# Properties

 Label 400.4.c.b.49.2 Level $400$ Weight $4$ Character 400.49 Analytic conductor $23.601$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

 Self dual: no Analytic conductor: $$23.6007640023$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 400.49 Dual form 400.4.c.b.49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+9.00000i q^{3} -26.0000i q^{7} -54.0000 q^{9} +O(q^{10})$$ $$q+9.00000i q^{3} -26.0000i q^{7} -54.0000 q^{9} +59.0000 q^{11} -28.0000i q^{13} +5.00000i q^{17} +109.000 q^{19} +234.000 q^{21} -194.000i q^{23} -243.000i q^{27} +32.0000 q^{29} -10.0000 q^{31} +531.000i q^{33} -198.000i q^{37} +252.000 q^{39} +117.000 q^{41} +388.000i q^{43} +68.0000i q^{47} -333.000 q^{49} -45.0000 q^{51} +18.0000i q^{53} +981.000i q^{57} +392.000 q^{59} -710.000 q^{61} +1404.00i q^{63} +253.000i q^{67} +1746.00 q^{69} +612.000 q^{71} +549.000i q^{73} -1534.00i q^{77} +414.000 q^{79} +729.000 q^{81} -121.000i q^{83} +288.000i q^{87} +81.0000 q^{89} -728.000 q^{91} -90.0000i q^{93} -1502.00i q^{97} -3186.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 108 q^{9}+O(q^{10})$$ 2 * q - 108 * q^9 $$2 q - 108 q^{9} + 118 q^{11} + 218 q^{19} + 468 q^{21} + 64 q^{29} - 20 q^{31} + 504 q^{39} + 234 q^{41} - 666 q^{49} - 90 q^{51} + 784 q^{59} - 1420 q^{61} + 3492 q^{69} + 1224 q^{71} + 828 q^{79} + 1458 q^{81} + 162 q^{89} - 1456 q^{91} - 6372 q^{99}+O(q^{100})$$ 2 * q - 108 * q^9 + 118 * q^11 + 218 * q^19 + 468 * q^21 + 64 * q^29 - 20 * q^31 + 504 * q^39 + 234 * q^41 - 666 * q^49 - 90 * q^51 + 784 * q^59 - 1420 * q^61 + 3492 * q^69 + 1224 * q^71 + 828 * q^79 + 1458 * q^81 + 162 * q^89 - 1456 * q^91 - 6372 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 9.00000i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 26.0000i − 1.40387i −0.712242 0.701934i $$-0.752320\pi$$
0.712242 0.701934i $$-0.247680\pi$$
$$8$$ 0 0
$$9$$ −54.0000 −2.00000
$$10$$ 0 0
$$11$$ 59.0000 1.61720 0.808599 0.588361i $$-0.200226\pi$$
0.808599 + 0.588361i $$0.200226\pi$$
$$12$$ 0 0
$$13$$ − 28.0000i − 0.597369i −0.954352 0.298685i $$-0.903452\pi$$
0.954352 0.298685i $$-0.0965479\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.00000i 0.0713340i 0.999364 + 0.0356670i $$0.0113556\pi$$
−0.999364 + 0.0356670i $$0.988644\pi$$
$$18$$ 0 0
$$19$$ 109.000 1.31612 0.658061 0.752965i $$-0.271377\pi$$
0.658061 + 0.752965i $$0.271377\pi$$
$$20$$ 0 0
$$21$$ 234.000 2.43157
$$22$$ 0 0
$$23$$ − 194.000i − 1.75877i −0.476108 0.879387i $$-0.657953\pi$$
0.476108 0.879387i $$-0.342047\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 243.000i − 1.73205i
$$28$$ 0 0
$$29$$ 32.0000 0.204905 0.102453 0.994738i $$-0.467331\pi$$
0.102453 + 0.994738i $$0.467331\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −0.0579372 −0.0289686 0.999580i $$-0.509222\pi$$
