LMFDB - Embedded newform 350.6.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,6,Mod(1,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-8,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$56.1343369345$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1129}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 282 $ x^2 - x - 282
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-16.3003$ of defining polynomial
Character	$\chi$	$=$	350.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +14.3003 q^{3} +16.0000 q^{4} -57.2012 q^{6} -49.0000 q^{7} -64.0000 q^{8} -38.5015 q^{9} +425.904 q^{11} +228.805 q^{12} -399.303 q^{13} +196.000 q^{14} +256.000 q^{16} -1751.52 q^{17} +154.006 q^{18} +2874.23 q^{19} -700.715 q^{21} -1703.62 q^{22} +2313.43 q^{23} -915.219 q^{24} +1597.21 q^{26} -4025.56 q^{27} -784.000 q^{28} -2127.93 q^{29} -10262.5 q^{31} -1024.00 q^{32} +6090.55 q^{33} +7006.08 q^{34} -616.024 q^{36} +7266.05 q^{37} -11496.9 q^{38} -5710.16 q^{39} -5893.44 q^{41} +2802.86 q^{42} -20157.7 q^{43} +6814.46 q^{44} -9253.72 q^{46} -20056.5 q^{47} +3660.88 q^{48} +2401.00 q^{49} -25047.2 q^{51} -6388.85 q^{52} +33954.9 q^{53} +16102.2 q^{54} +3136.00 q^{56} +41102.4 q^{57} +8511.71 q^{58} -4319.34 q^{59} -12253.6 q^{61} +41050.1 q^{62} +1886.57 q^{63} +4096.00 q^{64} -24362.2 q^{66} +17533.8 q^{67} -28024.3 q^{68} +33082.7 q^{69} +1658.18 q^{71} +2464.10 q^{72} +8246.91 q^{73} -29064.2 q^{74} +45987.7 q^{76} -20869.3 q^{77} +22840.6 q^{78} -9168.61 q^{79} -48210.8 q^{81} +23573.7 q^{82} -95203.2 q^{83} -11211.4 q^{84} +80631.0 q^{86} -30430.0 q^{87} -27257.8 q^{88} -14441.8 q^{89} +19565.9 q^{91} +37014.9 q^{92} -146757. q^{93} +80226.1 q^{94} -14643.5 q^{96} -62132.4 q^{97} -9604.00 q^{98} -16397.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} - 5 q^{3} + 32 q^{4} + 20 q^{6} - 98 q^{7} - 128 q^{8} + 91 q^{9} + 415 q^{11} - 80 q^{12} - 429 q^{13} + 392 q^{14} + 512 q^{16} - 1319 q^{17} - 364 q^{18} + 1918 q^{19} + 245 q^{21} - 1660 q^{22}+ \cdots - 17810 q^{99}+O(q^{100}) $ 2 * q - 8 * q^2 - 5 * q^3 + 32 * q^4 + 20 * q^6 - 98 * q^7 - 128 * q^8 + 91 * q^9 + 415 * q^11 - 80 * q^12 - 429 * q^13 + 392 * q^14 + 512 * q^16 - 1319 * q^17 - 364 * q^18 + 1918 * q^19 + 245 * q^21 - 1660 * q^22 + 1334 * q^23 + 320 * q^24 + 1716 * q^26 - 1835 * q^27 - 1568 * q^28 - 1131 * q^29 - 5472 * q^31 - 2048 * q^32 + 6301 * q^33 + 5276 * q^34 + 1456 * q^36 + 9156 * q^37 - 7672 * q^38 - 5137 * q^39 - 7822 * q^41 - 980 * q^42 - 2078 * q^43 + 6640 * q^44 - 5336 * q^46 + 8507 * q^47 - 1280 * q^48 + 4802 * q^49 - 33395 * q^51 - 6864 * q^52 + 34242 * q^53 + 7340 * q^54 + 6272 * q^56 + 59558 * q^57 + 4524 * q^58 + 6952 * q^59 - 45138 * q^61 + 21888 * q^62 - 4459 * q^63 + 8192 * q^64 - 25204 * q^66 + 54556 * q^67 - 21104 * q^68 + 51986 * q^69 - 62272 * q^71 - 5824 * q^72 - 40896 * q^73 - 36624 * q^74 + 30688 * q^76 - 20335 * q^77 + 20548 * q^78 + 62069 * q^79 - 121958 * q^81 + 31288 * q^82 - 189600 * q^83 + 3920 * q^84 + 8312 * q^86 - 49671 * q^87 - 26560 * q^88 + 64190 * q^89 + 21021 * q^91 + 21344 * q^92 - 239216 * q^93 - 34028 * q^94 + 5120 * q^96 - 155547 * q^97 - 19208 * q^98 - 17810 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	14.3003	0.917365	0.458682	−	0.888600i	$-0.348322\pi$
$3$	14.3003	0.917365	0.458682	+	0.888600i	$0.348322\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−57.2012	−0.648675
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−64.0000	−0.353553
$9$	−38.5015	−0.158442
$10$	0	0
$11$	425.904	1.06128	0.530640	−	0.847597i	$-0.321952\pi$
$11$	425.904	1.06128	0.530640	+	0.847597i	$0.321952\pi$
$12$	228.805	0.458682
$13$	−399.303	−0.655307	−0.327653	−	0.944798i	$-0.606258\pi$
$13$	−399.303	−0.655307	−0.327653	+	0.944798i	$0.606258\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	−1751.52	−1.46992	−0.734958	−	0.678112i	$-0.762798\pi$
$17$	−1751.52	−1.46992	−0.734958	+	0.678112i	$0.762798\pi$
$18$	154.006	0.112036
$19$	2874.23	1.82658	0.913289	−	0.407313i	$-0.133534\pi$
$19$	2874.23	1.82658	0.913289	+	0.407313i	$0.133534\pi$
$20$	0	0
$21$	−700.715	−0.346731
$22$	−1703.62	−0.750438
$23$	2313.43	0.911878	0.455939	−	0.890011i	$-0.349304\pi$
$23$	2313.43	0.911878	0.455939	+	0.890011i	$0.349304\pi$
$24$	−915.219	−0.324337
$25$	0	0
$26$	1597.21	0.463372
$27$	−4025.56	−1.06271
$28$	−784.000	−0.188982
$29$	−2127.93	−0.469853	−0.234926	−	0.972013i	$-0.575485\pi$
$29$	−2127.93	−0.469853	−0.234926	+	0.972013i	$0.575485\pi$
$30$	0	0
$31$	−10262.5	−1.91801	−0.959004	−	0.283393i	$-0.908540\pi$
$31$	−10262.5	−1.91801	−0.959004	+	0.283393i	$0.908540\pi$
$32$	−1024.00	−0.176777
$33$	6090.55	0.973580
$34$	7006.08	1.03939
$35$	0	0
$36$	−616.024	−0.0792212
$37$	7266.05	0.872557	0.436279	−	0.899812i	$-0.356296\pi$
$37$	7266.05	0.872557	0.436279	+	0.899812i	$0.356296\pi$
$38$	−11496.9	−1.29159
$39$	−5710.16	−0.601155
$40$	0	0
$41$	−5893.44	−0.547531	−0.273766	−	0.961796i	$-0.588269\pi$
$41$	−5893.44	−0.547531	−0.273766	+	0.961796i	$0.588269\pi$
$42$	2802.86	0.245176
$43$	−20157.7	−1.66253	−0.831267	−	0.555873i	$-0.812384\pi$
$43$	−20157.7	−1.66253	−0.831267	+	0.555873i	$0.812384\pi$
$44$	6814.46	0.530640
$45$	0	0
$46$	−9253.72	−0.644795
$47$	−20056.5	−1.32438	−0.662188	−	0.749338i	$-0.730372\pi$
$47$	−20056.5	−1.32438	−0.662188	+	0.749338i	$0.730372\pi$
$48$	3660.88	0.229341
$49$	2401.00	0.142857
$50$	0	0
$51$	−25047.2	−1.34845
$52$	−6388.85	−0.327653
$53$	33954.9	1.66040	0.830200	−	0.557466i	$-0.188226\pi$
$53$	33954.9	1.66040	0.830200	+	0.557466i	$0.188226\pi$
$54$	16102.2	0.751452
$55$	0	0
$56$	3136.00	0.133631
$57$	41102.4	1.67564
