LMFDB - Embedded newform 315.4.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.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:	$18.5856016518$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.14360.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 17x - 14 $ x^3 - 17*x - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.48565$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.48565 q^{2} +4.14976 q^{4} -5.00000 q^{5} +7.00000 q^{7} +13.4206 q^{8} +17.4283 q^{10} +6.90764 q^{11} -22.1364 q^{13} -24.3996 q^{14} -79.9776 q^{16} -88.3030 q^{17} +36.9560 q^{19} -20.7488 q^{20} -24.0776 q^{22} +95.5283 q^{23} +25.0000 q^{25} +77.1598 q^{26} +29.0483 q^{28} -269.029 q^{29} +197.114 q^{31} +171.409 q^{32} +307.793 q^{34} -35.0000 q^{35} +2.14546 q^{37} -128.816 q^{38} -67.1029 q^{40} -174.127 q^{41} -17.0345 q^{43} +28.6650 q^{44} -332.978 q^{46} +528.029 q^{47} +49.0000 q^{49} -87.1413 q^{50} -91.8608 q^{52} +641.114 q^{53} -34.5382 q^{55} +93.9441 q^{56} +937.742 q^{58} +642.975 q^{59} +142.967 q^{61} -687.070 q^{62} +42.3480 q^{64} +110.682 q^{65} +478.797 q^{67} -366.436 q^{68} +121.998 q^{70} -105.550 q^{71} +986.512 q^{73} -7.47834 q^{74} +153.358 q^{76} +48.3534 q^{77} -1099.86 q^{79} +399.888 q^{80} +606.947 q^{82} +1236.62 q^{83} +441.515 q^{85} +59.3763 q^{86} +92.7045 q^{88} +711.698 q^{89} -154.955 q^{91} +396.420 q^{92} -1840.52 q^{94} -184.780 q^{95} -636.553 q^{97} -170.797 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 13 q^{4} - 15 q^{5} + 21 q^{7} + 15 q^{8} - 15 q^{10} + 74 q^{11} + 44 q^{13} + 21 q^{14} - 79 q^{16} + 52 q^{17} + 168 q^{19} - 65 q^{20} + 184 q^{22} + 124 q^{23} + 75 q^{25} + 446 q^{26}+ \cdots + 147 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 13 * q^4 - 15 * q^5 + 21 * q^7 + 15 * q^8 - 15 * q^10 + 74 * q^11 + 44 * q^13 + 21 * q^14 - 79 * q^16 + 52 * q^17 + 168 * q^19 - 65 * q^20 + 184 * q^22 + 124 * q^23 + 75 * q^25 + 446 * q^26 + 91 * q^28 - 332 * q^29 + 320 * q^31 + 183 * q^32 + 582 * q^34 - 105 * q^35 - 54 * q^37 + 460 * q^38 - 75 * q^40 - 362 * q^41 - 16 * q^43 + 264 * q^44 - 336 * q^46 + 730 * q^47 + 147 * q^49 + 75 * q^50 + 1202 * q^52 - 110 * q^53 - 370 * q^55 + 105 * q^56 + 450 * q^58 + 180 * q^59 + 1222 * q^61 - 464 * q^62 - 391 * q^64 - 220 * q^65 + 204 * q^67 - 918 * q^68 - 105 * q^70 + 136 * q^71 + 310 * q^73 - 502 * q^74 + 1796 * q^76 + 518 * q^77 - 1034 * q^79 + 395 * q^80 - 6 * q^82 + 1660 * q^83 - 260 * q^85 - 764 * q^86 - 20 * q^88 - 242 * q^89 + 308 * q^91 - 96 * q^92 - 1108 * q^94 - 840 * q^95 + 100 * q^97 + 147 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−3.48565	−1.23236	−0.616182	−	0.787604i	$-0.711321\pi$
$2$	−3.48565	−1.23236	−0.616182	+	0.787604i	$0.711321\pi$
$3$	0	0
$3$	0	0
$4$	4.14976	0.518720
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	13.4206	0.593112
$9$	0	0
$10$	17.4283	0.551130
$11$	6.90764	0.189339	0.0946696	−	0.995509i	$-0.469821\pi$
$11$	6.90764	0.189339	0.0946696	+	0.995509i	$0.469821\pi$
$12$	0	0
$13$	−22.1364	−0.472272	−0.236136	−	0.971720i	$-0.575881\pi$
$13$	−22.1364	−0.472272	−0.236136	+	0.971720i	$0.575881\pi$
$14$	−24.3996	−0.465790
$15$	0	0
$16$	−79.9776	−1.24965
$17$	−88.3030	−1.25980	−0.629901	−	0.776676i	$-0.716904\pi$
$17$	−88.3030	−1.25980	−0.629901	+	0.776676i	$0.716904\pi$
$18$	0	0
$19$	36.9560	0.446225	0.223113	−	0.974793i	$-0.428378\pi$
$19$	36.9560	0.446225	0.223113	+	0.974793i	$0.428378\pi$
$20$	−20.7488	−0.231979
$21$	0	0
$22$	−24.0776	−0.233335
$23$	95.5283	0.866045	0.433022	−	0.901383i	$-0.357447\pi$
$23$	95.5283	0.866045	0.433022	+	0.901383i	$0.357447\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	77.1598	0.582010
$27$	0	0
$28$	29.0483	0.196058
$29$	−269.029	−1.72267	−0.861336	−	0.508035i	$-0.830372\pi$
$29$	−269.029	−1.72267	−0.861336	+	0.508035i	$0.830372\pi$
$30$	0	0
$31$	197.114	1.14202	0.571012	−	0.820942i	$-0.306551\pi$
$31$	197.114	1.14202	0.571012	+	0.820942i	$0.306551\pi$
$32$	171.409	0.946911
$33$	0	0
$34$	307.793	1.55253
$35$	−35.0000	−0.169031
$36$	0	0
$37$	2.14546	0.00953276	0.00476638	−	0.999989i	$-0.498483\pi$
$37$	2.14546	0.00953276	0.00476638	+	0.999989i	$0.498483\pi$
$38$	−128.816	−0.549912
$39$	0	0
$40$	−67.1029	−0.265248
$41$	−174.127	−0.663271	−0.331636	−	0.943408i	$-0.607600\pi$
$41$	−174.127	−0.663271	−0.331636	+	0.943408i	$0.607600\pi$
$42$	0	0
$43$	−17.0345	−0.0604125	−0.0302062	−	0.999544i	$-0.509616\pi$
$43$	−17.0345	−0.0604125	−0.0302062	+	0.999544i	$0.509616\pi$
$44$	28.6650	0.0982140
$45$	0	0
$46$	−332.978	−1.06728
$47$	528.029	1.63874	0.819371	−	0.573264i	$-0.194323\pi$
$47$	528.029	1.63874	0.819371	+	0.573264i	$0.194323\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−87.1413	−0.246473
$51$	0	0
$52$	−91.8608	−0.244977
$53$	641.114	1.66158	0.830790	−	0.556586i	$-0.187889\pi$
$53$	641.114	1.66158	0.830790	+	0.556586i	$0.187889\pi$
$54$	0	0
$55$	−34.5382	−0.0846750
$56$	93.9441	0.224175
$57$	0	0
$58$	937.742	2.12296
$59$	642.975	1.41878	0.709391	−	0.704815i	$-0.248970\pi$
$59$	642.975	1.41878	0.709391	+	0.704815i	$0.248970\pi$
$60$	0	0
$61$	142.967	0.300083	0.150042	−	0.988680i	$-0.452059\pi$
$61$	142.967	0.300083	0.150042	+	0.988680i	$0.452059\pi$