−0.0289686 + 0.999580i $$0.509222\pi$$
$$32$$ 0 0
$$33$$ 531.000i 2.80107i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 198.000i − 0.879757i −0.898057 0.439878i $$-0.855022\pi$$
0.898057 0.439878i $$-0.144978\pi$$
$$38$$ 0 0
$$39$$ 252.000 1.03467
$$40$$ 0 0
$$41$$ 117.000 0.445667 0.222833 0.974857i $$-0.428469\pi$$
0.222833 + 0.974857i $$0.428469\pi$$
$$42$$ 0 0
$$43$$ 388.000i 1.37603i 0.725695 + 0.688017i $$0.241518\pi$$
−0.725695 + 0.688017i $$0.758482\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 68.0000i 0.211039i 0.994417 + 0.105519i $$0.0336505\pi$$
−0.994417 + 0.105519i $$0.966350\pi$$
$$48$$ 0 0
$$49$$ −333.000 −0.970845
$$50$$ 0 0
$$51$$ −45.0000 −0.123554
$$52$$ 0 0
$$53$$ 18.0000i 0.0466508i 0.999728 + 0.0233254i $$0.00742537\pi$$
−0.999728 + 0.0233254i $$0.992575\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 981.000i 2.27959i
$$58$$ 0 0
$$59$$ 392.000 0.864984 0.432492 0.901638i $$-0.357634\pi$$
0.432492 + 0.901638i $$0.357634\pi$$
$$60$$ 0 0
$$61$$ −710.000 −1.49027 −0.745133 0.666916i $$-0.767614\pi$$
−0.745133 + 0.666916i $$0.767614\pi$$
$$62$$ 0 0
$$63$$ 1404.00i 2.80774i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 253.000i 0.461326i 0.973034 + 0.230663i $$0.0740896\pi$$
−0.973034 + 0.230663i $$0.925910\pi$$
$$68$$ 0 0
$$69$$ 1746.00 3.04629
$$70$$ 0 0
$$71$$ 612.000 1.02297 0.511486 0.859292i $$-0.329095\pi$$
0.511486 + 0.859292i $$0.329095\pi$$
$$72$$ 0 0
$$73$$ 549.000i 0.880214i 0.897945 + 0.440107i $$0.145059\pi$$
−0.897945 + 0.440107i $$0.854941\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 1534.00i − 2.27033i
$$78$$ 0 0
$$79$$ 414.000 0.589603 0.294802 0.955559i $$-0.404746\pi$$
0.294802 + 0.955559i $$0.404746\pi$$
$$80$$ 0 0
$$81$$ 729.000 1.00000
$$82$$ 0 0
$$83$$ − 121.000i − 0.160018i −0.996794 0.0800089i $$-0.974505\pi$$
0.996794 0.0800089i $$-0.0254949\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 288.000i 0.354906i
$$88$$ 0 0
$$89$$ 81.0000 0.0964717 0.0482359 0.998836i $$-0.484640\pi$$
0.0482359 + 0.998836i $$0.484640\pi$$
$$90$$ 0 0
$$91$$ −728.000 −0.838628
$$92$$ 0 0
$$93$$ − 90.0000i − 0.100350i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 1502.00i − 1.57222i −0.618089 0.786108i $$-0.712093\pi$$
0.618089 0.786108i $$-0.287907\pi$$
$$98$$ 0 0
$$99$$ −3186.00 −3.23439
$$100$$ 0 0
$$101$$ −234.000 −0.230533 −0.115267 0.993335i $$-0.536772\pi$$
−0.115267 + 0.993335i $$0.536772\pi$$
$$102$$ 0 0
$$103$$ − 1172.00i − 1.12117i −0.828097 0.560585i $$-0.810576\pi$$
0.828097 0.560585i $$-0.189424\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 1125.00i − 1.01643i −0.861231 0.508214i $$-0.830306\pi$$
0.861231 0.508214i $$-0.169694\pi$$
$$108$$ 0 0
$$109$$ 1234.00 1.08436 0.542182 0.840261i $$-0.317598\pi$$