$58$	8511.71	0.332236
$59$	−4319.34	−0.161543	−0.0807713	−	0.996733i	$-0.525738\pi$
$59$	−4319.34	−0.161543	−0.0807713	+	0.996733i	$0.525738\pi$
$60$	0	0
$61$	−12253.6	−0.421638	−0.210819	−	0.977525i	$-0.567613\pi$
$61$	−12253.6	−0.421638	−0.210819	+	0.977525i	$0.567613\pi$
$62$	41050.1	1.35624
$63$	1886.57	0.0598856
$64$	4096.00	0.125000
$65$	0	0
$66$	−24362.2	−0.688425
$67$	17533.8	0.477188	0.238594	−	0.971119i	$-0.423313\pi$
$67$	17533.8	0.477188	0.238594	+	0.971119i	$0.423313\pi$
$68$	−28024.3	−0.734958
$69$	33082.7	0.836524
$70$	0	0
$71$	1658.18	0.0390379	0.0195189	−	0.999809i	$-0.493787\pi$
$71$	1658.18	0.0390379	0.0195189	+	0.999809i	$0.493787\pi$
$72$	2464.10	0.0560178
$73$	8246.91	0.181127	0.0905637	−	0.995891i	$-0.471133\pi$
$73$	8246.91	0.181127	0.0905637	+	0.995891i	$0.471133\pi$
$74$	−29064.2	−0.616991
$75$	0	0
$76$	45987.7	0.913289
$77$	−20869.3	−0.401126
$78$	22840.6	0.425081
$79$	−9168.61	−0.165286	−0.0826430	−	0.996579i	$-0.526336\pi$
$79$	−9168.61	−0.165286	−0.0826430	+	0.996579i	$0.526336\pi$
$80$	0	0
$81$	−48210.8	−0.816454
$82$	23573.7	0.387163
$83$	−95203.2	−1.51690	−0.758449	−	0.651732i	$-0.774043\pi$
$83$	−95203.2	−1.51690	−0.758449	+	0.651732i	$0.774043\pi$
$84$	−11211.4	−0.173366
$85$	0	0
$86$	80631.0	1.17559
$87$	−30430.0	−0.431026
$88$	−27257.8	−0.375219
$89$	−14441.8	−0.193262	−0.0966311	−	0.995320i	$-0.530807\pi$
$89$	−14441.8	−0.193262	−0.0966311	+	0.995320i	$0.530807\pi$
$90$	0	0
$91$	19565.9	0.247683
$92$	37014.9	0.455939
$93$	−146757.	−1.75951
$94$	80226.1	0.936475
$95$	0	0
$96$	−14643.5	−0.162169
$97$	−62132.4	−0.670485	−0.335242	−	0.942132i	$-0.608818\pi$
$97$	−62132.4	−0.670485	−0.335242	+	0.942132i	$0.608818\pi$
$98$	−9604.00	−0.101015
$99$	−16397.9	−0.168152
$100$	0	0
$101$	−108138.	−1.05481	−0.527404	−	0.849615i	$-0.676835\pi$
$101$	−108138.	−1.05481	−0.527404	+	0.849615i	$0.676835\pi$
$102$	100189.	0.953498
$103$	−138034.	−1.28201	−0.641005	−	0.767536i	$-0.721482\pi$
$103$	−138034.	−1.28201	−0.641005	+	0.767536i	$0.721482\pi$
$104$	25555.4	0.231686
$105$	0	0
$106$	−135820.	−1.17408
$107$	−6189.61	−0.0522641	−0.0261321	−	0.999658i	$-0.508319\pi$
$107$	−6189.61	−0.0522641	−0.0261321	+	0.999658i	$0.508319\pi$
$108$	−64408.9	−0.531357
$109$	−68652.9	−0.553468	−0.276734	−	0.960947i	$-0.589252\pi$
$109$	−68652.9	−0.553468	−0.276734	+	0.960947i	$0.589252\pi$
$110$	0	0
$111$	103907.	0.800453
$112$	−12544.0	−0.0944911
$113$	62835.9	0.462926	0.231463	−	0.972844i	$-0.425649\pi$
$113$	62835.9	0.462926	0.231463	+	0.972844i	$0.425649\pi$
$114$	−164410.	−1.18485
$115$	0	0
$116$	−34046.8	−0.234926
$117$	15373.8	0.103828
$118$	17277.4	0.114228
$119$	85824.4	0.555576
$120$	0	0
$121$	20343.1	0.126315
$122$	49014.5	0.298143
$123$	−84277.9	−0.502286
$124$	−164201.	−0.959004
$125$	0	0
$126$	−7546.29	−0.0423455
$127$	198049.	1.08959	0.544794	−	0.838570i	$-0.316608\pi$
$127$	198049.	1.08959	0.544794	+	0.838570i	$0.316608\pi$
$128$	−16384.0	−0.0883883
$129$	−288262.	−1.52515
$130$	0	0
$131$	−285132.	−1.45167	−0.725835	−	0.687868i	$-0.758547\pi$
$131$	−285132.	−1.45167	−0.725835	+	0.687868i	$0.758547\pi$
$132$	97448.8	0.486790
$133$	−140837.	−0.690381
$134$	−70135.3	−0.337423
$135$	0	0
$136$	112097.	0.519694
$137$	−406979.	−1.85255	−0.926277	−	0.376844i	$-0.877009\pi$
$137$	−406979.	−1.85255	−0.926277	+	0.376844i	$0.877009\pi$
$138$	−132331.	−0.591512
$139$	13530.4	0.0593985	0.0296992	−	0.999559i	$-0.490545\pi$
$139$	13530.4	0.0593985	0.0296992	+	0.999559i	$0.490545\pi$
$140$	0	0
$141$	−286814.	−1.21494
$142$	−6632.72	−0.0276039
$143$	−170065.	−0.695464
$144$	−9856.38	−0.0396106
$145$	0	0
$146$	−32987.6	−0.128076
$147$	34335.0	0.131052
$148$	116257.	0.436279
$149$	77129.8	0.284614	0.142307	−	0.989823i	$-0.454548\pi$
$149$	77129.8	0.284614	0.142307	+	0.989823i	$0.454548\pi$
$150$	0	0
$151$	−420464.	−1.50067	−0.750336	−	0.661056i	$-0.770108\pi$
$151$	−420464.	−1.50067	−0.750336	+	0.661056i	$0.770108\pi$
$152$	−183951.	−0.645793
$153$	67436.1	0.232897
$154$	83477.2	0.283639
$155$	0	0
$156$	−91362.5	−0.300577
$157$	481908.	1.56032	0.780162	−	0.625577i	$-0.215136\pi$
$157$	481908.	1.56032	0.780162	+	0.625577i	$0.215136\pi$
$158$	36674.4	0.116875
$159$	485565.	1.52319
$160$	0	0
$161$	−113358.	−0.344657
$162$	192843.	0.577320
$163$	−282039.	−0.831458	−0.415729	−	0.909488i	$-0.636474\pi$
$163$	−282039.	−0.831458	−0.415729	+	0.909488i	$0.636474\pi$
$164$	−94295.0	−0.273766
$165$	0	0
$166$	380813.	1.07261
$167$	−131971.	−0.366173	−0.183086	−	0.983097i	$-0.558609\pi$
$167$	−131971.	−0.366173	−0.183086	+	0.983097i	$0.558609\pi$
$168$	44845.7	0.122588
$169$	−211850.	−0.570573
$170$	0	0
$171$	−110662.	−0.289407
$172$	−322524.	−0.831267
$173$	−55501.5	−0.140990	−0.0704952	−	0.997512i	$-0.522458\pi$
$173$	−55501.5	−0.140990	−0.0704952	+	0.997512i	$0.522458\pi$
$174$	121720.	0.304782
$175$	0	0
$176$	109031.	0.265320
$177$	−61767.8	−0.148193
$178$	57767.3	0.136657
$179$	20768.3	0.0484471	0.0242236	−	0.999707i	$-0.492289\pi$
$179$	20768.3	0.0484471	0.0242236	+	0.999707i	$0.492289\pi$
$180$	0	0
$181$	−255866.	−0.580520	−0.290260	−	0.956948i	$-0.593742\pi$
$181$	−255866.	−0.580520	−0.290260	+	0.956948i	$0.593742\pi$
$182$	−78263.4	−0.175138
$183$	−175230.	−0.386796
$184$	−148059.	−0.322397
$185$	0	0
$186$	587029.	1.24416
$187$	−745979.	−1.55999
$188$	−320904.	−0.662188
$189$	197252.	0.401668
$190$	0	0
$191$	649260.	1.28776	0.643881	−	0.765126i	$-0.277323\pi$
$191$	649260.	1.28776	0.643881	+	0.765126i	$0.277323\pi$
$192$	58574.0	0.114671
$193$	768642.	1.48536	0.742678	−	0.669649i	$-0.233555\pi$
$193$	768642.	1.48536	0.742678	+	0.669649i	$0.233555\pi$