$62$	−687.070	−1.40739
$63$	0	0
$64$	42.3480	0.0827109
$65$	110.682	0.211206
$66$	0	0
$67$	478.797	0.873050	0.436525	−	0.899692i	$-0.356209\pi$
$67$	478.797	0.873050	0.436525	+	0.899692i	$0.356209\pi$
$68$	−366.436	−0.653484
$69$	0	0
$70$	121.998	0.208307
$71$	−105.550	−0.176430	−0.0882150	−	0.996101i	$-0.528116\pi$
$71$	−105.550	−0.176430	−0.0882150	+	0.996101i	$0.528116\pi$
$72$	0	0
$73$	986.512	1.58168	0.790839	−	0.612024i	$-0.209644\pi$
$73$	986.512	1.58168	0.790839	+	0.612024i	$0.209644\pi$
$74$	−7.47834	−0.0117478
$75$	0	0
$76$	153.358	0.231466
$77$	48.3534	0.0715635
$78$	0	0
$79$	−1099.86	−1.56638	−0.783190	−	0.621783i	$-0.786409\pi$
$79$	−1099.86	−1.56638	−0.783190	+	0.621783i	$0.786409\pi$
$80$	399.888	0.558860
$81$	0	0
$82$	606.947	0.817391
$83$	1236.62	1.63538	0.817691	−	0.575657i	$-0.195254\pi$
$83$	1236.62	1.63538	0.817691	+	0.575657i	$0.195254\pi$
$84$	0	0
$85$	441.515	0.563400
$86$	59.3763	0.0744501
$87$	0	0
$88$	92.7045	0.112299
$89$	711.698	0.847638	0.423819	−	0.905747i	$-0.360689\pi$
$89$	711.698	0.847638	0.423819	+	0.905747i	$0.360689\pi$
$90$	0	0
$91$	−154.955	−0.178502
$92$	396.420	0.449235
$93$	0	0
$94$	−1840.52	−2.01953
$95$	−184.780	−0.199558
$96$	0	0
$97$	−636.553	−0.666311	−0.333156	−	0.942872i	$-0.608113\pi$
$97$	−636.553	−0.666311	−0.333156	+	0.942872i	$0.608113\pi$
$98$	−170.797	−0.176052
$99$	0	0
$100$	103.744	0.103744
$101$	−1742.05	−1.71624	−0.858121	−	0.513448i	$-0.828368\pi$
$101$	−1742.05	−1.71624	−0.858121	+	0.513448i	$0.828368\pi$
$102$	0	0
$103$	1454.62	1.39154	0.695769	−	0.718266i	$-0.255064\pi$
$103$	1454.62	1.39154	0.695769	+	0.718266i	$0.255064\pi$
$104$	−297.083	−0.280110
$105$	0	0
$106$	−2234.70	−2.04767
$107$	1181.67	1.06763	0.533813	−	0.845603i	$-0.320759\pi$
$107$	1181.67	1.06763	0.533813	+	0.845603i	$0.320759\pi$
$108$	0	0
$109$	2204.43	1.93712	0.968559	−	0.248784i	$-0.0800310\pi$
$109$	2204.43	1.93712	0.968559	+	0.248784i	$0.0800310\pi$
$110$	120.388	0.104350
$111$	0	0
$112$	−559.843	−0.472323
$113$	−236.886	−0.197207	−0.0986034	−	0.995127i	$-0.531438\pi$
$113$	−236.886	−0.197207	−0.0986034	+	0.995127i	$0.531438\pi$
$114$	0	0
$115$	−477.641	−0.387307
$116$	−1116.41	−0.893585
$117$	0	0
$118$	−2241.19	−1.74846
$119$	−618.121	−0.476160
$120$	0	0
$121$	−1283.28	−0.964151
$122$	−498.334	−0.369811
$123$	0	0
$124$	817.976	0.592390
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1667.21	−1.16489	−0.582446	−	0.812869i	$-0.697904\pi$
$127$	−1667.21	−1.16489	−0.582446	+	0.812869i	$0.697904\pi$
$128$	−1518.88	−1.04884
$129$	0	0
$130$	−385.799	−0.260283
$131$	−891.722	−0.594733	−0.297367	−	0.954763i	$-0.596108\pi$
$131$	−891.722	−0.594733	−0.297367	+	0.954763i	$0.596108\pi$
$132$	0	0
$133$	258.692	0.168657
$134$	−1668.92	−1.07591
$135$	0	0
$136$	−1185.08	−0.747203
$137$	400.425	0.249713	0.124856	−	0.992175i	$-0.460153\pi$
$137$	400.425	0.249713	0.124856	+	0.992175i	$0.460153\pi$
$138$	0	0
$139$	515.050	0.314287	0.157144	−	0.987576i	$-0.449771\pi$
$139$	515.050	0.314287	0.157144	+	0.987576i	$0.449771\pi$
$140$	−145.242	−0.0876797
$141$	0	0
$142$	367.912	0.217426
$143$	−152.910	−0.0894195
$144$	0	0
$145$	1345.15	0.770403
$146$	−3438.64	−1.94920
$147$	0	0
$148$	8.90316	0.00494483
$149$	−218.374	−0.120066	−0.0600332	−	0.998196i	$-0.519121\pi$
$149$	−218.374	−0.120066	−0.0600332	+	0.998196i	$0.519121\pi$
$150$	0	0
$151$	−175.011	−0.0943190	−0.0471595	−	0.998887i	$-0.515017\pi$
$151$	−175.011	−0.0943190	−0.0471595	+	0.998887i	$0.515017\pi$
$152$	495.971	0.264661
$153$	0	0
$154$	−168.543	−0.0881922
$155$	−985.570	−0.510728
$156$	0	0
$157$	−919.642	−0.467487	−0.233743	−	0.972298i	$-0.575098\pi$
$157$	−919.642	−0.467487	−0.233743	+	0.972298i	$0.575098\pi$
$158$	3833.73	1.93035
$159$	0	0
$160$	−857.046	−0.423471
$161$	668.698	0.327334
$162$	0	0
$163$	2368.51	1.13813	0.569067	−	0.822291i	$-0.307305\pi$
$163$	2368.51	1.13813	0.569067	+	0.822291i	$0.307305\pi$
$164$	−722.587	−0.344052
$165$	0	0
$166$	−4310.43	−2.01539
$167$	−1079.37	−0.500144	−0.250072	−	0.968227i	$-0.580454\pi$
$167$	−1079.37	−0.500144	−0.250072	+	0.968227i	$0.580454\pi$
$168$	0	0
$169$	−1706.98	−0.776959
$170$	−1538.97	−0.694314
$171$	0	0
$172$	−70.6891	−0.0313372
$173$	881.271	0.387294	0.193647	−	0.981071i	$-0.437968\pi$
$173$	881.271	0.387294	0.193647	+	0.981071i	$0.437968\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	−552.456	−0.236608
$177$	0	0
$178$	−2480.73	−1.04460
$179$	3377.72	1.41041	0.705203	−	0.709006i	$-0.250856\pi$
$179$	3377.72	1.41041	0.705203	+	0.709006i	$0.250856\pi$
$180$	0	0
$181$	1435.58	0.589533	0.294767	−	0.955569i	$-0.404758\pi$
$181$	1435.58	0.589533	0.294767	+	0.955569i	$0.404758\pi$
$182$	540.118	0.219979
$183$	0	0
$184$	1282.05	0.513661
$185$	−10.7273	−0.00426318
$186$	0	0
$187$	−609.965	−0.238530
$188$	2191.19	0.850049
$189$	0	0
$190$	644.078	0.245928
$191$	1588.14	0.601642	0.300821	−	0.953681i	$-0.402739\pi$
$191$	1588.14	0.601642	0.300821	+	0.953681i	$0.402739\pi$
$192$	0	0
$193$	−977.704	−0.364646	−0.182323	−	0.983239i	$-0.558362\pi$
$193$	−977.704	−0.364646	−0.182323	+	0.983239i	$0.558362\pi$
$194$	2218.80	0.821138
$195$	0	0
$196$	203.338	0.0741029