0.542182 + 0.840261i $$0.317598\pi$$
$$110$$ 0 0
$$111$$ 1782.00 1.52378
$$112$$ 0 0
$$113$$ − 567.000i − 0.472025i −0.971750 0.236013i $$-0.924159\pi$$
0.971750 0.236013i $$-0.0758407\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1512.00i 1.19474i
$$118$$ 0 0
$$119$$ 130.000 0.100144
$$120$$ 0 0
$$121$$ 2150.00 1.61533
$$122$$ 0 0
$$123$$ 1053.00i 0.771917i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2358.00i 1.64755i 0.566918 + 0.823774i $$0.308136\pi$$
−0.566918 + 0.823774i $$0.691864\pi$$
$$128$$ 0 0
$$129$$ −3492.00 −2.38336
$$130$$ 0 0
$$131$$ 1692.00 1.12848 0.564239 0.825611i $$-0.309169\pi$$
0.564239 + 0.825611i $$0.309169\pi$$
$$132$$ 0 0
$$133$$ − 2834.00i − 1.84766i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 229.000i 0.142809i 0.997447 + 0.0714043i $$0.0227481\pi$$
−0.997447 + 0.0714043i $$0.977252\pi$$
$$138$$ 0 0
$$139$$ 2781.00 1.69699 0.848494 0.529205i $$-0.177510\pi$$
0.848494 + 0.529205i $$0.177510\pi$$
$$140$$ 0 0
$$141$$ −612.000 −0.365530
$$142$$ 0 0
$$143$$ − 1652.00i − 0.966064i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 2997.00i − 1.68155i
$$148$$ 0 0
$$149$$ −1472.00 −0.809335 −0.404668 0.914464i $$-0.632613\pi$$
−0.404668 + 0.914464i $$0.632613\pi$$
$$150$$ 0 0
$$151$$ −1322.00 −0.712469 −0.356235 0.934397i $$-0.615940\pi$$
−0.356235 + 0.934397i $$0.615940\pi$$
$$152$$ 0 0
$$153$$ − 270.000i − 0.142668i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 298.000i 0.151484i 0.997127 + 0.0757420i $$0.0241325\pi$$
−0.997127 + 0.0757420i $$0.975867\pi$$
$$158$$ 0 0
$$159$$ −162.000 −0.0808015
$$160$$ 0 0
$$161$$ −5044.00 −2.46909
$$162$$ 0 0
$$163$$ − 341.000i − 0.163860i −0.996638 0.0819300i $$-0.973892\pi$$
0.996638 0.0819300i $$-0.0261084\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 684.000i − 0.316943i −0.987364 0.158472i $$-0.949343\pi$$
0.987364 0.158472i $$-0.0506566\pi$$
$$168$$ 0 0
$$169$$ 1413.00 0.643150
$$170$$ 0 0
$$171$$ −5886.00 −2.63224
$$172$$ 0 0
$$173$$ 2344.00i 1.03012i 0.857154 + 0.515061i $$0.172231\pi$$
−0.857154 + 0.515061i $$0.827769\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 3528.00i 1.49820i
$$178$$ 0 0
$$179$$ −1111.00 −0.463911 −0.231955 0.972726i $$-0.574512\pi$$
−0.231955 + 0.972726i $$0.574512\pi$$
$$180$$ 0 0
$$181$$ 2042.00 0.838567 0.419284 0.907855i $$-0.362281\pi$$
0.419284 + 0.907855i $$0.362281\pi$$
$$182$$ 0 0
$$183$$ − 6390.00i − 2.58122i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 295.000i 0.115361i
$$188$$ 0 0
$$189$$ −6318.00 −2.43157
$$190$$ 0 0
$$191$$ −5270.00 −1.99646 −0.998230 0.0594735i $$-0.981058\pi$$
−0.998230 + 0.0594735i $$0.981058\pi$$
$$192$$ 0 0
$$193$$ − 613.000i − 0.228625i −0.993445 0.114313i $$-0.963533\pi$$