$194$	248530.	0.474104
$195$	0	0
$196$	38416.0	0.0714286
$197$	166160.	0.305044	0.152522	−	0.988300i	$-0.451261\pi$
$197$	166160.	0.305044	0.152522	+	0.988300i	$0.451261\pi$
$198$	65591.7	0.118901
$199$	−74756.1	−0.133818	−0.0669089	−	0.997759i	$-0.521314\pi$
$199$	−74756.1	−0.133818	−0.0669089	+	0.997759i	$0.521314\pi$
$200$	0	0
$201$	250739.	0.437756
$202$	432551.	0.745862
$203$	104268.	0.177588
$204$	−400756.	−0.674225
$205$	0	0
$206$	552134.	0.906518
$207$	−89070.5	−0.144480
$208$	−102222.	−0.163827
$209$	1.22415e6	1.93851
$210$	0	0
$211$	−1.08646e6	−1.67999	−0.839997	−	0.542591i	$-0.817443\pi$
$211$	−1.08646e6	−1.67999	−0.839997	+	0.542591i	$0.817443\pi$
$212$	543278.	0.830200
$213$	23712.5	0.0358120
$214$	24758.4	0.0369563
$215$	0	0
$216$	257636.	0.375726
$217$	502864.	0.724939
$218$	274612.	0.391361
$219$	117933.	0.166160
$220$	0	0
$221$	699387.	0.963246
$222$	−415627.	−0.566006
$223$	430809.	0.580126	0.290063	−	0.957008i	$-0.406324\pi$
$223$	430809.	0.580126	0.290063	+	0.957008i	$0.406324\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	−251343.	−0.327338
$227$	228724.	0.294610	0.147305	−	0.989091i	$-0.452940\pi$
$227$	228724.	0.294610	0.147305	+	0.989091i	$0.452940\pi$
$228$	657638.	0.837819
$229$	−1.01797e6	−1.28276	−0.641378	−	0.767225i	$-0.721637\pi$
$229$	−1.01797e6	−1.28276	−0.641378	+	0.767225i	$0.721637\pi$
$230$	0	0
$231$	−298437.	−0.367979
$232$	136187.	0.166118
$233$	236710.	0.285645	0.142822	−	0.989748i	$-0.454382\pi$
$233$	236710.	0.285645	0.142822	+	0.989748i	$0.454382\pi$
$234$	−61495.1	−0.0734177
$235$	0	0
$236$	−69109.4	−0.0807713
$237$	−131114.	−0.151627
$238$	−343298.	−0.392852
$239$	1.73994e6	1.97034	0.985169	−	0.171587i	$-0.0548894\pi$
$239$	1.73994e6	1.97034	0.985169	+	0.171587i	$0.0548894\pi$
$240$	0	0
$241$	−332840.	−0.369142	−0.184571	−	0.982819i	$-0.559090\pi$
$241$	−332840.	−0.369142	−0.184571	+	0.982819i	$0.559090\pi$
$242$	−81372.4	−0.0893180
$243$	288781.	0.313728
$244$	−196058.	−0.210819
$245$	0	0
$246$	337112.	0.355170
$247$	−1.14769e6	−1.19697
$248$	656802.	0.678118
$249$	−1.36143e6	−1.39155
$250$	0	0
$251$	1.88585e6	1.88939	0.944696	−	0.327948i	$-0.106357\pi$
$251$	1.88585e6	1.88939	0.944696	+	0.327948i	$0.106357\pi$
$252$	30185.2	0.0299428
$253$	985298.	0.967757
$254$	−792194.	−0.770455
$255$	0	0
$256$	65536.0	0.0625000
$257$	−432608.	−0.408566	−0.204283	−	0.978912i	$-0.565486\pi$
$257$	−432608.	−0.408566	−0.204283	+	0.978912i	$0.565486\pi$
$258$	1.15305e6	1.07844
$259$	−356036.	−0.329796
$260$	0	0
$261$	81928.4	0.0744446
$262$	1.14053e6	1.02649
$263$	1.20388e6	1.07323	0.536614	−	0.843828i	$-0.319703\pi$
$263$	1.20388e6	1.07323	0.536614	+	0.843828i	$0.319703\pi$
$264$	−389795.	−0.344213
$265$	0	0
$266$	563350.	0.488173
$267$	−206522.	−0.177292
$268$	280541.	0.238594
$269$	548133.	0.461855	0.230927	−	0.972971i	$-0.425824\pi$
$269$	548133.	0.461855	0.230927	+	0.972971i	$0.425824\pi$
$270$	0	0
$271$	−906590.	−0.749873	−0.374936	−	0.927051i	$-0.622335\pi$
$271$	−906590.	−0.749873	−0.374936	+	0.927051i	$0.622335\pi$
$272$	−448389.	−0.367479
$273$	279798.	0.227215
$274$	1.62792e6	1.30995
$275$	0	0
$276$	529324.	0.418262
$277$	−1.35338e6	−1.05979	−0.529895	−	0.848064i	$-0.677769\pi$
$277$	−1.35338e6	−1.05979	−0.529895	+	0.848064i	$0.677769\pi$
$278$	−54121.8	−0.0420011
$279$	395123.	0.303894
$280$	0	0
$281$	752309.	0.568369	0.284185	−	0.958770i	$-0.408277\pi$
$281$	752309.	0.568369	0.284185	+	0.958770i	$0.408277\pi$
$282$	1.14726e6	0.859089
$283$	820710.	0.609149	0.304575	−	0.952489i	$-0.401486\pi$
$283$	820710.	0.609149	0.304575	+	0.952489i	$0.401486\pi$
$284$	26530.9	0.0195189
$285$	0	0
$286$	680259.	0.491767
$287$	288778.	0.206947
$288$	39425.5	0.0280089
$289$	1.64796e6	1.16065
$290$	0	0
$291$	−888512.	−0.615079
$292$	131951.	0.0905637
$293$	1.92239e6	1.30819	0.654097	−	0.756411i	$-0.273049\pi$
$293$	1.92239e6	1.30819	0.654097	+	0.756411i	$0.273049\pi$
$294$	−137340.	−0.0926678
$295$	0	0
$296$	−465027.	−0.308496
$297$	−1.71450e6	−1.12784
$298$	−308519.	−0.201253
$299$	−923760.	−0.597559
$300$	0	0
$301$	987729.	0.628379
$302$	1.68185e6	1.06114
$303$	−1.54640e6	−0.967643
$304$	735804.	0.456644
$305$	0	0
$306$	−269744.	−0.164683
$307$	−223229.	−0.135177	−0.0675887	−	0.997713i	$-0.521531\pi$
$307$	−223229.	−0.135177	−0.0675887	+	0.997713i	$0.521531\pi$
$308$	−333909.	−0.200563
$309$	−1.97392e6	−1.17607
$310$	0	0
$311$	2.29818e6	1.34736	0.673678	−	0.739025i	$-0.264714\pi$
$311$	2.29818e6	1.34736	0.673678	+	0.739025i	$0.264714\pi$
$312$	365450.	0.212540
$313$	1.62881e6	0.939744	0.469872	−	0.882735i	$-0.344300\pi$
$313$	1.62881e6	0.939744	0.469872	+	0.882735i	$0.344300\pi$
$314$	−1.92763e6	−1.10332
$315$	0	0
$316$	−146698.	−0.0826430
$317$	−1.13620e6	−0.635049	−0.317524	−	0.948250i	$-0.602851\pi$
$317$	−1.13620e6	−0.635049	−0.317524	+	0.948250i	$0.602851\pi$
$318$	−1.94226e6	−1.07706
$319$	−906293.	−0.498645
$320$	0	0
$321$	−88513.3	−0.0479453
$322$	453432.	0.243710
$323$	−5.03428e6	−2.68492
$324$	−771372.	−0.408227
$325$	0	0
$326$	1.12816e6	0.587930
$327$	−981757.	−0.507732
$328$	377180.	0.193582
$329$	982770.	0.500567
$330$	0	0
$331$	−3.77549e6	−1.89410	−0.947051	−	0.321084i	$-0.895953\pi$
$331$	−3.77549e6	−1.89410	−0.947051	+	0.321084i	$0.895953\pi$
$332$	−1.52325e6	−0.758449
$333$	−279754.	−0.138250
$334$	527882.	0.258923
$335$	0	0
$336$	−179383.	−0.0866828
$337$	−3.74913e6	−1.79827	−0.899136	−	0.437669i	$-0.855804\pi$
$337$	−3.74913e6	−1.79827	−0.899136	+	0.437669i	$0.855804\pi$
$338$	847400.	0.403456
$339$	898571.	0.424672
$340$	0	0
$341$	−4.37085e6	−2.03554
$342$	442649.	0.204642
$343$	−117649.	−0.0539949