$197$	−359.682	−0.130083	−0.0650413	−	0.997883i	$-0.520718\pi$
$197$	−359.682	−0.130083	−0.0650413	+	0.997883i	$0.520718\pi$
$198$	0	0
$199$	2818.38	1.00397	0.501983	−	0.864877i	$-0.332604\pi$
$199$	2818.38	1.00397	0.501983	+	0.864877i	$0.332604\pi$
$200$	335.515	0.118622
$201$	0	0
$202$	6072.18	2.11503
$203$	−1883.21	−0.651109
$204$	0	0
$205$	870.637	0.296624
$206$	−5070.31	−1.71488
$207$	0	0
$208$	1770.42	0.590174
$209$	255.278	0.0844879
$210$	0	0
$211$	−1009.64	−0.329415	−0.164708	−	0.986342i	$-0.552668\pi$
$211$	−1009.64	−0.329415	−0.164708	+	0.986342i	$0.552668\pi$
$212$	2660.47	0.861895
$213$	0	0
$214$	−4118.88	−1.31570
$215$	85.1725	0.0270173
$216$	0	0
$217$	1379.80	0.431644
$218$	−7683.86	−2.38723
$219$	0	0
$220$	−143.325	−0.0439226
$221$	1954.71	0.594969
$222$	0	0
$223$	1277.28	0.383555	0.191778	−	0.981438i	$-0.438575\pi$
$223$	1277.28	0.383555	0.191778	+	0.981438i	$0.438575\pi$
$224$	1199.86	0.357899
$225$	0	0
$226$	825.702	0.243030
$227$	−1399.87	−0.409307	−0.204654	−	0.978834i	$-0.565607\pi$
$227$	−1399.87	−0.409307	−0.204654	+	0.978834i	$0.565607\pi$
$228$	0	0
$229$	3182.00	0.918222	0.459111	−	0.888379i	$-0.348168\pi$
$229$	3182.00	0.918222	0.459111	+	0.888379i	$0.348168\pi$
$230$	1664.89	0.477303
$231$	0	0
$232$	−3610.53	−1.02174
$233$	3027.84	0.851332	0.425666	−	0.904880i	$-0.360040\pi$
$233$	3027.84	0.851332	0.425666	+	0.904880i	$0.360040\pi$
$234$	0	0
$235$	−2640.14	−0.732868
$236$	2668.19	0.735951
$237$	0	0
$238$	2154.55	0.586802
$239$	4995.69	1.35207	0.676034	−	0.736870i	$-0.263697\pi$
$239$	4995.69	1.35207	0.676034	+	0.736870i	$0.263697\pi$
$240$	0	0
$241$	−3756.52	−1.00406	−0.502030	−	0.864850i	$-0.667413\pi$
$241$	−3756.52	−1.00406	−0.502030	+	0.864850i	$0.667413\pi$
$242$	4473.08	1.18818
$243$	0	0
$244$	593.280	0.155659
$245$	−245.000	−0.0638877
$246$	0	0
$247$	−818.072	−0.210740
$248$	2645.39	0.677347
$249$	0	0
$250$	435.706	0.110226
$251$	−6565.46	−1.65103	−0.825514	−	0.564381i	$-0.809115\pi$
$251$	−6565.46	−1.65103	−0.825514	+	0.564381i	$0.809115\pi$
$252$	0	0
$253$	659.875	0.163976
$254$	5811.33	1.43557
$255$	0	0
$256$	4955.51	1.20984
$257$	6879.44	1.66976	0.834879	−	0.550433i	$-0.185537\pi$
$257$	6879.44	1.66976	0.834879	+	0.550433i	$0.185537\pi$
$258$	0	0
$259$	15.0182	0.00360304
$260$	459.304	0.109557
$261$	0	0
$262$	3108.23	0.732928
$263$	−3080.15	−0.722169	−0.361084	−	0.932533i	$-0.617593\pi$
$263$	−3080.15	−0.722169	−0.361084	+	0.932533i	$0.617593\pi$
$264$	0	0
$265$	−3205.57	−0.743081
$266$	−901.709	−0.207847
$267$	0	0
$268$	1986.89	0.452869
$269$	−6710.33	−1.52095	−0.760476	−	0.649366i	$-0.775034\pi$
$269$	−6710.33	−1.52095	−0.760476	+	0.649366i	$0.775034\pi$
$270$	0	0
$271$	7842.95	1.75803	0.879014	−	0.476796i	$-0.158202\pi$
$271$	7842.95	1.75803	0.879014	+	0.476796i	$0.158202\pi$
$272$	7062.26	1.57431
$273$	0	0
$274$	−1395.74	−0.307737
$275$	172.691	0.0378678
$276$	0	0
$277$	5446.87	1.18148	0.590742	−	0.806861i	$-0.298835\pi$
$277$	5446.87	1.18148	0.590742	+	0.806861i	$0.298835\pi$
$278$	−1795.28	−0.387316
$279$	0	0
$280$	−469.721	−0.100254
$281$	−2126.76	−0.451501	−0.225751	−	0.974185i	$-0.572483\pi$
$281$	−2126.76	−0.451501	−0.225751	+	0.974185i	$0.572483\pi$
$282$	0	0
$283$	−3426.38	−0.719707	−0.359853	−	0.933009i	$-0.617173\pi$
$283$	−3426.38	−0.719707	−0.359853	+	0.933009i	$0.617173\pi$
$284$	−438.009	−0.0915178
$285$	0	0
$286$	532.991	0.110197
$287$	−1218.89	−0.250693
$288$	0	0
$289$	2884.42	0.587099
$290$	−4688.71	−0.949416
$291$	0	0
$292$	4093.79	0.820448
$293$	1749.82	0.348894	0.174447	−	0.984667i	$-0.444186\pi$
$293$	1749.82	0.348894	0.174447	+	0.984667i	$0.444186\pi$
$294$	0	0
$295$	−3214.87	−0.634499
$296$	28.7934	0.00565399
$297$	0	0
$298$	761.176	0.147966
$299$	−2114.65	−0.409008
$300$	0	0
$301$	−119.241	−0.0228338
$302$	610.026	0.116235
$303$	0	0
$304$	−2955.65	−0.557625
$305$	−714.836	−0.134201
$306$	0	0
$307$	7970.33	1.48173	0.740864	−	0.671655i	$-0.234416\pi$
$307$	7970.33	1.48173	0.740864	+	0.671655i	$0.234416\pi$
$308$	200.655	0.0371214
$309$	0	0
$310$	3435.35	0.629403
$311$	2560.72	0.466898	0.233449	−	0.972369i	$-0.424999\pi$
$311$	2560.72	0.466898	0.233449	+	0.972369i	$0.424999\pi$
$312$	0	0
$313$	−4861.11	−0.877848	−0.438924	−	0.898524i	$-0.644640\pi$
$313$	−4861.11	−0.877848	−0.438924	+	0.898524i	$0.644640\pi$
$314$	3205.55	0.576114
$315$	0	0
$316$	−4564.16	−0.812513
$317$	−8166.16	−1.44687	−0.723434	−	0.690394i	$-0.757437\pi$
$317$	−8166.16	−1.44687	−0.723434	+	0.690394i	$0.757437\pi$
$318$	0	0
$319$	−1858.36	−0.326169
$320$	−211.740	−0.0369894
$321$	0	0
$322$	−2330.85	−0.403395
$323$	−3263.32	−0.562155
$324$	0	0
$325$	−553.410	−0.0944543
$326$	−8255.79	−1.40259
$327$	0	0
$328$	−2336.89	−0.393394
$329$	3696.20	0.619386
$330$	0	0
$331$	2974.89	0.494002	0.247001	−	0.969015i	$-0.420555\pi$
$331$	2974.89	0.494002	0.247001	+	0.969015i	$0.420555\pi$
$332$	5131.68	0.848306
$333$	0	0
$334$	3762.30	0.616359
$335$	−2393.98	−0.390440
$336$	0	0
$337$	3496.34	0.565157	0.282578	−	0.959244i	$-0.408810\pi$
$337$	3496.34	0.565157	0.282578	+	0.959244i	$0.408810\pi$
$338$	5949.94	0.957497
$339$	0	0
$340$	1832.18	0.292247
$341$	1361.59	0.216230
$342$	0	0
$343$	343.000	0.0539949
$344$	−228.613	−0.0358314