0.993445 0.114313i $$-0.0364666\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1174.00i 0.424589i 0.977206 + 0.212295i $$0.0680936\pi$$
−0.977206 + 0.212295i $$0.931906\pi$$
$$198$$ 0 0
$$199$$ 3428.00 1.22113 0.610564 0.791967i $$-0.290943\pi$$
0.610564 + 0.791967i $$0.290943\pi$$
$$200$$ 0 0
$$201$$ −2277.00 −0.799041
$$202$$ 0 0
$$203$$ − 832.000i − 0.287660i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 10476.0i 3.51755i
$$208$$ 0 0
$$209$$ 6431.00 2.12843
$$210$$ 0 0
$$211$$ −2339.00 −0.763144 −0.381572 0.924339i $$-0.624617\pi$$
−0.381572 + 0.924339i $$0.624617\pi$$
$$212$$ 0 0
$$213$$ 5508.00i 1.77184i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 260.000i 0.0813362i
$$218$$ 0 0
$$219$$ −4941.00 −1.52457
$$220$$ 0 0
$$221$$ 140.000 0.0426128
$$222$$ 0 0
$$223$$ 3932.00i 1.18075i 0.807131 + 0.590373i $$0.201019\pi$$
−0.807131 + 0.590373i $$0.798981\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 6084.00i − 1.77890i −0.457037 0.889448i $$-0.651089\pi$$
0.457037 0.889448i $$-0.348911\pi$$
$$228$$ 0 0
$$229$$ −4996.00 −1.44168 −0.720841 0.693101i $$-0.756244\pi$$
−0.720841 + 0.693101i $$0.756244\pi$$
$$230$$ 0 0
$$231$$ 13806.0 3.93233
$$232$$ 0 0
$$233$$ − 3222.00i − 0.905924i −0.891530 0.452962i $$-0.850367\pi$$
0.891530 0.452962i $$-0.149633\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 3726.00i 1.02122i
$$238$$ 0 0
$$239$$ 2736.00 0.740490 0.370245 0.928934i $$-0.379274\pi$$
0.370245 + 0.928934i $$0.379274\pi$$
$$240$$ 0 0
$$241$$ 1673.00 0.447168 0.223584 0.974685i $$-0.428224\pi$$
0.223584 + 0.974685i $$0.428224\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 3052.00i − 0.786211i
$$248$$ 0 0
$$249$$ 1089.00 0.277159
$$250$$ 0 0
$$251$$ −5355.00 −1.34663 −0.673316 0.739355i $$-0.735131\pi$$
−0.673316 + 0.739355i $$0.735131\pi$$
$$252$$ 0 0
$$253$$ − 11446.0i − 2.84428i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 5490.00i − 1.33252i −0.745721 0.666258i $$-0.767895\pi$$
0.745721 0.666258i $$-0.232105\pi$$
$$258$$ 0 0
$$259$$ −5148.00 −1.23506
$$260$$ 0 0
$$261$$ −1728.00 −0.409810
$$262$$ 0 0
$$263$$ 3150.00i 0.738545i 0.929321 + 0.369272i $$0.120393\pi$$
−0.929321 + 0.369272i $$0.879607\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 729.000i 0.167094i
$$268$$ 0 0
$$269$$ −176.000 −0.0398919 −0.0199459 0.999801i $$-0.506349\pi$$
−0.0199459 + 0.999801i $$0.506349\pi$$
$$270$$ 0 0
$$271$$ −2394.00 −0.536624 −0.268312 0.963332i $$-0.586466\pi$$
−0.268312 + 0.963332i $$0.586466\pi$$
$$272$$ 0 0
$$273$$ − 6552.00i − 1.45255i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 6256.00i − 1.35699i −0.734604 0.678496i $$-0.762632\pi$$
0.734604 0.678496i $$-0.237368\pi$$
$$278$$ 0 0
$$279$$ 540.000 0.115874
$$280$$ 0 0
$$281$$ −4802.00 −1.01944 −0.509721 0.860340i $$-0.670251\pi$$