$344$	1.29010e6	0.587795
$345$	0	0
$346$	222006.	0.0996953
$347$	1.14843e6	0.512012	0.256006	−	0.966675i	$-0.417593\pi$
$347$	1.14843e6	0.512012	0.256006	+	0.966675i	$0.417593\pi$
$348$	−486880.	−0.215513
$349$	4.03275e6	1.77230	0.886151	−	0.463396i	$-0.153369\pi$
$349$	4.03275e6	1.77230	0.886151	+	0.463396i	$0.153369\pi$
$350$	0	0
$351$	1.60742e6	0.696403
$352$	−436126.	−0.187610
$353$	−652660.	−0.278773	−0.139386	−	0.990238i	$-0.544513\pi$
$353$	−652660.	−0.278773	−0.139386	+	0.990238i	$0.544513\pi$
$354$	247071.	0.104789
$355$	0	0
$356$	−231069.	−0.0966311
$357$	1.22732e6	0.509666
$358$	−83073.2	−0.0342573
$359$	−2.84604e6	−1.16548	−0.582741	−	0.812658i	$-0.698020\pi$
$359$	−2.84604e6	−1.16548	−0.582741	+	0.812658i	$0.698020\pi$
$360$	0	0
$361$	5.78512e6	2.33639
$362$	1.02347e6	0.410489
$363$	290912.	0.115877
$364$	313054.	0.123841
$365$	0	0
$366$	700921.	0.273506
$367$	2.65232e6	1.02792	0.513962	−	0.857813i	$-0.328177\pi$
$367$	2.65232e6	1.02792	0.513962	+	0.857813i	$0.328177\pi$
$368$	592238.	0.227969
$369$	226906.	0.0867521
$370$	0	0
$371$	−1.66379e6	−0.627572
$372$	−2.34812e6	−0.879756
$373$	−2.83283e6	−1.05426	−0.527132	−	0.849784i	$-0.676733\pi$
$373$	−2.83283e6	−1.05426	−0.527132	+	0.849784i	$0.676733\pi$
$374$	2.98392e6	1.10308
$375$	0	0
$376$	1.28362e6	0.468237
$377$	849688.	0.307898
$378$	−789009.	−0.284022
$379$	−1.20678e6	−0.431551	−0.215775	−	0.976443i	$-0.569228\pi$
$379$	−1.20678e6	−0.431551	−0.215775	+	0.976443i	$0.569228\pi$
$380$	0	0
$381$	2.83215e6	0.999550
$382$	−2.59704e6	−0.910585
$383$	2.22793e6	0.776076	0.388038	−	0.921643i	$-0.373153\pi$
$383$	2.22793e6	0.776076	0.388038	+	0.921643i	$0.373153\pi$
$384$	−234296.	−0.0810843
$385$	0	0
$386$	−3.07457e6	−1.05031
$387$	776103.	0.263416
$388$	−994119.	−0.335242
$389$	−3.74004e6	−1.25315	−0.626574	−	0.779362i	$-0.715543\pi$
$389$	−3.74004e6	−1.25315	−0.626574	+	0.779362i	$0.715543\pi$
$390$	0	0
$391$	−4.05202e6	−1.34038
$392$	−153664.	−0.0505076
$393$	−4.07748e6	−1.33171
$394$	−664642.	−0.215699
$395$	0	0
$396$	−262367.	−0.0840758
$397$	2.90335e6	0.924533	0.462267	−	0.886741i	$-0.347036\pi$
$397$	2.90335e6	0.924533	0.462267	+	0.886741i	$0.347036\pi$
$398$	299024.	0.0946235
$399$	−2.01402e6	−0.633331
$400$	0	0
$401$	1.43777e6	0.446506	0.223253	−	0.974761i	$-0.428332\pi$
$401$	1.43777e6	0.446506	0.223253	+	0.974761i	$0.428332\pi$
$402$	−1.00296e6	−0.309540
$403$	4.09786e6	1.25688
$404$	−1.73020e6	−0.527404
$405$	0	0
$406$	−417074.	−0.125573
$407$	3.09464e6	0.926027
$408$	1.60302e6	0.476749
$409$	−4.78825e6	−1.41536	−0.707682	−	0.706531i	$-0.750259\pi$
$409$	−4.78825e6	−1.41536	−0.707682	+	0.706531i	$0.750259\pi$
$410$	0	0
$411$	−5.81992e6	−1.69947
$412$	−2.20854e6	−0.641005
$413$	211648.	0.0610574
$414$	356282.	0.102163
$415$	0	0
$416$	408887.	0.115843
$417$	193489.	0.0544900
$418$	−4.89659e6	−1.37073
$419$	−2.08411e6	−0.579943	−0.289972	−	0.957035i	$-0.593646\pi$
$419$	−2.08411e6	−0.579943	−0.289972	+	0.957035i	$0.593646\pi$
$420$	0	0
$421$	−1.00260e6	−0.275690	−0.137845	−	0.990454i	$-0.544018\pi$
$421$	−1.00260e6	−0.275690	−0.137845	+	0.990454i	$0.544018\pi$
$422$	4.34584e6	1.18793
$423$	772206.	0.209837
$424$	−2.17311e6	−0.587040
$425$	0	0
$426$	−94849.9	−0.0253229
$427$	600427.	0.159364
$428$	−99033.8	−0.0261321
$429$	−2.43198e6	−0.637994
$430$	0	0
$431$	1.45620e6	0.377595	0.188798	−	0.982016i	$-0.439541\pi$
$431$	1.45620e6	0.377595	0.188798	+	0.982016i	$0.439541\pi$
$432$	−1.03054e6	−0.265678
$433$	1.10726e6	0.283812	0.141906	−	0.989880i	$-0.454677\pi$
$433$	1.10726e6	0.283812	0.141906	+	0.989880i	$0.454677\pi$
$434$	−2.01146e6	−0.512609
$435$	0	0
$436$	−1.09845e6	−0.276734
$437$	6.64934e6	1.66562
$438$	−471733.	−0.117493
$439$	4.15410e6	1.02876	0.514382	−	0.857561i	$-0.328021\pi$
$439$	4.15410e6	1.02876	0.514382	+	0.857561i	$0.328021\pi$
$440$	0	0
$441$	−92442.1	−0.0226346
$442$	−2.79755e6	−0.681118
$443$	7.20799e6	1.74504	0.872519	−	0.488581i	$-0.162485\pi$
$443$	7.20799e6	1.74504	0.872519	+	0.488581i	$0.162485\pi$
$444$	1.66251e6	0.400227
$445$	0	0
$446$	−1.72324e6	−0.410211
$447$	1.10298e6	0.261095
$448$	−200704.	−0.0472456
$449$	5.58367e6	1.30709	0.653543	−	0.756890i	$-0.273282\pi$
$449$	5.58367e6	1.30709	0.653543	+	0.756890i	$0.273282\pi$
$450$	0	0
$451$	−2.51004e6	−0.581084
$452$	1.00537e6	0.231463
$453$	−6.01275e6	−1.37666
$454$	−914895.	−0.208320
$455$	0	0
$456$	−2.63055e6	−0.592427
$457$	3.54344e6	0.793661	0.396831	−	0.917892i	$-0.370110\pi$
$457$	3.54344e6	0.793661	0.396831	+	0.917892i	$0.370110\pi$
$458$	4.07186e6	0.907046
$459$	7.05084e6	1.56210
$460$	0	0
$461$	5.88069e6	1.28877	0.644386	−	0.764700i	$-0.277113\pi$
$461$	5.88069e6	1.28877	0.644386	+	0.764700i	$0.277113\pi$
$462$	1.19375e6	0.260200
$463$	−6.65045e6	−1.44178	−0.720890	−	0.693050i	$-0.756267\pi$
$463$	−6.65045e6	−1.44178	−0.720890	+	0.693050i	$0.756267\pi$
$464$	−544749.	−0.117463
$465$	0	0
$466$	−946839.	−0.201981
$467$	−4.34743e6	−0.922445	−0.461222	−	0.887285i	$-0.652589\pi$
$467$	−4.34743e6	−0.922445	−0.461222	+	0.887285i	$0.652589\pi$
$468$	245980.	0.0519141
$469$	−859158.	−0.180360
$470$	0	0
$471$	6.89143e6	1.43139
$472$	276438.	0.0571139
$473$	−8.58526e6	−1.76441
$474$	524456.	0.107217
$475$	0	0
$476$	1.37319e6	0.277788
$477$	−1.30731e6	−0.263078
$478$	−6.95978e6	−1.39324
$479$	−929680.	−0.185138	−0.0925688	−	0.995706i	$-0.529508\pi$
$479$	−929680.	−0.185138	−0.0925688	+	0.995706i	$0.529508\pi$
$480$	0	0
$481$	−2.90136e6	−0.571792
$482$	1.33136e6	0.261023
$483$	−1.62105e6	−0.316176
$484$	325490.	0.0631573
$485$	0	0
$486$	−1.15513e6	−0.221839
$487$	5.44887e6	1.04108	0.520540	−	0.853837i	$-0.325731\pi$