$345$	0	0
$346$	−3071.80	−0.477287
$347$	−3959.08	−0.612491	−0.306246	−	0.951953i	$-0.599073\pi$
$347$	−3959.08	−0.612491	−0.306246	+	0.951953i	$0.599073\pi$
$348$	0	0
$349$	5581.65	0.856099	0.428050	−	0.903755i	$-0.359201\pi$
$349$	5581.65	0.856099	0.428050	+	0.903755i	$0.359201\pi$
$350$	−609.989	−0.0931579
$351$	0	0
$352$	1184.03	0.179287
$353$	9896.43	1.49216	0.746082	−	0.665854i	$-0.231933\pi$
$353$	9896.43	1.49216	0.746082	+	0.665854i	$0.231933\pi$
$354$	0	0
$355$	527.752	0.0789019
$356$	2953.38	0.439687
$357$	0	0
$358$	−11773.5	−1.73813
$359$	−11917.6	−1.75205	−0.876025	−	0.482265i	$-0.839814\pi$
$359$	−11917.6	−1.75205	−0.876025	+	0.482265i	$0.839814\pi$
$360$	0	0
$361$	−5493.26	−0.800883
$362$	−5003.92	−0.726519
$363$	0	0
$364$	−643.025	−0.0925926
$365$	−4932.56	−0.707348
$366$	0	0
$367$	−7101.58	−1.01008	−0.505040	−	0.863096i	$-0.668522\pi$
$367$	−7101.58	−1.01008	−0.505040	+	0.863096i	$0.668522\pi$
$368$	−7640.12	−1.08225
$369$	0	0
$370$	37.3917	0.00525379
$371$	4487.80	0.628018
$372$	0	0
$373$	294.316	0.0408555	0.0204277	−	0.999791i	$-0.493497\pi$
$373$	294.316	0.0408555	0.0204277	+	0.999791i	$0.493497\pi$
$374$	2126.12	0.293955
$375$	0	0
$376$	7086.45	0.971957
$377$	5955.34	0.813569
$378$	0	0
$379$	−9436.57	−1.27896	−0.639478	−	0.768810i	$-0.720849\pi$
$379$	−9436.57	−1.27896	−0.639478	+	0.768810i	$0.720849\pi$
$380$	−766.792	−0.103515
$381$	0	0
$382$	−5535.70	−0.741442
$383$	3160.82	0.421699	0.210849	−	0.977519i	$-0.432377\pi$
$383$	3160.82	0.421699	0.210849	+	0.977519i	$0.432377\pi$
$384$	0	0
$385$	−241.767	−0.0320042
$386$	3407.93	0.449376
$387$	0	0
$388$	−2641.55	−0.345629
$389$	−7822.76	−1.01961	−0.509807	−	0.860289i	$-0.670283\pi$
$389$	−7822.76	−1.01961	−0.509807	+	0.860289i	$0.670283\pi$
$390$	0	0
$391$	−8435.43	−1.09104
$392$	657.609	0.0847302
$393$	0	0
$394$	1253.73	0.160309
$395$	5499.30	0.700506
$396$	0	0
$397$	−7935.18	−1.00316	−0.501581	−	0.865111i	$-0.667248\pi$
$397$	−7935.18	−1.00316	−0.501581	+	0.865111i	$0.667248\pi$
$398$	−9823.87	−1.23725
$399$	0	0
$400$	−1999.44	−0.249930
$401$	488.380	0.0608193	0.0304097	−	0.999538i	$-0.490319\pi$
$401$	488.380	0.0608193	0.0304097	+	0.999538i	$0.490319\pi$
$402$	0	0
$403$	−4363.39	−0.539345
$404$	−7229.09	−0.890249
$405$	0	0
$406$	6564.20	0.802403
$407$	14.8201	0.00180492
$408$	0	0
$409$	11230.6	1.35775	0.678874	−	0.734254i	$-0.262468\pi$
$409$	11230.6	1.35775	0.678874	+	0.734254i	$0.262468\pi$
$410$	−3034.74	−0.365549
$411$	0	0
$412$	6036.34	0.721818
$413$	4500.82	0.536249
$414$	0	0
$415$	−6183.10	−0.731365
$416$	−3794.38	−0.447199
$417$	0	0
$418$	−889.811	−0.104120
$419$	7369.62	0.859259	0.429629	−	0.903005i	$-0.358644\pi$
$419$	7369.62	0.859259	0.429629	+	0.903005i	$0.358644\pi$
$420$	0	0
$421$	11972.5	1.38599	0.692997	−	0.720941i	$-0.256290\pi$
$421$	11972.5	1.38599	0.692997	+	0.720941i	$0.256290\pi$
$422$	3519.26	0.405959
$423$	0	0
$424$	8604.12	0.985503
$425$	−2207.57	−0.251960
$426$	0	0
$427$	1000.77	0.113421
$428$	4903.63	0.553799
$429$	0	0
$430$	−296.882	−0.0332951
$431$	3568.60	0.398825	0.199412	−	0.979916i	$-0.436097\pi$
$431$	3568.60	0.398825	0.199412	+	0.979916i	$0.436097\pi$
$432$	0	0
$433$	2291.60	0.254335	0.127168	−	0.991881i	$-0.459411\pi$
$433$	2291.60	0.254335	0.127168	+	0.991881i	$0.459411\pi$
$434$	−4809.49	−0.531943
$435$	0	0
$436$	9147.85	1.00482
$437$	3530.34	0.386451
$438$	0	0
$439$	−7329.66	−0.796870	−0.398435	−	0.917197i	$-0.630446\pi$
$439$	−7329.66	−0.796870	−0.398435	+	0.917197i	$0.630446\pi$
$440$	−463.523	−0.0502218
$441$	0	0
$442$	−6813.44	−0.733218
$443$	−8297.38	−0.889889	−0.444944	−	0.895558i	$-0.646777\pi$
$443$	−8297.38	−0.889889	−0.444944	+	0.895558i	$0.646777\pi$
$444$	0	0
$445$	−3558.49	−0.379075
$446$	−4452.14	−0.472679
$447$	0	0
$448$	296.436	0.0312618
$449$	−9758.62	−1.02570	−0.512848	−	0.858479i	$-0.671410\pi$
$449$	−9758.62	−1.02570	−0.512848	+	0.858479i	$0.671410\pi$
$450$	0	0
$451$	−1202.81	−0.125583
$452$	−983.021	−0.102295
$453$	0	0
$454$	4879.47	0.504416
$455$	774.774	0.0798285
$456$	0	0
$457$	−11745.0	−1.20220	−0.601102	−	0.799172i	$-0.705272\pi$
$457$	−11745.0	−1.20220	−0.601102	+	0.799172i	$0.705272\pi$
$458$	−11091.4	−1.13158
$459$	0	0
$460$	−1982.10	−0.200904
$461$	10748.6	1.08593	0.542963	−	0.839756i	$-0.317302\pi$
$461$	10748.6	1.08593	0.542963	+	0.839756i	$0.317302\pi$
$462$	0	0
$463$	−9862.51	−0.989957	−0.494978	−	0.868905i	$-0.664824\pi$
$463$	−9862.51	−0.989957	−0.494978	+	0.868905i	$0.664824\pi$
$464$	21516.3	2.15274
$465$	0	0
$466$	−10554.0	−1.04915
$467$	4660.78	0.461831	0.230916	−	0.972974i	$-0.425828\pi$
$467$	4660.78	0.461831	0.230916	+	0.972974i	$0.425828\pi$
$468$	0	0
$469$	3351.58	0.329982
$470$	9202.62	0.903160
$471$	0	0
$472$	8629.10	0.841497
$473$	−117.668	−0.0114384
$474$	0	0
$475$	923.899	0.0892451
$476$	−2565.05	−0.246994
$477$	0	0
$478$	−17413.2	−1.66624
$479$	16293.2	1.55419	0.777094	−	0.629385i	$-0.216693\pi$
$479$	16293.2	1.55419	0.777094	+	0.629385i	$0.216693\pi$
$480$	0	0
$481$	−47.4928	−0.00450205
$482$	13093.9	1.23737
$483$	0	0
$484$	−5325.33	−0.500124
$485$	3182.77	0.297984
$486$	0	0
$487$	−3515.00	−0.327063	−0.163531	−	0.986538i	$-0.552289\pi$