−0.509721 + 0.860340i $$0.670251\pi$$
$$282$$ 0 0
$$283$$ 2123.00i 0.445934i 0.974826 + 0.222967i $$0.0715742\pi$$
−0.974826 + 0.222967i $$0.928426\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 3042.00i − 0.625657i
$$288$$ 0 0
$$289$$ 4888.00 0.994911
$$290$$ 0 0
$$291$$ 13518.0 2.72316
$$292$$ 0 0
$$293$$ 8834.00i 1.76139i 0.473682 + 0.880696i $$0.342925\pi$$
−0.473682 + 0.880696i $$0.657075\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 14337.0i − 2.80107i
$$298$$ 0 0
$$299$$ −5432.00 −1.05064
$$300$$ 0 0
$$301$$ 10088.0 1.93177
$$302$$ 0 0
$$303$$ − 2106.00i − 0.399296i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 1369.00i − 0.254505i −0.991870 0.127252i $$-0.959384\pi$$
0.991870 0.127252i $$-0.0406158\pi$$
$$308$$ 0 0
$$309$$ 10548.0 1.94192
$$310$$ 0 0
$$311$$ 10426.0 1.90098 0.950489 0.310758i $$-0.100583\pi$$
0.950489 + 0.310758i $$0.100583\pi$$
$$312$$ 0 0
$$313$$ 3574.00i 0.645413i 0.946499 + 0.322707i $$0.104593\pi$$
−0.946499 + 0.322707i $$0.895407\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 9036.00i 1.60099i 0.599343 + 0.800493i $$0.295429\pi$$
−0.599343 + 0.800493i $$0.704571\pi$$
$$318$$ 0 0
$$319$$ 1888.00 0.331372
$$320$$ 0 0
$$321$$ 10125.0 1.76051
$$322$$ 0 0
$$323$$ 545.000i 0.0938842i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 11106.0i 1.87817i
$$328$$ 0 0
$$329$$ 1768.00 0.296271
$$330$$ 0 0
$$331$$ −10233.0 −1.69926 −0.849632 0.527376i $$-0.823176\pi$$
−0.849632 + 0.527376i $$0.823176\pi$$
$$332$$ 0 0
$$333$$ 10692.0i 1.75951i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 4627.00i 0.747919i 0.927445 + 0.373960i $$0.122000\pi$$
−0.927445 + 0.373960i $$0.878000\pi$$
$$338$$ 0 0
$$339$$ 5103.00 0.817572
$$340$$ 0 0
$$341$$ −590.000 −0.0936959
$$342$$ 0 0
$$343$$ − 260.000i − 0.0409291i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 4901.00i − 0.758212i −0.925353 0.379106i $$-0.876232\pi$$
0.925353 0.379106i $$-0.123768\pi$$
$$348$$ 0 0
$$349$$ 4482.00 0.687438 0.343719 0.939072i $$-0.388313\pi$$
0.343719 + 0.939072i $$0.388313\pi$$
$$350$$ 0 0
$$351$$ −6804.00 −1.03467
$$352$$ 0 0
$$353$$ 1210.00i 0.182441i 0.995831 + 0.0912207i $$0.0290769\pi$$
−0.995831 + 0.0912207i $$0.970923\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 1170.00i 0.173454i
$$358$$ 0 0
$$359$$ −9882.00 −1.45279 −0.726396 0.687277i $$-0.758806\pi$$
−0.726396 + 0.687277i $$0.758806\pi$$
$$360$$ 0 0
$$361$$ 5022.00 0.732177
$$362$$ 0 0
$$363$$ 19350.0i 2.79783i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 11260.0i 1.60155i 0.598968 + 0.800773i $$0.295578\pi$$
−0.598968 + 0.800773i $$0.704422\pi$$
$$368$$ 0 0
$$369$$ −6318.00 −0.891333
$$370$$ 0 0
$$371$$ 468.000 0.0654915
$$372$$ 0 0
$$373$$ 3230.00i 0.448373i 0.974546 + 0.224186i $$0.0719724\pi$$