$487$	5.44887e6	1.04108	0.520540	+	0.853837i	$0.325731\pi$
$488$	784232.	0.149072
$489$	−4.03324e6	−0.762750
$490$	0	0
$491$	−670070.	−0.125434	−0.0627172	−	0.998031i	$-0.519977\pi$
$491$	−670070.	−0.125434	−0.0627172	+	0.998031i	$0.519977\pi$
$492$	−1.34845e6	−0.251143
$493$	3.72711e6	0.690644
$494$	4.59076e6	0.846384
$495$	0	0
$496$	−2.62721e6	−0.479502
$497$	−81250.9	−0.0147549
$498$	5.44574e6	0.983973
$499$	6.58017e6	1.18300	0.591501	−	0.806304i	$-0.298535\pi$
$499$	6.58017e6	1.18300	0.591501	+	0.806304i	$0.298535\pi$
$500$	0	0
$501$	−1.88722e6	−0.335914
$502$	−7.54339e6	−1.33600
$503$	1.47714e6	0.260317	0.130159	−	0.991493i	$-0.458451\pi$
$503$	1.47714e6	0.260317	0.130159	+	0.991493i	$0.458451\pi$
$504$	−120741.	−0.0211727
$505$	0	0
$506$	−3.94119e6	−0.684308
$507$	−3.02952e6	−0.523424
$508$	3.16878e6	0.544794
$509$	5.16474e6	0.883597	0.441799	−	0.897114i	$-0.354341\pi$
$509$	5.16474e6	0.883597	0.441799	+	0.897114i	$0.354341\pi$
$510$	0	0
$511$	−404099.	−0.0684597
$512$	−262144.	−0.0441942
$513$	−1.15704e7	−1.94113
$514$	1.73043e6	0.288899
$515$	0	0
$516$	−4.61219e6	−0.762575
$517$	−8.54215e6	−1.40553
$518$	1.42415e6	0.233201
$519$	−793688.	−0.129340
$520$	0	0
$521$	294103.	0.0474684	0.0237342	−	0.999718i	$-0.492444\pi$
$521$	294103.	0.0474684	0.0237342	+	0.999718i	$0.492444\pi$
$522$	−327714.	−0.0526403
$523$	−1.39098e6	−0.222365	−0.111182	−	0.993800i	$-0.535464\pi$
$523$	−1.39098e6	−0.222365	−0.111182	+	0.993800i	$0.535464\pi$
$524$	−4.56212e6	−0.725835
$525$	0	0
$526$	−4.81550e6	−0.758887
$527$	1.79750e7	2.81931
$528$	1.55918e6	0.243395
$529$	−1.08439e6	−0.168479
$530$	0	0
$531$	166301.	0.0255952
$532$	−2.25340e6	−0.345191
$533$	2.35327e6	0.358801
$534$	826090.	0.125364
$535$	0	0
$536$	−1.12216e6	−0.168712
$537$	296993.	0.0444437
$538$	−2.19253e6	−0.326581
$539$	1.02260e6	0.151611
$540$	0	0
$541$	2.36315e6	0.347134	0.173567	−	0.984822i	$-0.444471\pi$
$541$	2.36315e6	0.347134	0.173567	+	0.984822i	$0.444471\pi$
$542$	3.62636e6	0.530240
$543$	−3.65897e6	−0.532548
$544$	1.79356e6	0.259847
$545$	0	0
$546$	−1.11919e6	−0.160665
$547$	−4.52298e6	−0.646333	−0.323167	−	0.946342i	$-0.604747\pi$
$547$	−4.52298e6	−0.646333	−0.323167	+	0.946342i	$0.604747\pi$
$548$	−6.51167e6	−0.926277
$549$	471782.	0.0668053
$550$	0	0
$551$	−6.11616e6	−0.858223
$552$	−2.11729e6	−0.295756
$553$	449262.	0.0624722
$554$	5.41351e6	0.749384
$555$	0	0
$556$	216487.	0.0296992
$557$	8.40670e6	1.14812	0.574060	−	0.818813i	$-0.305367\pi$
$557$	8.40670e6	1.14812	0.574060	+	0.818813i	$0.305367\pi$
$558$	−1.58049e6	−0.214885
$559$	8.04905e6	1.08947
$560$	0	0
$561$	−1.06677e7	−1.43108
$562$	−3.00924e6	−0.401898
$563$	2.22527e6	0.295877	0.147938	−	0.988997i	$-0.452736\pi$
$563$	2.22527e6	0.295877	0.147938	+	0.988997i	$0.452736\pi$
$564$	−4.58903e6	−0.607468
$565$	0	0
$566$	−3.28284e6	−0.430733
$567$	2.36233e6	0.308590
$568$	−106124.	−0.0138020
$569$	1.58379e6	0.205078	0.102539	−	0.994729i	$-0.467303\pi$
$569$	1.58379e6	0.205078	0.102539	+	0.994729i	$0.467303\pi$
$570$	0	0
$571$	1.25695e7	1.61335	0.806676	−	0.590994i	$-0.201264\pi$
$571$	1.25695e7	1.61335	0.806676	+	0.590994i	$0.201264\pi$
$572$	−2.72104e6	−0.347732
$573$	9.28462e6	1.18135
$574$	−1.15511e6	−0.146334
$575$	0	0
$576$	−157702.	−0.0198053
$577$	−1.37636e7	−1.72104	−0.860521	−	0.509414i	$-0.829862\pi$
$577$	−1.37636e7	−1.72104	−0.860521	+	0.509414i	$0.829862\pi$
$578$	−6.59185e6	−0.820706
$579$	1.09918e7	1.36261
$580$	0	0
$581$	4.66496e6	0.573334
$582$	3.55405e6	0.434927
$583$	1.44615e7	1.76215
$584$	−527802.	−0.0640382
$585$	0	0
$586$	−7.68956e6	−0.925033
$587$	−1.43475e6	−0.171862	−0.0859311	−	0.996301i	$-0.527386\pi$
$587$	−1.43475e6	−0.171862	−0.0859311	+	0.996301i	$0.527386\pi$
$588$	549360.	0.0655260
$589$	−2.94969e7	−3.50339
$590$	0	0
$591$	2.37614e6	0.279836
$592$	1.86011e6	0.218139
$593$	−2.58486e6	−0.301856	−0.150928	−	0.988545i	$-0.548226\pi$
$593$	−2.58486e6	−0.301856	−0.150928	+	0.988545i	$0.548226\pi$
$594$	6.85800e6	0.797501
$595$	0	0
$596$	1.23408e6	0.142307
$597$	−1.06903e6	−0.122760
$598$	3.69504e6	0.422538
$599$	−5.51209e6	−0.627696	−0.313848	−	0.949473i	$-0.601618\pi$
$599$	−5.51209e6	−0.627696	−0.313848	+	0.949473i	$0.601618\pi$
$600$	0	0
$601$	6.92526e6	0.782077	0.391039	−	0.920374i	$-0.372116\pi$
$601$	6.92526e6	0.782077	0.391039	+	0.920374i	$0.372116\pi$
$602$	−3.95092e6	−0.444331
$603$	−675078.	−0.0756068
$604$	−6.72742e6	−0.750336
$605$	0	0
$606$	6.18560e6	0.684227
$607$	−1.19865e7	−1.32045	−0.660225	−	0.751067i	$-0.729539\pi$
$607$	−1.19865e7	−1.32045	−0.660225	+	0.751067i	$0.729539\pi$
$608$	−2.94322e6	−0.322896
$609$	1.49107e6	0.162913
$610$	0	0
$611$	8.00864e6	0.867872
$612$	1.07898e6	0.116448
$613$	4.84046e6	0.520278	0.260139	−	0.965571i	$-0.416232\pi$
$613$	4.84046e6	0.520278	0.260139	+	0.965571i	$0.416232\pi$
$614$	892915.	0.0955849
$615$	0	0
$616$	1.33563e6	0.141819
$617$	9.94864e6	1.05208	0.526042	−	0.850458i	$-0.323675\pi$
$617$	9.94864e6	1.05208	0.526042	+	0.850458i	$0.323675\pi$
$618$	7.89568e6	0.831608
$619$	−1.38913e7	−1.45719	−0.728597	−	0.684943i	$-0.759827\pi$
$619$	−1.38913e7	−1.45719	−0.728597	+	0.684943i	$0.759827\pi$
$620$	0	0
$621$	−9.31284e6	−0.969065
$622$	−9.19270e6	−0.952724
$623$	707649.	0.0730463
$624$	−1.46180e6	−0.150289
$625$	0	0
$626$	−6.51524e6	−0.664499
$627$	1.75057e7	1.77832
$628$	7.71053e6	0.780162
$629$	−1.27266e7	−1.28259
$630$	0	0
$631$	−7.29060e6	−0.728937	−0.364469	−	0.931216i	$-0.618749\pi$
$631$	−7.29060e6	−0.728937	−0.364469	+	0.931216i	$0.618749\pi$
$632$	586791.	0.0584374
$633$	−1.55367e7	−1.54117
$634$	4.54480e6	0.449047
$635$	0	0
$636$	7.76904e6	0.761596
