + 1124 * q^19 - 1402 * q^22 + 3584 * q^28 - 2582 * q^31 + 1558 * q^34 - 2040 * q^37 - 1922 * q^43 - 5862 * q^46 + 6782 * q^49 + 834 * q^52 + 7234 * q^58 - 11656 * q^61 + 19734 * q^64 - 1288 * q^67 + 25828 * q^73 - 2520 * q^76 + 25690 * q^79 + 11692 * q^82 + 32994 * q^88 + 40676 * q^91 + 35266 * q^94 + 27792 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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