−0.974546 + 0.224186i $$0.928028\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 896.000i − 0.122404i
$$378$$ 0 0
$$379$$ 11575.0 1.56878 0.784390 0.620268i $$-0.212976\pi$$
0.784390 + 0.620268i $$0.212976\pi$$
$$380$$ 0 0
$$381$$ −21222.0 −2.85364
$$382$$ 0 0
$$383$$ − 18.0000i − 0.00240145i −0.999999 0.00120073i $$-0.999618\pi$$
0.999999 0.00120073i $$-0.000382203\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 20952.0i − 2.75207i
$$388$$ 0 0
$$389$$ −10710.0 −1.39593 −0.697967 0.716130i $$-0.745912\pi$$
−0.697967 + 0.716130i $$0.745912\pi$$
$$390$$ 0 0
$$391$$ 970.000 0.125460
$$392$$ 0 0
$$393$$ 15228.0i 1.95458i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 3788.00i − 0.478877i −0.970912 0.239439i $$-0.923037\pi$$
0.970912 0.239439i $$-0.0769634\pi$$
$$398$$ 0 0
$$399$$ 25506.0 3.20024
$$400$$ 0 0
$$401$$ −10539.0 −1.31245 −0.656225 0.754565i $$-0.727848\pi$$
−0.656225 + 0.754565i $$0.727848\pi$$
$$402$$ 0 0
$$403$$ 280.000i 0.0346099i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 11682.0i − 1.42274i
$$408$$ 0 0
$$409$$ −5581.00 −0.674725 −0.337363 0.941375i $$-0.609535\pi$$
−0.337363 + 0.941375i $$0.609535\pi$$
$$410$$ 0 0
$$411$$ −2061.00 −0.247352
$$412$$ 0 0
$$413$$ − 10192.0i − 1.21432i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 25029.0i 2.93927i
$$418$$ 0 0
$$419$$ 5193.00 0.605476 0.302738 0.953074i $$-0.402099\pi$$
0.302738 + 0.953074i $$0.402099\pi$$
$$420$$ 0 0
$$421$$ 4788.00 0.554282 0.277141 0.960829i $$-0.410613\pi$$
0.277141 + 0.960829i $$0.410613\pi$$
$$422$$ 0 0
$$423$$ − 3672.00i − 0.422077i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 18460.0i 2.09214i
$$428$$ 0 0
$$429$$ 14868.0 1.67327
$$430$$ 0 0
$$431$$ −8006.00 −0.894746 −0.447373 0.894348i $$-0.647640\pi$$
−0.447373 + 0.894348i $$0.647640\pi$$
$$432$$ 0 0
$$433$$ 2395.00i 0.265811i 0.991129 + 0.132906i $$0.0424308\pi$$
−0.991129 + 0.132906i $$0.957569\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 21146.0i − 2.31476i
$$438$$ 0 0
$$439$$ 1864.00 0.202651 0.101326 0.994853i $$-0.467692\pi$$
0.101326 + 0.994853i $$0.467692\pi$$
$$440$$ 0 0
$$441$$ 17982.0 1.94169
$$442$$ 0 0
$$443$$ 5463.00i 0.585903i 0.956127 + 0.292951i $$0.0946374\pi$$
−0.956127 + 0.292951i $$0.905363\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 13248.0i − 1.40181i
$$448$$ 0 0
$$449$$ −12969.0 −1.36313 −0.681565 0.731758i $$-0.738700\pi$$
−0.681565 + 0.731758i $$0.738700\pi$$
$$450$$ 0 0
$$451$$ 6903.00 0.720731
$$452$$ 0 0
$$453$$ − 11898.0i − 1.23403i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18313.0i 1.87450i 0.348659 + 0.937249i $$0.386637\pi$$
−0.348659 + 0.937249i $$0.613363\pi$$
$$458$$ 0 0
$$459$$ 1215.00 0.123554
$$460$$ 0 0
$$461$$ 12492.0 1.26206 0.631031 0.775758i $$-0.282632\pi$$