$637$	−958727.	−0.0936152
$638$	3.62517e6	0.352595
$639$	−63842.4	−0.00618525
$640$	0	0
$641$	1.58175e7	1.52052	0.760259	−	0.649620i	$-0.225072\pi$
$641$	1.58175e7	1.52052	0.760259	+	0.649620i	$0.225072\pi$
$642$	354053.	0.0339024
$643$	7.35926e6	0.701951	0.350975	−	0.936385i	$-0.385850\pi$
$643$	7.35926e6	0.701951	0.350975	+	0.936385i	$0.385850\pi$
$644$	−1.81373e6	−0.172329
$645$	0	0
$646$	2.01371e7	1.89852
$647$	−1.08644e7	−1.02034	−0.510172	−	0.860072i	$-0.670418\pi$
$647$	−1.08644e7	−1.02034	−0.510172	+	0.860072i	$0.670418\pi$
$648$	3.08549e6	0.288660
$649$	−1.83962e6	−0.171442
$650$	0	0
$651$	7.19111e6	0.665033
$652$	−4.51263e6	−0.415729
$653$	2.85006e6	0.261560	0.130780	−	0.991411i	$-0.458252\pi$
$653$	2.85006e6	0.261560	0.130780	+	0.991411i	$0.458252\pi$
$654$	3.92703e6	0.359021
$655$	0	0
$656$	−1.50872e6	−0.136883
$657$	−317518.	−0.0286982
$658$	−3.93108e6	−0.353954
$659$	−214583.	−0.0192479	−0.00962393	−	0.999954i	$-0.503063\pi$
$659$	−214583.	−0.0192479	−0.00962393	+	0.999954i	$0.503063\pi$
$660$	0	0
$661$	2.26579e6	0.201704	0.100852	−	0.994901i	$-0.467843\pi$
$661$	2.26579e6	0.201704	0.100852	+	0.994901i	$0.467843\pi$
$662$	1.51020e7	1.33933
$663$	1.00014e7	0.883648
$664$	6.09301e6	0.536304
$665$	0	0
$666$	1.11901e6	0.0977575
$667$	−4.92281e6	−0.428448
$668$	−2.11153e6	−0.183086
$669$	6.16070e6	0.532187
$670$	0	0
$671$	−5.21886e6	−0.447476
$672$	717532.	0.0612940
$673$	9.52991e6	0.811057	0.405528	−	0.914082i	$-0.367088\pi$
$673$	9.52991e6	0.811057	0.405528	+	0.914082i	$0.367088\pi$
$674$	1.49965e7	1.27157
$675$	0	0
$676$	−3.38960e6	−0.285287
$677$	−2.14827e7	−1.80143	−0.900714	−	0.434412i	$-0.856956\pi$
$677$	−2.14827e7	−1.80143	−0.900714	+	0.434412i	$0.856956\pi$
$678$	−3.59429e6	−0.300288
$679$	3.04449e6	0.253419
$680$	0	0
$681$	3.27082e6	0.270264
$682$	1.74834e7	1.43935
$683$	2.89672e6	0.237605	0.118802	−	0.992918i	$-0.462095\pi$
$683$	2.89672e6	0.237605	0.118802	+	0.992918i	$0.462095\pi$
$684$	−1.77060e6	−0.144704
$685$	0	0
$686$	470596.	0.0381802
$687$	−1.45572e7	−1.17676
$688$	−5.16038e6	−0.415634
$689$	−1.35583e7	−1.08807
$690$	0	0
$691$	−2.09710e6	−0.167080	−0.0835399	−	0.996504i	$-0.526623\pi$
$691$	−2.09710e6	−0.167080	−0.0835399	+	0.996504i	$0.526623\pi$
$692$	−888024.	−0.0704952
$693$	803499.	0.0635553
$694$	−4.59371e6	−0.362047
$695$	0	0
$696$	1.94752e6	0.152391
$697$	1.03225e7	0.804825
$698$	−1.61310e7	−1.25321
$699$	3.38502e6	0.262041
$700$	0	0
$701$	6.59970e6	0.507258	0.253629	−	0.967302i	$-0.418376\pi$
$701$	6.59970e6	0.507258	0.253629	+	0.967302i	$0.418376\pi$
$702$	−6.42967e6	−0.492432
$703$	2.08843e7	1.59379
$704$	1.74450e6	0.132660
$705$	0	0
$706$	2.61064e6	0.197122
$707$	5.29874e6	0.398680
$708$	−988285.	−0.0740967
$709$	−2.22011e7	−1.65866	−0.829332	−	0.558756i	$-0.811279\pi$
$709$	−2.22011e7	−1.65866	−0.829332	+	0.558756i	$0.811279\pi$
$710$	0	0
$711$	353005.	0.0261883
$712$	924277.	0.0683285
$713$	−2.37416e7	−1.74899
$714$	−4.90926e6	−0.360388
$715$	0	0
$716$	332293.	0.0242236
$717$	2.48817e7	1.80752
$718$	1.13842e7	0.824120
$719$	1.87540e7	1.35292	0.676460	−	0.736479i	$-0.263513\pi$
$719$	1.87540e7	1.35292	0.676460	+	0.736479i	$0.263513\pi$
$720$	0	0
$721$	6.76365e6	0.484555
$722$	−2.31405e7	−1.65207
$723$	−4.75972e6	−0.338638
$724$	−4.09386e6	−0.290260
$725$	0	0
$726$	−1.16365e6	−0.0819371
$727$	−1.38593e7	−0.972537	−0.486269	−	0.873809i	$-0.661642\pi$
$727$	−1.38593e7	−0.972537	−0.486269	+	0.873809i	$0.661642\pi$
$728$	−1.25222e6	−0.0875690
$729$	1.58449e7	1.10426
$730$	0	0
$731$	3.53067e7	2.44379
$732$	−2.80369e6	−0.193398
$733$	−1.08280e7	−0.744372	−0.372186	−	0.928158i	$-0.621392\pi$
$733$	−1.08280e7	−0.744372	−0.372186	+	0.928158i	$0.621392\pi$
$734$	−1.06093e7	−0.726851
$735$	0	0
$736$	−2.36895e6	−0.161199
$737$	7.46772e6	0.506430
$738$	−907624.	−0.0613430
$739$	−906866.	−0.0610846	−0.0305423	−	0.999533i	$-0.509723\pi$
$739$	−906866.	−0.0610846	−0.0305423	+	0.999533i	$0.509723\pi$
$740$	0	0
$741$	−1.64123e7	−1.09806
$742$	6.65516e6	0.443761
$743$	1.76804e7	1.17495	0.587475	−	0.809242i	$-0.300122\pi$
$743$	1.76804e7	1.17495	0.587475	+	0.809242i	$0.300122\pi$
$744$	9.39247e6	0.622082
$745$	0	0
$746$	1.13313e7	0.745477
$747$	3.66547e6	0.240341
$748$	−1.19357e7	−0.779996
$749$	303291.	0.0197540
$750$	0	0
$751$	1.96975e7	1.27441	0.637207	−	0.770692i	$-0.280089\pi$
$751$	1.96975e7	1.27441	0.637207	+	0.770692i	$0.280089\pi$
$752$	−5.13447e6	−0.331094
$753$	2.69682e7	1.73326
$754$	−3.39875e6	−0.217717
$755$	0	0
$756$	3.15604e6	0.200834
$757$	1.88790e7	1.19740	0.598699	−	0.800974i	$-0.295685\pi$
$757$	1.88790e7	1.19740	0.598699	+	0.800974i	$0.295685\pi$
$758$	4.82714e6	0.305152
$759$	1.40901e7	0.887786
$760$	0	0
$761$	9.74849e6	0.610205	0.305102	−	0.952320i	$-0.401309\pi$
$761$	9.74849e6	0.610205	0.305102	+	0.952320i	$0.401309\pi$
$762$	−1.13286e7	−0.706788
$763$	3.36399e6	0.209191
$764$	1.03882e7	0.643881
$765$	0	0
$766$	−8.91171e6	−0.548768
$767$	1.72473e6	0.105860
$768$	937184.	0.0573353
$769$	6.11920e6	0.373146	0.186573	−	0.982441i	$-0.440262\pi$
$769$	6.11920e6	0.373146	0.186573	+	0.982441i	$0.440262\pi$
$770$	0	0
$771$	−6.18642e6	−0.374804
$772$	1.22983e7	0.742678
$773$	−1.47893e6	−0.0890223	−0.0445111	−	0.999009i	$-0.514173\pi$
$773$	−1.47893e6	−0.0890223	−0.0445111	+	0.999009i	$0.514173\pi$
$774$	−3.10441e6	−0.186263
$775$	0	0
$776$	3.97648e6	0.237052
$777$	−5.09143e6	−0.302543
$778$	1.49602e7	0.886109
$779$	−1.69391e7	−1.00011
$780$	0	0
$781$	706226.	0.0414301
$782$	1.62081e7	0.947795
$783$	8.56609e6	0.499319
$784$	614656.	0.0357143
$785$	0	0
$786$	1.63099e7	0.941662
$787$	−2.39360e7	−1.37758	−0.688788	−	0.724963i	$-0.741857\pi$