$487$	−3515.00	−0.327063	−0.163531	+	0.986538i	$0.552289\pi$
$488$	1918.70	0.177983
$489$	0	0
$490$	853.984	0.0787328
$491$	2516.79	0.231326	0.115663	−	0.993288i	$-0.463101\pi$
$491$	2516.79	0.231326	0.115663	+	0.993288i	$0.463101\pi$
$492$	0	0
$493$	23756.1	2.17023
$494$	2851.51	0.259708
$495$	0	0
$496$	−15764.7	−1.42713
$497$	−738.853	−0.0666842
$498$	0	0
$499$	−8747.48	−0.784751	−0.392376	−	0.919805i	$-0.628347\pi$
$499$	−8747.48	−0.784751	−0.392376	+	0.919805i	$0.628347\pi$
$500$	−518.720	−0.0463957
$501$	0	0
$502$	22884.9	2.03467
$503$	−11426.1	−1.01285	−0.506426	−	0.862284i	$-0.669034\pi$
$503$	−11426.1	−1.01285	−0.506426	+	0.862284i	$0.669034\pi$
$504$	0	0
$505$	8710.25	0.767526
$506$	−2300.09	−0.202078
$507$	0	0
$508$	−6918.54	−0.604254
$509$	8078.44	0.703478	0.351739	−	0.936098i	$-0.385590\pi$
$509$	8078.44	0.703478	0.351739	+	0.936098i	$0.385590\pi$
$510$	0	0
$511$	6905.58	0.597818
$512$	−5122.12	−0.442125
$513$	0	0
$514$	−23979.3	−2.05775
$515$	−7273.12	−0.622314
$516$	0	0
$517$	3647.43	0.310278
$518$	−52.3484	−0.00444026
$519$	0	0
$520$	1485.42	0.125269
$521$	−7226.14	−0.607645	−0.303822	−	0.952729i	$-0.598263\pi$
$521$	−7226.14	−0.607645	−0.303822	+	0.952729i	$0.598263\pi$
$522$	0	0
$523$	9333.06	0.780318	0.390159	−	0.920748i	$-0.372420\pi$
$523$	9333.06	0.780318	0.390159	+	0.920748i	$0.372420\pi$
$524$	−3700.43	−0.308500
$525$	0	0
$526$	10736.3	0.889974
$527$	−17405.8	−1.43872
$528$	0	0
$529$	−3041.35	−0.249967
$530$	11173.5	0.915746
$531$	0	0
$532$	1073.51	0.0874859
$533$	3854.55	0.313244
$534$	0	0
$535$	−5908.33	−0.477457
$536$	6425.73	0.517816
$537$	0	0
$538$	23389.9	1.87437
$539$	338.474	0.0270484
$540$	0	0
$541$	15263.1	1.21296	0.606482	−	0.795097i	$-0.292580\pi$
$541$	15263.1	1.21296	0.606482	+	0.795097i	$0.292580\pi$
$542$	−27337.8	−2.16653
$543$	0	0
$544$	−15135.9	−1.19292
$545$	−11022.1	−0.866305
$546$	0	0
$547$	−13226.0	−1.03382	−0.516912	−	0.856039i	$-0.672918\pi$
$547$	−13226.0	−1.03382	−0.516912	+	0.856039i	$0.672918\pi$
$548$	1661.67	0.129531
$549$	0	0
$550$	−601.940	−0.0466669
$551$	−9942.24	−0.768700
$552$	0	0
$553$	−7699.03	−0.592036
$554$	−18985.9	−1.45602
$555$	0	0
$556$	2137.33	0.163027
$557$	−6993.63	−0.532010	−0.266005	−	0.963972i	$-0.585704\pi$
$557$	−6993.63	−0.532010	−0.266005	+	0.963972i	$0.585704\pi$
$558$	0	0
$559$	377.082	0.0285311
$560$	2799.22	0.211229
$561$	0	0
$562$	7413.14	0.556413
$563$	−392.197	−0.0293590	−0.0146795	−	0.999892i	$-0.504673\pi$
$563$	−392.197	−0.0293590	−0.0146795	+	0.999892i	$0.504673\pi$
$564$	0	0
$565$	1184.43	0.0881935
$566$	11943.2	0.886940
$567$	0	0
$568$	−1416.55	−0.104643
$569$	−8811.72	−0.649221	−0.324610	−	0.945848i	$-0.605233\pi$
$569$	−8811.72	−0.649221	−0.324610	+	0.945848i	$0.605233\pi$
$570$	0	0
$571$	−24775.6	−1.81581	−0.907905	−	0.419175i	$-0.862319\pi$
$571$	−24775.6	−1.81581	−0.907905	+	0.419175i	$0.862319\pi$
$572$	−634.541	−0.0463837
$573$	0	0
$574$	4248.63	0.308945
$575$	2388.21	0.173209
$576$	0	0
$577$	−8850.62	−0.638572	−0.319286	−	0.947658i	$-0.603443\pi$
$577$	−8850.62	−0.638572	−0.319286	+	0.947658i	$0.603443\pi$
$578$	−10054.1	−0.723520
$579$	0	0
$580$	5582.04	0.399623
$581$	8656.35	0.618117
$582$	0	0
$583$	4428.58	0.314602
$584$	13239.6	0.938112
$585$	0	0
$586$	−6099.28	−0.429964
$587$	46.0232	0.00323608	0.00161804	−	0.999999i	$-0.499485\pi$
$587$	46.0232	0.00323608	0.00161804	+	0.999999i	$0.499485\pi$
$588$	0	0
$589$	7284.54	0.509600
$590$	11205.9	0.781933
$591$	0	0
$592$	−171.589	−0.0119126
$593$	2729.93	0.189047	0.0945235	−	0.995523i	$-0.469867\pi$
$593$	2729.93	0.189047	0.0945235	+	0.995523i	$0.469867\pi$
$594$	0	0
$595$	3090.60	0.212945
$596$	−906.200	−0.0622809
$597$	0	0
$598$	7370.94	0.504047
$599$	5505.07	0.375511	0.187756	−	0.982216i	$-0.439879\pi$
$599$	5505.07	0.375511	0.187756	+	0.982216i	$0.439879\pi$
$600$	0	0
$601$	−7446.97	−0.505438	−0.252719	−	0.967540i	$-0.581325\pi$
$601$	−7446.97	−0.505438	−0.252719	+	0.967540i	$0.581325\pi$
$602$	415.634	0.0281395
$603$	0	0
$604$	−726.253	−0.0489252
$605$	6416.42	0.431181
$606$	0	0
$607$	−24071.4	−1.60960	−0.804799	−	0.593547i	$-0.797727\pi$
$607$	−24071.4	−1.60960	−0.804799	+	0.593547i	$0.797727\pi$
$608$	6334.59	0.422536
$609$	0	0
$610$	2491.67	0.165385
$611$	−11688.7	−0.773932
$612$	0	0
$613$	−4108.61	−0.270710	−0.135355	−	0.990797i	$-0.543218\pi$
$613$	−4108.61	−0.270710	−0.135355	+	0.990797i	$0.543218\pi$
$614$	−27781.8	−1.82603
$615$	0	0
$616$	648.932	0.0424451
$617$	−3542.46	−0.231141	−0.115571	−	0.993299i	$-0.536870\pi$
$617$	−3542.46	−0.231141	−0.115571	+	0.993299i	$0.536870\pi$
$618$	0	0
$619$	6484.81	0.421077	0.210538	−	0.977586i	$-0.432478\pi$
$619$	6484.81	0.421077	0.210538	+	0.977586i	$0.432478\pi$
$620$	−4089.88	−0.264925
$621$	0	0
$622$	−8925.79	−0.575388
$623$	4981.88	0.320377
$624$	0	0
$625$	625.000	0.0400000
$626$	16944.1	1.08183
$627$	0	0
$628$	−3816.29	−0.242495
$629$	−189.451	−0.0120094
$630$	0	0
$631$	3250.84	0.205094	0.102547	−	0.994728i	$-0.467301\pi$
$631$	3250.84	0.205094	0.102547	+	0.994728i	$0.467301\pi$
$632$	−14760.8	−0.929038
$633$	0	0
$634$	28464.4	1.78307
$635$	8336.07	0.520956
$636$	0	0
$637$	−1084.68	−0.0674674
$638$	6477.58	0.401959
$639$	0	0
$640$	7594.42	0.469056