0.631031 + 0.775758i $$0.282632\pi$$
$$462$$ 0 0
$$463$$ 4428.00i 0.444464i 0.974994 + 0.222232i $$0.0713342\pi$$
−0.974994 + 0.222232i $$0.928666\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 1084.00i 0.107412i 0.998557 + 0.0537061i $$0.0171034\pi$$
−0.998557 + 0.0537061i $$0.982897\pi$$
$$468$$ 0 0
$$469$$ 6578.00 0.647641
$$470$$ 0 0
$$471$$ −2682.00 −0.262378
$$472$$ 0 0
$$473$$ 22892.0i 2.22532i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 972.000i − 0.0933015i
$$478$$ 0 0
$$479$$ −13082.0 −1.24787 −0.623937 0.781474i $$-0.714468\pi$$
−0.623937 + 0.781474i $$0.714468\pi$$
$$480$$ 0 0
$$481$$ −5544.00 −0.525540
$$482$$ 0 0
$$483$$ − 45396.0i − 4.27658i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 3014.00i 0.280446i 0.990120 + 0.140223i $$0.0447820\pi$$
−0.990120 + 0.140223i $$0.955218\pi$$
$$488$$ 0 0
$$489$$ 3069.00 0.283814
$$490$$ 0 0
$$491$$ 3564.00 0.327579 0.163789 0.986495i $$-0.447628\pi$$
0.163789 + 0.986495i $$0.447628\pi$$
$$492$$ 0 0
$$493$$ 160.000i 0.0146167i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 15912.0i − 1.43612i
$$498$$ 0 0
$$499$$ −15796.0 −1.41709 −0.708543 0.705667i $$-0.750647\pi$$
−0.708543 + 0.705667i $$0.750647\pi$$
$$500$$ 0 0
$$501$$ 6156.00 0.548962
$$502$$ 0 0
$$503$$ − 10908.0i − 0.966926i −0.875365 0.483463i $$-0.839379\pi$$
0.875365 0.483463i $$-0.160621\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 12717.0i 1.11397i
$$508$$ 0 0
$$509$$ −21946.0 −1.91108 −0.955540 0.294863i $$-0.904726\pi$$
−0.955540 + 0.294863i $$0.904726\pi$$
$$510$$ 0 0
$$511$$ 14274.0 1.23570
$$512$$ 0 0
$$513$$ − 26487.0i − 2.27959i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 4012.00i 0.341291i
$$518$$ 0 0
$$519$$ −21096.0 −1.78422
$$520$$ 0 0
$$521$$ −6395.00 −0.537754 −0.268877 0.963174i $$-0.586653\pi$$
−0.268877 + 0.963174i $$0.586653\pi$$
$$522$$ 0 0
$$523$$ − 5615.00i − 0.469459i −0.972061 0.234729i $$-0.924580\pi$$
0.972061 0.234729i $$-0.0754204\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 50.0000i − 0.00413289i
$$528$$ 0 0
$$529$$ −25469.0 −2.09329
$$530$$ 0 0
$$531$$ −21168.0 −1.72997
$$532$$ 0 0
$$533$$ − 3276.00i − 0.266228i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 9999.00i − 0.803517i
$$538$$ 0 0
$$539$$ −19647.0 −1.57005
$$540$$ 0 0
$$541$$ −4112.00 −0.326781 −0.163391 0.986561i $$-0.552243\pi$$
−0.163391 + 0.986561i $$0.552243\pi$$
$$542$$ 0 0
$$543$$ 18378.0i 1.45244i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 2167.00i 0.169386i 0.996407 + 0.0846931i $$0.0269910\pi$$
−0.996407 + 0.0846931i $$0.973009\pi$$
$$548$$ 0 0
$$549$$ 38340.0 2.98053
$$550$$ 0 0
$$551$$ 3488.00 0.269680
$$552$$ 0 0
$$553$$ − 10764.0i − 0.827725i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 19444.0i − 1.47912i −0.673092 0.739559i $$-0.735034\pi$$