$787$	−2.39360e7	−1.37758	−0.688788	+	0.724963i	$0.741857\pi$
$788$	2.65857e6	0.152522
$789$	1.72158e7	0.984542
$790$	0	0
$791$	−3.07896e6	−0.174970
$792$	1.04947e6	0.0594506
$793$	4.89291e6	0.276302
$794$	−1.16134e7	−0.653744
$795$	0	0
$796$	−1.19610e6	−0.0669089
$797$	−6.21796e6	−0.346738	−0.173369	−	0.984857i	$-0.555465\pi$
$797$	−6.21796e6	−0.346738	−0.173369	+	0.984857i	$0.555465\pi$
$798$	8.05607e6	0.447833
$799$	3.51294e7	1.94672
$800$	0	0
$801$	556032.	0.0306209
$802$	−5.75106e6	−0.315727
$803$	3.51239e6	0.192227
$804$	4.01182e6	0.218878
$805$	0	0
$806$	−1.63915e7	−0.888750
$807$	7.83847e6	0.423689
$808$	6.92081e6	0.372931
$809$	−4.98100e6	−0.267575	−0.133787	−	0.991010i	$-0.542714\pi$
$809$	−4.98100e6	−0.267575	−0.133787	+	0.991010i	$0.542714\pi$
$810$	0	0
$811$	−3.09722e7	−1.65356	−0.826780	−	0.562525i	$-0.809830\pi$
$811$	−3.09722e7	−1.65356	−0.826780	+	0.562525i	$0.809830\pi$
$812$	1.66830e6	0.0887938
$813$	−1.29645e7	−0.687907
$814$	−1.23786e7	−0.654800
$815$	0	0
$816$	−6.41210e6	−0.337112
$817$	−5.79381e7	−3.03675
$818$	1.91530e7	1.00081
$819$	−753315.	−0.0392434
$820$	0	0
$821$	−2.87667e7	−1.48947	−0.744734	−	0.667361i	$-0.767424\pi$
$821$	−2.87667e7	−1.48947	−0.744734	+	0.667361i	$0.767424\pi$
$822$	2.32797e7	1.20170
$823$	−3.55713e7	−1.83063	−0.915313	−	0.402743i	$-0.868057\pi$
$823$	−3.55713e7	−1.83063	−0.915313	+	0.402743i	$0.868057\pi$
$824$	8.83415e6	0.453259
$825$	0	0
$826$	−846590.	−0.0431741
$827$	−3.01542e7	−1.53315	−0.766575	−	0.642155i	$-0.778040\pi$
$827$	−3.01542e7	−1.53315	−0.766575	+	0.642155i	$0.778040\pi$
$828$	−1.42513e6	−0.0722400
$829$	−3.11697e7	−1.57524	−0.787618	−	0.616164i	$-0.788686\pi$
$829$	−3.11697e7	−1.57524	−0.787618	+	0.616164i	$0.788686\pi$
$830$	0	0
$831$	−1.93537e7	−0.972213
$832$	−1.63555e6	−0.0819133
$833$	−4.20540e6	−0.209988
$834$	−773958.	−0.0385303
$835$	0	0
$836$	1.95864e7	0.969255
$837$	4.13124e7	2.03829
$838$	8.33644e6	0.410082
$839$	3.63050e7	1.78058	0.890291	−	0.455393i	$-0.150501\pi$
$839$	3.63050e7	1.78058	0.890291	+	0.455393i	$0.150501\pi$
$840$	0	0
$841$	−1.59831e7	−0.779238
$842$	4.01039e6	0.194942
$843$	1.07582e7	0.521402
$844$	−1.73834e7	−0.839997
$845$	0	0
$846$	−3.08883e6	−0.148377
$847$	−996812.	−0.0477425
$848$	8.69245e6	0.415100
$849$	1.17364e7	0.558812
$850$	0	0
$851$	1.68095e7	0.795666
$852$	379400.	0.0179060
$853$	−1.00191e7	−0.471472	−0.235736	−	0.971817i	$-0.575750\pi$
$853$	−1.00191e7	−0.471472	−0.235736	+	0.971817i	$0.575750\pi$
$854$	−2.40171e6	−0.112687
$855$	0	0
$856$	396135.	0.0184782
$857$	1.17397e7	0.546014	0.273007	−	0.962012i	$-0.411982\pi$
$857$	1.17397e7	0.546014	0.273007	+	0.962012i	$0.411982\pi$
$858$	9.72791e6	0.451130
$859$	−1.69498e7	−0.783756	−0.391878	−	0.920017i	$-0.628175\pi$
$859$	−1.69498e7	−0.783756	−0.391878	+	0.920017i	$0.628175\pi$
$860$	0	0
$861$	4.12962e6	0.189846
$862$	−5.82478e6	−0.267000
$863$	2.11590e7	0.967091	0.483546	−	0.875319i	$-0.339349\pi$
$863$	2.11590e7	0.967091	0.483546	+	0.875319i	$0.339349\pi$
$864$	4.12217e6	0.187863
$865$	0	0
$866$	−4.42905e6	−0.200686
$867$	2.35664e7	1.06474
$868$	8.04583e6	0.362469
$869$	−3.90495e6	−0.175415
$870$	0	0
$871$	−7.00131e6	−0.312705
$872$	4.39379e6	0.195681
$873$	2.39219e6	0.106233
$874$	−2.65973e7	−1.17777
$875$	0	0
$876$	1.88693e6	0.0830799
$877$	−1.01611e7	−0.446109	−0.223055	−	0.974806i	$-0.571603\pi$
$877$	−1.01611e7	−0.446109	−0.223055	+	0.974806i	$0.571603\pi$
$878$	−1.66164e7	−0.727446
$879$	2.74907e7	1.20009
$880$	0	0
$881$	−3.50223e7	−1.52021	−0.760106	−	0.649799i	$-0.774853\pi$
$881$	−3.50223e7	−1.52021	−0.760106	+	0.649799i	$0.774853\pi$
$882$	369768.	0.0160051
$883$	2.15600e7	0.930566	0.465283	−	0.885162i	$-0.345953\pi$
$883$	2.15600e7	0.930566	0.465283	+	0.885162i	$0.345953\pi$
$884$	1.11902e7	0.481623
$885$	0	0
$886$	−2.88319e7	−1.23393
$887$	1.29121e7	0.551048	0.275524	−	0.961294i	$-0.411149\pi$
$887$	1.29121e7	0.551048	0.275524	+	0.961294i	$0.411149\pi$
$888$	−6.65003e6	−0.283003
$889$	−9.70438e6	−0.411826
$890$	0	0
$891$	−2.05332e7	−0.866486
$892$	6.89294e6	0.290063
$893$	−5.76472e7	−2.41907
$894$	−4.41192e6	−0.184622
$895$	0	0
$896$	802816.	0.0334077
$897$	−1.32100e7	−0.548180
$898$	−2.23347e7	−0.924249
$899$	2.18379e7	0.901181
$900$	0	0
$901$	−5.94727e7	−2.44065
$902$	1.00401e7	0.410888
$903$	1.41248e7	0.576453
$904$	−4.02149e6	−0.163669
$905$	0	0
$906$	2.40510e7	0.973448
$907$	−2.63522e7	−1.06365	−0.531825	−	0.846854i	$-0.678494\pi$
$907$	−2.63522e7	−1.06365	−0.531825	+	0.846854i	$0.678494\pi$
$908$	3.65958e6	0.147305
$909$	4.16346e6	0.167126
$910$	0	0
$911$	−4.82227e6	−0.192511	−0.0962555	−	0.995357i	$-0.530687\pi$
$911$	−4.82227e6	−0.192511	−0.0962555	+	0.995357i	$0.530687\pi$
$912$	1.05222e7	0.418909
$913$	−4.05474e7	−1.60985
$914$	−1.41738e7	−0.561203
$915$	0	0
$916$	−1.62874e7	−0.641378
$917$	1.39715e7	0.548680
$918$	−2.82034e7	−1.10457
$919$	−121562.	−0.00474798	−0.00237399	−	0.999997i	$-0.500756\pi$
$919$	−121562.	−0.00474798	−0.00237399	+	0.999997i	$0.500756\pi$
$920$	0	0
$921$	−3.19224e6	−0.124007
$922$	−2.35228e7	−0.911299
$923$	−662117.	−0.0255818
$924$	−4.77499e6	−0.183989
$925$	0	0
$926$	2.66018e7	1.01949
$927$	5.31450e6	0.203125
$928$	2.17900e6	0.0830590
$929$	1.74856e7	0.664725	0.332362	−	0.943152i	$-0.392154\pi$
$929$	1.74856e7	0.664725	0.332362	+	0.943152i	$0.392154\pi$
$930$	0	0
$931$	6.90104e6	0.260940
$932$	3.78736e6	0.142822
$933$	3.28646e7	1.23602
$934$	1.73897e7	0.652267
$935$	0	0
$936$	−983921.	−0.0367088
$937$	1.97882e7	0.736304	0.368152	−	0.929766i	$-0.379991\pi$
$937$	1.97882e7	0.736304	0.368152	+	0.929766i	$0.379991\pi$