$641$	−2800.61	−0.172570	−0.0862852	−	0.996270i	$-0.527500\pi$
$641$	−2800.61	−0.172570	−0.0862852	+	0.996270i	$0.527500\pi$
$642$	0	0
$643$	18910.6	1.15982	0.579908	−	0.814682i	$-0.303089\pi$
$643$	18910.6	1.15982	0.579908	+	0.814682i	$0.303089\pi$
$644$	2774.94	0.169795
$645$	0	0
$646$	11374.8	0.692780
$647$	24522.7	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$647$	24522.7	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$648$	0	0
$649$	4441.43	0.268631
$650$	1928.99	0.116402
$651$	0	0
$652$	9828.74	0.590373
$653$	15299.6	0.916875	0.458438	−	0.888727i	$-0.348409\pi$
$653$	15299.6	0.916875	0.458438	+	0.888727i	$0.348409\pi$
$654$	0	0
$655$	4458.61	0.265973
$656$	13926.3	0.828857
$657$	0	0
$658$	−12883.7	−0.763309
$659$	2203.13	0.130230	0.0651151	−	0.997878i	$-0.479259\pi$
$659$	2203.13	0.130230	0.0651151	+	0.997878i	$0.479259\pi$
$660$	0	0
$661$	−3162.36	−0.186084	−0.0930421	−	0.995662i	$-0.529659\pi$
$661$	−3162.36	−0.186084	−0.0930421	+	0.995662i	$0.529659\pi$
$662$	−10369.4	−0.608790
$663$	0	0
$664$	16596.2	0.969965
$665$	−1293.46	−0.0754258
$666$	0	0
$667$	−25699.9	−1.49191
$668$	−4479.12	−0.259435
$669$	0	0
$670$	8344.59	0.481164
$671$	987.565	0.0568175
$672$	0	0
$673$	−4443.07	−0.254484	−0.127242	−	0.991872i	$-0.540613\pi$
$673$	−4443.07	−0.254484	−0.127242	+	0.991872i	$0.540613\pi$
$674$	−12187.0	−0.696479
$675$	0	0
$676$	−7083.56	−0.403025
$677$	−4456.32	−0.252984	−0.126492	−	0.991968i	$-0.540372\pi$
$677$	−4456.32	−0.252984	−0.126492	+	0.991968i	$0.540372\pi$
$678$	0	0
$679$	−4455.87	−0.251842
$680$	5925.39	0.334159
$681$	0	0
$682$	−4746.03	−0.266474
$683$	10046.8	0.562858	0.281429	−	0.959582i	$-0.409192\pi$
$683$	10046.8	0.562858	0.281429	+	0.959582i	$0.409192\pi$
$684$	0	0
$685$	−2002.13	−0.111675
$686$	−1195.58	−0.0665414
$687$	0	0
$688$	1362.38	0.0754944
$689$	−14191.9	−0.784717
$690$	0	0
$691$	31811.2	1.75131	0.875655	−	0.482938i	$-0.160430\pi$
$691$	31811.2	1.75131	0.875655	+	0.482938i	$0.160430\pi$
$692$	3657.06	0.200897
$693$	0	0
$694$	13800.0	0.754812
$695$	−2575.25	−0.140554
$696$	0	0
$697$	15376.0	0.835590
$698$	−19455.7	−1.05503
$699$	0	0
$700$	726.208	0.0392116
$701$	13907.2	0.749312	0.374656	−	0.927164i	$-0.377761\pi$
$701$	13907.2	0.749312	0.374656	+	0.927164i	$0.377761\pi$
$702$	0	0
$703$	79.2877	0.00425376
$704$	292.524	0.0156604
$705$	0	0
$706$	−34495.5	−1.83889
$707$	−12194.3	−0.648678
$708$	0	0
$709$	−228.952	−0.0121276	−0.00606381	−	0.999982i	$-0.501930\pi$
$709$	−228.952	−0.0121276	−0.00606381	+	0.999982i	$0.501930\pi$
$710$	−1839.56	−0.0972358
$711$	0	0
$712$	9551.40	0.502744
$713$	18830.0	0.989043
$714$	0	0
$715$	764.551	0.0399896
$716$	14016.7	0.731606
$717$	0	0
$718$	41540.5	2.15916
$719$	−36162.2	−1.87569	−0.937846	−	0.347052i	$-0.887183\pi$
$719$	−36162.2	−1.87569	−0.937846	+	0.347052i	$0.887183\pi$
$720$	0	0
$721$	10182.4	0.525952
$722$	19147.6	0.986979
$723$	0	0
$724$	5957.30	0.305803
$725$	−6725.73	−0.344534
$726$	0	0
$727$	22268.3	1.13602	0.568010	−	0.823021i	$-0.307713\pi$
$727$	22268.3	1.13602	0.568010	+	0.823021i	$0.307713\pi$
$728$	−2079.58	−0.105872
$729$	0	0
$730$	17193.2	0.871710
$731$	1504.20	0.0761077
$732$	0	0
$733$	−2333.20	−0.117570	−0.0587848	−	0.998271i	$-0.518723\pi$
$733$	−2333.20	−0.117570	−0.0587848	+	0.998271i	$0.518723\pi$
$734$	24753.6	1.24479
$735$	0	0
$736$	16374.4	0.820067
$737$	3307.35	0.165302
$738$	0	0
$739$	−4829.15	−0.240383	−0.120192	−	0.992751i	$-0.538351\pi$
$739$	−4829.15	−0.240383	−0.120192	+	0.992751i	$0.538351\pi$
$740$	−44.5158	−0.00221140
$741$	0	0
$742$	−15642.9	−0.773947
$743$	25459.0	1.25707	0.628533	−	0.777783i	$-0.283656\pi$
$743$	25459.0	1.25707	0.628533	+	0.777783i	$0.283656\pi$
$744$	0	0
$745$	1091.87	0.0536953
$746$	−1025.88	−0.0503488
$747$	0	0
$748$	−2531.21	−0.123730
$749$	8271.66	0.403525
$750$	0	0
$751$	−5707.08	−0.277303	−0.138651	−	0.990341i	$-0.544277\pi$
$751$	−5707.08	−0.277303	−0.138651	+	0.990341i	$0.544277\pi$
$752$	−42230.4	−2.04785
$753$	0	0
$754$	−20758.2	−1.00261
$755$	875.054	0.0421808
$756$	0	0
$757$	−1900.91	−0.0912677	−0.0456339	−	0.998958i	$-0.514531\pi$
$757$	−1900.91	−0.0912677	−0.0456339	+	0.998958i	$0.514531\pi$
$758$	32892.6	1.57614
$759$	0	0
$760$	−2479.85	−0.118360
$761$	11583.8	0.551791	0.275896	−	0.961188i	$-0.411026\pi$
$761$	11583.8	0.551791	0.275896	+	0.961188i	$0.411026\pi$
$762$	0	0
$763$	15431.0	0.732162
$764$	6590.40	0.312084
$765$	0	0
$766$	−11017.5	−0.519686
$767$	−14233.1	−0.670051
$768$	0	0
$769$	−26059.7	−1.22202	−0.611012	−	0.791622i	$-0.709237\pi$
$769$	−26059.7	−1.22202	−0.611012	+	0.791622i	$0.709237\pi$
$770$	842.716	0.0394408
$771$	0	0
$772$	−4057.24	−0.189149
$773$	16213.6	0.754413	0.377206	−	0.926129i	$-0.376885\pi$
$773$	16213.6	0.754413	0.377206	+	0.926129i	$0.376885\pi$
$774$	0	0
$775$	4927.85	0.228405
$776$	−8542.92	−0.395197
$777$	0	0
$778$	27267.4	1.25653
$779$	−6435.04	−0.295968
$780$	0	0
$781$	−729.103	−0.0334051
$782$	29403.0	1.34456
$783$	0	0
$784$	−3918.90	−0.178521
$785$	4598.21	0.209066
$786$	0	0
$787$	−1371.34	−0.0621131	−0.0310565	−	0.999518i	$-0.509887\pi$
$787$	−1371.34	−0.0621131	−0.0310565	+	0.999518i	$0.509887\pi$
$788$	−1492.59	−0.0674765
$789$	0	0
$790$	−19168.7	−0.863279
$791$	−1658.20	−0.0745371
$792$	0	0
$793$	−3164.78	−0.141721
$794$	27659.3	1.23626
$795$	0	0
$796$	11695.6	0.520778
$797$	−7991.49	−0.355173	−0.177587	−	0.984105i	$-0.556829\pi$
$797$	−7991.49	−0.355173	−0.177587	+	0.984105i	$0.556829\pi$
$798$	0	0
$799$	−46626.5	−2.06449
$800$	4285.23	0.189382
$801$	0	0
$802$	−1702.32	−0.0749515
$803$	6814.47	0.299474
$804$	0	0
$805$	−3343.49	−0.146388
$806$	15209.3	0.664669
$807$	0	0
$808$	−23379.3	−1.01792
$809$	17661.4	0.767542	0.383771	−	0.923428i	$-0.374625\pi$
$809$	17661.4	0.767542	0.383771	+	0.923428i	$0.374625\pi$
$810$	0	0
$811$	−24180.6	−1.04697	−0.523486	−	0.852034i	$-0.675369\pi$
$811$	−24180.6	−1.04697	−0.523486	+	0.852034i	$0.675369\pi$
$812$	−7814.85	−0.337743
$813$	0	0
$814$	−51.6576	−0.00222432
$815$	−11842.5	−0.508989
$816$	0	0
$817$	−629.526	−0.0269576
$818$	−39146.1	−1.67324
$819$	0	0
$820$	3612.93	0.153865
$821$	23340.9	0.992208	0.496104	−	0.868263i	$-0.334763\pi$
$821$	23340.9	0.992208	0.496104	+	0.868263i	$0.334763\pi$
$822$	0	0
$823$	20630.4	0.873792	0.436896	−	0.899512i	$-0.356078\pi$
$823$	20630.4	0.873792	0.436896	+	0.899512i	$0.356078\pi$
$824$	19521.9	0.825337
$825$	0	0
$826$	−15688.3	−0.660854
$827$	−24113.7	−1.01393	−0.506963	−	0.861968i	$-0.669232\pi$
$827$	−24113.7	−1.01393	−0.506963	+	0.861968i	$0.669232\pi$
$828$	0	0
$829$	41738.9	1.74868	0.874338	−	0.485317i	$-0.161296\pi$
$829$	41738.9	1.74868	0.874338	+	0.485317i	$0.161296\pi$
$830$	21552.1	0.901308
$831$	0	0
$832$	−937.432	−0.0390620
$833$	−4326.85	−0.179972
$834$	0	0
$835$	5396.84	0.223671
$836$	1059.34	0.0438256
$837$	0	0
$838$	−25687.9	−1.05892
$839$	−30403.0	−1.25105	−0.625523	−	0.780205i	$-0.715114\pi$
$839$	−30403.0	−1.25105	−0.625523	+	0.780205i	$0.715114\pi$
$840$	0	0
$841$	47987.8	1.96760
$842$	−41731.9	−1.70805
$843$	0	0
$844$	−4189.77	−0.170874
$845$	8534.90	0.347467
$846$	0	0
$847$	−8982.99	−0.364415
$848$	−51274.7	−2.07639
$849$	0	0
$850$	7694.84	0.310507
$851$	204.952	0.00825579
$852$	0	0
$853$	5900.43	0.236843	0.118421	−	0.992963i	$-0.462217\pi$
$853$	5900.43	0.236843	0.118421	+	0.992963i	$0.462217\pi$
$854$	−3488.33	−0.139776
$855$	0	0
$856$	15858.7	0.633221
$857$	18226.2	0.726480	0.363240	−	0.931696i	$-0.381670\pi$
$857$	18226.2	0.726480	0.363240	+	0.931696i	$0.381670\pi$
$858$	0	0
$859$	−19944.2	−0.792186	−0.396093	−	0.918210i	$-0.629634\pi$
$859$	−19944.2	−0.792186	−0.396093	+	0.918210i	$0.629634\pi$
$860$	353.446	0.0140144
$861$	0	0
$862$	−12438.9	−0.491497
$863$	−30830.3	−1.21608	−0.608038	−	0.793908i	$-0.708043\pi$
$863$	−30830.3	−1.21608	−0.608038	+	0.793908i	$0.708043\pi$
$864$	0	0
$865$	−4406.36	−0.173203
$866$	−7987.70	−0.313433
$867$	0	0
$868$	5725.83	0.223903
$869$	−7597.44	−0.296577
$870$	0	0
$871$	−10598.8	−0.412317
$872$	29584.7	1.14893
$873$	0	0
$874$	−12305.5	−0.476248
$875$	−875.000	−0.0338062
$876$	0	0
$877$	−45885.6	−1.76676	−0.883379	−	0.468659i	$-0.844737\pi$
$877$	−45885.6	−1.76676	−0.883379	+	0.468659i	$0.844737\pi$
$878$	25548.6	0.982033
$879$	0	0
$880$	2762.28	0.105814
$881$	−41132.6	−1.57298	−0.786489	−	0.617605i	$-0.788103\pi$
$881$	−41132.6	−1.57298	−0.786489	+	0.617605i	$0.788103\pi$
$882$	0	0
$883$	19850.0	0.756520	0.378260	−	0.925699i	$-0.376523\pi$
$883$	19850.0	0.756520	0.378260	+	0.925699i	$0.376523\pi$
$884$	8111.58	0.308622
$885$	0	0
$886$	28921.8	1.09667
$887$	−29029.3	−1.09888	−0.549441	−	0.835532i	$-0.685159\pi$
$887$	−29029.3	−1.09888	−0.549441	+	0.835532i	$0.685159\pi$
$888$	0	0
$889$	−11670.5	−0.440288
$890$	12403.6	0.467159
$891$	0	0
$892$	5300.40	0.198958
$893$	19513.8	0.731248
$894$	0	0
$895$	−16888.6	−0.630752
$896$	−10632.2	−0.396425
$897$	0	0
$898$	34015.2	1.26403
$899$	−53029.4	−1.96733
$900$	0	0
$901$	−56612.3	−2.09326
$902$	4192.57	0.154764
$903$	0	0
$904$	−3179.15	−0.116966
$905$	−7177.88	−0.263647
$906$	0	0
$907$	−21029.1	−0.769856	−0.384928	−	0.922947i	$-0.625774\pi$
$907$	−21029.1	−0.769856	−0.384928	+	0.922947i	$0.625774\pi$
$908$	−5809.14	−0.212316
$909$	0	0
$910$	−2700.59	−0.0983777
$911$	19225.5	0.699198	0.349599	−	0.936899i	$-0.386318\pi$
$911$	19225.5	0.699198	0.349599	+	0.936899i	$0.386318\pi$
$912$	0	0
$913$	8542.13	0.309642
$914$	40938.9	1.48155
$915$	0	0
$916$	13204.6	0.476300
$917$	−6242.05	−0.224788
$918$	0	0
$919$	21316.8	0.765154	0.382577	−	0.923924i	$-0.375037\pi$
$919$	21316.8	0.765154	0.382577	+	0.923924i	$0.375037\pi$
$920$	−6410.23	−0.229716
$921$	0	0
$922$	−37465.9	−1.33826
$923$	2336.50	0.0833229
$924$	0	0
$925$	53.6366	0.00190655
$926$	34377.3	1.21999
$927$	0	0
$928$	−46114.1	−1.63122
$929$	−19989.6	−0.705959	−0.352979	−	0.935631i	$-0.614831\pi$
$929$	−19989.6	−0.705959	−0.352979	+	0.935631i	$0.614831\pi$
$930$	0	0
$931$	1810.84	0.0637465
$932$	12564.8	0.441603
$933$	0	0
$934$	−16245.9	−0.569144
$935$	3049.82	0.106674
$936$	0	0
$937$	55676.4	1.94116	0.970580	−	0.240779i	$-0.0774027\pi$
$937$	55676.4	1.94116	0.970580	+	0.240779i	$0.0774027\pi$
$938$	−11682.4	−0.406658
$939$	0	0
$940$	−10956.0	−0.380153
$941$	108.842	0.00377062	0.00188531	−	0.999998i	$-0.499400\pi$
$941$	108.842	0.00377062	0.00188531	+	0.999998i	$0.499400\pi$
$942$	0	0
$943$	−16634.1	−0.574422
$944$	−51423.6	−1.77298
$945$	0	0