$938$	3.43663e6	0.127534
$939$	2.32925e7	0.862088
$940$	0	0
$941$	2.05948e7	0.758199	0.379100	−	0.925356i	$-0.376234\pi$
$941$	2.05948e7	0.758199	0.379100	+	0.925356i	$0.376234\pi$
$942$	−2.75657e7	−1.01214
$943$	−1.36340e7	−0.499282
$944$	−1.10575e6	−0.0403857
$945$	0	0
$946$	3.43410e7	1.24763
$947$	−6.90803e6	−0.250311	−0.125155	−	0.992137i	$-0.539943\pi$
$947$	−6.90803e6	−0.250311	−0.125155	+	0.992137i	$0.539943\pi$
$948$	−2.09782e6	−0.0758137
$949$	−3.29302e6	−0.118694
$950$	0	0
$951$	−1.62480e7	−0.582571
$952$	−5.49276e6	−0.196426
$953$	−3.72040e7	−1.32696	−0.663480	−	0.748194i	$-0.730921\pi$
$953$	−3.72040e7	−1.32696	−0.663480	+	0.748194i	$0.730921\pi$
$954$	5.22926e6	0.186024
$955$	0	0
$956$	2.78391e7	0.985169
$957$	−1.29603e7	−0.457440
$958$	3.71872e6	0.130912
$959$	1.99420e7	0.700199
$960$	0	0
$961$	7.66904e7	2.67875
$962$	1.16054e7	0.404318
$963$	238309.	0.00828085
$964$	−5.32545e6	−0.184571
$965$	0	0
$966$	6.48421e6	0.223571
$967$	−1.88878e7	−0.649555	−0.324778	−	0.945790i	$-0.605289\pi$
$967$	−1.88878e7	−0.649555	−0.324778	+	0.945790i	$0.605289\pi$
$968$	−1.30196e6	−0.0446590
$969$	−7.19916e7	−2.46305
$970$	0	0
$971$	−4.20566e7	−1.43148	−0.715741	−	0.698366i	$-0.753911\pi$
$971$	−4.20566e7	−1.43148	−0.715741	+	0.698366i	$0.753911\pi$
$972$	4.62050e6	0.156864
$973$	−662992.	−0.0224505
$974$	−2.17955e7	−0.736155
$975$	0	0
$976$	−3.13693e6	−0.105409
$977$	−4.61174e7	−1.54571	−0.772855	−	0.634583i	$-0.781172\pi$
$977$	−4.61174e7	−1.54571	−0.772855	+	0.634583i	$0.781172\pi$
$978$	1.61330e7	0.539346
$979$	−6.15083e6	−0.205105
$980$	0	0
$981$	2.64324e6	0.0876928
$982$	2.68028e6	0.0886954
$983$	−3.25827e7	−1.07548	−0.537741	−	0.843110i	$-0.680722\pi$
$983$	−3.25827e7	−1.07548	−0.537741	+	0.843110i	$0.680722\pi$
$984$	5.39378e6	0.177585
$985$	0	0
$986$	−1.49084e7	−0.488359
$987$	1.40539e7	0.459202
$988$	−1.83631e7	−0.598484
$989$	−4.66335e7	−1.51603
$990$	0	0
$991$	1.68209e7	0.544084	0.272042	−	0.962285i	$-0.412301\pi$
$991$	1.68209e7	0.544084	0.272042	+	0.962285i	$0.412301\pi$
$992$	1.05088e7	0.339059
$993$	−5.39906e7	−1.73758
$994$	325003.	0.0104333
$995$	0	0
$996$	−2.17829e7	−0.695774
$997$	−7.67044e6	−0.244389	−0.122195	−	0.992506i	$-0.538993\pi$
$997$	−7.67044e6	−0.244389	−0.122195	+	0.992506i	$0.538993\pi$
$998$	−2.63207e7	−0.836509
$999$	−2.92499e7	−0.927279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.p.1.2		2
5.2	odd	4		350.6.c.k.99.1		4
5.3	odd	4		350.6.c.k.99.4		4
5.4	even	2		70.6.a.h.1.1	✓	2
15.14	odd	2		630.6.a.s.1.1		2
20.19	odd	2		560.6.a.k.1.2		2
35.34	odd	2		490.6.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.h.1.1	✓	2	5.4	even	2
350.6.a.p.1.2		2	1.1	even	1	trivial
350.6.c.k.99.1		4	5.2	odd	4
350.6.c.k.99.4		4	5.3	odd	4
490.6.a.u.1.2		2	35.34	odd	2
560.6.a.k.1.2		2	20.19	odd	2
630.6.a.s.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +14.3003 q^{3} +16.0000 q^{4} -57.2012 q^{6} -49.0000 q^{7} -64.0000 q^{8} -38.5015 q^{9} +425.904 q^{11} +228.805 q^{12} -399.303 q^{13} +196.000 q^{14} +256.000 q^{16} -1751.52 q^{17} +154.006 q^{18} +2874.23 q^{19} -700.715 q^{21} -1703.62 q^{22} +2313.43 q^{23} -915.219 q^{24} +1597.21 q^{26} -4025.56 q^{27} -784.000 q^{28} -2127.93 q^{29} -10262.5 q^{31} -1024.00 q^{32} +6090.55 q^{33} +7006.08 q^{34} -616.024 q^{36} +7266.05 q^{37} -11496.9 q^{38} -5710.16 q^{39} -5893.44 q^{41} +2802.86 q^{42} -20157.7 q^{43} +6814.46 q^{44} -9253.72 q^{46} -20056.5 q^{47} +3660.88 q^{48} +2401.00 q^{49} -25047.2 q^{51} -6388.85 q^{52} +33954.9 q^{53} +16102.2 q^{54} +3136.00 q^{56} +41102.4 q^{57} +8511.71 q^{58} -4319.34 q^{59} -12253.6 q^{61} +41050.1 q^{62} +1886.57 q^{63} +4096.00 q^{64} -24362.2 q^{66} +17533.8 q^{67} -28024.3 q^{68} +33082.7 q^{69} +1658.18 q^{71} +2464.10 q^{72} +8246.91 q^{73} -29064.2 q^{74} +45987.7 q^{76} -20869.3 q^{77} +22840.6 q^{78} -9168.61 q^{79} -48210.8 q^{81} +23573.7 q^{82} -95203.2 q^{83} -11211.4 q^{84} +80631.0 q^{86} -30430.0 q^{87} -27257.8 q^{88} -14441.8 q^{89} +19565.9 q^{91} +37014.9 q^{92} -146757. q^{93} +80226.1 q^{94} -14643.5 q^{96} -62132.4 q^{97} -9604.00 q^{98} -16397.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} - 5 q^{3} + 32 q^{4} + 20 q^{6} - 98 q^{7} - 128 q^{8} + 91 q^{9} + 415 q^{11} - 80 q^{12} - 429 q^{13} + 392 q^{14} + 512 q^{16} - 1319 q^{17} - 364 q^{18} + 1918 q^{19} + 245 q^{21} - 1660 q^{22}+ \cdots - 17810 q^{99}+O(q^{100}) \) 2 * q - 8 * q^2 - 5 * q^3 + 32 * q^4 + 20 * q^6 - 98 * q^7 - 128 * q^8 + 91 * q^9 + 415 * q^11 - 80 * q^12 - 429 * q^13 + 392 * q^14 + 512 * q^16 - 1319 * q^17 - 364 * q^18 + 1918 * q^19 + 245 * q^21 - 1660 * q^22 + 1334 * q^23 + 320 * q^24 + 1716 * q^26 - 1835 * q^27 - 1568 * q^28 - 1131 * q^29 - 5472 * q^31 - 2048 * q^32 + 6301 * q^33 + 5276 * q^34 + 1456 * q^36 + 9156 * q^37 - 7672 * q^38 - 5137 * q^39 - 7822 * q^41 - 980 * q^42 - 2078 * q^43 + 6640 * q^44 - 5336 * q^46 + 8507 * q^47 - 1280 * q^48 + 4802 * q^49 - 33395 * q^51 - 6864 * q^52 + 34242 * q^53 + 7340 * q^54 + 6272 * q^56 + 59558 * q^57 + 4524 * q^58 + 6952 * q^59 - 45138 * q^61 + 21888 * q^62 - 4459 * q^63 + 8192 * q^64 - 25204 * q^66 + 54556 * q^67 - 21104 * q^68 + 51986 * q^69 - 62272 * q^71 - 5824 * q^72 - 40896 * q^73 - 36624 * q^74 + 30688 * q^76 - 20335 * q^77 + 20548 * q^78 + 62069 * q^79 - 121958 * q^81 + 31288 * q^82 - 189600 * q^83 + 3920 * q^84 + 8312 * q^86 - 49671 * q^87 - 26560 * q^88 + 64190 * q^89 + 21021 * q^91 + 21344 * q^92 - 239216 * q^93 - 34028 * q^94 + 5120 * q^96 - 155547 * q^97 - 19208 * q^98 - 17810 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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