$946$	410.150	0.0140963
$947$	7785.95	0.267169	0.133585	−	0.991037i	$-0.457351\pi$
$947$	7785.95	0.267169	0.133585	+	0.991037i	$0.457351\pi$
$948$	0	0
$949$	−21837.8	−0.746982
$950$	−3220.39	−0.109982
$951$	0	0
$952$	−8295.55	−0.282416
$953$	−41445.5	−1.40876	−0.704381	−	0.709822i	$-0.748776\pi$
$953$	−41445.5	−1.40876	−0.704381	+	0.709822i	$0.748776\pi$
$954$	0	0
$955$	−7940.69	−0.269063
$956$	20730.9	0.701345
$957$	0	0
$958$	−56792.5	−1.91532
$959$	2802.98	0.0943825
$960$	0	0
$961$	9062.92	0.304217
$962$	165.543	0.00554817
$963$	0	0
$964$	−15588.7	−0.520826
$965$	4888.52	0.163075
$966$	0	0
$967$	39155.0	1.30211	0.651055	−	0.759030i	$-0.274327\pi$
$967$	39155.0	1.30211	0.651055	+	0.759030i	$0.274327\pi$
$968$	−17222.4	−0.571849
$969$	0	0
$970$	−11094.0	−0.367224
$971$	43440.8	1.43572	0.717859	−	0.696189i	$-0.245122\pi$
$971$	43440.8	1.43572	0.717859	+	0.696189i	$0.245122\pi$
$972$	0	0
$973$	3605.35	0.118789
$974$	12252.0	0.403061
$975$	0	0
$976$	−11434.2	−0.374999
$977$	−11297.8	−0.369957	−0.184978	−	0.982743i	$-0.559222\pi$
$977$	−11297.8	−0.369957	−0.184978	+	0.982743i	$0.559222\pi$
$978$	0	0
$979$	4916.15	0.160491
$980$	−1016.69	−0.0331398
$981$	0	0
$982$	−8772.66	−0.285078
$983$	−10865.9	−0.352563	−0.176282	−	0.984340i	$-0.556407\pi$
$983$	−10865.9	−0.352563	−0.176282	+	0.984340i	$0.556407\pi$
$984$	0	0
$985$	1798.41	0.0581747
$986$	−82805.5	−2.67451
$987$	0	0
$988$	−3394.80	−0.109315
$989$	−1627.28	−0.0523199
$990$	0	0
$991$	13884.1	0.445048	0.222524	−	0.974927i	$-0.428570\pi$
$991$	13884.1	0.445048	0.222524	+	0.974927i	$0.428570\pi$
$992$	33787.1	1.08139
$993$	0	0
$994$	2575.38	0.0821792
$995$	−14091.9	−0.448987
$996$	0	0
$997$	31665.7	1.00588	0.502940	−	0.864321i	$-0.332252\pi$
$997$	31665.7	1.00588	0.502940	+	0.864321i	$0.332252\pi$
$998$	30490.7	0.967099
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.a.p.1.1		3
3.2	odd	2		35.4.a.c.1.3	✓	3
5.4	even	2		1575.4.a.ba.1.3		3
7.6	odd	2		2205.4.a.bm.1.1		3
12.11	even	2		560.4.a.u.1.2		3
15.2	even	4		175.4.b.e.99.5		6
15.8	even	4		175.4.b.e.99.2		6
15.14	odd	2		175.4.a.f.1.1		3
21.2	odd	6		245.4.e.m.116.1		6
21.5	even	6		245.4.e.n.116.1		6
21.11	odd	6		245.4.e.m.226.1		6
21.17	even	6		245.4.e.n.226.1		6
21.20	even	2		245.4.a.l.1.3		3
24.5	odd	2		2240.4.a.bt.1.2		3
24.11	even	2		2240.4.a.bv.1.2		3
105.104	even	2		1225.4.a.y.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.c.1.3	✓	3	3.2	odd	2
175.4.a.f.1.1		3	15.14	odd	2
175.4.b.e.99.2		6	15.8	even	4
175.4.b.e.99.5		6	15.2	even	4
245.4.a.l.1.3		3	21.20	even	2
245.4.e.m.116.1		6	21.2	odd	6
245.4.e.m.226.1		6	21.11	odd	6
245.4.e.n.116.1		6	21.5	even	6
245.4.e.n.226.1		6	21.17	even	6
315.4.a.p.1.1		3	1.1	even	1	trivial
560.4.a.u.1.2		3	12.11	even	2
1225.4.a.y.1.1		3	105.104	even	2
1575.4.a.ba.1.3		3	5.4	even	2
2205.4.a.bm.1.1		3	7.6	odd	2
2240.4.a.bt.1.2		3	24.5	odd	2
2240.4.a.bv.1.2		3	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q-3.48565 q^{2} +4.14976 q^{4} -5.00000 q^{5} +7.00000 q^{7} +13.4206 q^{8} +17.4283 q^{10} +6.90764 q^{11} -22.1364 q^{13} -24.3996 q^{14} -79.9776 q^{16} -88.3030 q^{17} +36.9560 q^{19} -20.7488 q^{20} -24.0776 q^{22} +95.5283 q^{23} +25.0000 q^{25} +77.1598 q^{26} +29.0483 q^{28} -269.029 q^{29} +197.114 q^{31} +171.409 q^{32} +307.793 q^{34} -35.0000 q^{35} +2.14546 q^{37} -128.816 q^{38} -67.1029 q^{40} -174.127 q^{41} -17.0345 q^{43} +28.6650 q^{44} -332.978 q^{46} +528.029 q^{47} +49.0000 q^{49} -87.1413 q^{50} -91.8608 q^{52} +641.114 q^{53} -34.5382 q^{55} +93.9441 q^{56} +937.742 q^{58} +642.975 q^{59} +142.967 q^{61} -687.070 q^{62} +42.3480 q^{64} +110.682 q^{65} +478.797 q^{67} -366.436 q^{68} +121.998 q^{70} -105.550 q^{71} +986.512 q^{73} -7.47834 q^{74} +153.358 q^{76} +48.3534 q^{77} -1099.86 q^{79} +399.888 q^{80} +606.947 q^{82} +1236.62 q^{83} +441.515 q^{85} +59.3763 q^{86} +92.7045 q^{88} +711.698 q^{89} -154.955 q^{91} +396.420 q^{92} -1840.52 q^{94} -184.780 q^{95} -636.553 q^{97} -170.797 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 13 q^{4} - 15 q^{5} + 21 q^{7} + 15 q^{8} - 15 q^{10} + 74 q^{11} + 44 q^{13} + 21 q^{14} - 79 q^{16} + 52 q^{17} + 168 q^{19} - 65 q^{20} + 184 q^{22} + 124 q^{23} + 75 q^{25} + 446 q^{26}+ \cdots + 147 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 13 * q^4 - 15 * q^5 + 21 * q^7 + 15 * q^8 - 15 * q^10 + 74 * q^11 + 44 * q^13 + 21 * q^14 - 79 * q^16 + 52 * q^17 + 168 * q^19 - 65 * q^20 + 184 * q^22 + 124 * q^23 + 75 * q^25 + 446 * q^26 + 91 * q^28 - 332 * q^29 + 320 * q^31 + 183 * q^32 + 582 * q^34 - 105 * q^35 - 54 * q^37 + 460 * q^38 - 75 * q^40 - 362 * q^41 - 16 * q^43 + 264 * q^44 - 336 * q^46 + 730 * q^47 + 147 * q^49 + 75 * q^50 + 1202 * q^52 - 110 * q^53 - 370 * q^55 + 105 * q^56 + 450 * q^58 + 180 * q^59 + 1222 * q^61 - 464 * q^62 - 391 * q^64 - 220 * q^65 + 204 * q^67 - 918 * q^68 - 105 * q^70 + 136 * q^71 + 310 * q^73 - 502 * q^74 + 1796 * q^76 + 518 * q^77 - 1034 * q^79 + 395 * q^80 - 6 * q^82 + 1660 * q^83 - 260 * q^85 - 764 * q^86 - 20 * q^88 - 242 * q^89 + 308 * q^91 - 96 * q^92 - 1108 * q^94 - 840 * q^95 + 100 * q^97 + 147 * q^98

Introduction

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