0.673092 0.739559i $$-0.264966\pi$$
$$558$$ 0 0
$$559$$ 10864.0 0.822000
$$560$$ 0 0
$$561$$ −2655.00 −0.199811
$$562$$ 0 0
$$563$$ 20416.0i 1.52830i 0.645040 + 0.764149i $$0.276841\pi$$
−0.645040 + 0.764149i $$0.723159\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 18954.0i − 1.40387i
$$568$$ 0 0
$$569$$ 3127.00 0.230388 0.115194 0.993343i $$-0.463251\pi$$
0.115194 + 0.993343i $$0.463251\pi$$
$$570$$ 0 0
$$571$$ 22580.0 1.65489 0.827446 0.561545i $$-0.189793\pi$$
0.827446 + 0.561545i $$0.189793\pi$$
$$572$$ 0 0
$$573$$ − 47430.0i − 3.45797i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 829.000i − 0.0598123i −0.999553 0.0299062i $$-0.990479\pi$$
0.999553 0.0299062i $$-0.00952085\pi$$
$$578$$ 0 0
$$579$$ 5517.00 0.395991
$$580$$ 0 0
$$581$$ −3146.00 −0.224644
$$582$$ 0 0
$$583$$ 1062.00i 0.0754435i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 7119.00i − 0.500567i −0.968173 0.250283i $$-0.919476\pi$$
0.968173 0.250283i $$-0.0805238\pi$$
$$588$$ 0 0
$$589$$ −1090.00 −0.0762524
$$590$$ 0 0
$$591$$ −10566.0 −0.735410
$$592$$ 0 0
$$593$$ 8217.00i 0.569025i 0.958672 + 0.284512i $$0.0918317\pi$$
−0.958672 + 0.284512i $$0.908168\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 30852.0i 2.11506i
$$598$$ 0 0
$$599$$ −90.0000 −0.00613907 −0.00306953 0.999995i $$-0.500977\pi$$
−0.00306953 + 0.999995i $$0.500977\pi$$
$$600$$ 0 0
$$601$$ −17117.0 −1.16176 −0.580879 0.813990i $$-0.697291\pi$$
−0.580879 + 0.813990i $$0.697291\pi$$
$$602$$ 0 0
$$603$$ − 13662.0i − 0.922653i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 15120.0i 1.01104i 0.862815 + 0.505520i $$0.168699\pi$$
−0.862815 + 0.505520i $$0.831301\pi$$
$$608$$ 0 0
$$609$$ 7488.00 0.498241
$$610$$ 0 0
$$611$$ 1904.00 0.126068
$$612$$ 0 0
$$613$$ 6570.00i 0.432887i 0.976295 + 0.216444i $$0.0694457\pi$$
−0.976295 + 0.216444i $$0.930554\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18846.0i 1.22968i 0.788653 + 0.614839i $$0.210779\pi$$
−0.788653 + 0.614839i $$0.789221\pi$$
$$618$$ 0 0
$$619$$ 16316.0 1.05944 0.529722 0.848172i $$-0.322296\pi$$
0.529722 + 0.848172i $$0.322296\pi$$
$$620$$ 0 0
$$621$$ −47142.0 −3.04629
$$622$$ 0 0
$$623$$ − 2106.00i − 0.135434i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 57879.0i 3.68655i
$$628$$ 0 0
$$629$$ 990.000 0.0627566
$$630$$ 0 0
$$631$$ −20170.0 −1.27251 −0.636256 0.771478i $$-0.719518\pi$$
−0.636256 + 0.771478i $$0.719518\pi$$
$$632$$ 0 0
$$633$$ − 21051.0i − 1.32180i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 9324.00i 0.579953i
$$638$$ 0 0
$$639$$ −33048.0 −2.04594
$$640$$ 0 0
$$641$$ −12726.0 −0.784160 −0.392080 0.919931i $$-0.628244\pi$$
−0.392080 + 0.919931i $$0.628244\pi$$
$$642$$ 0 0
$$643$$ 2196.00i 0.134684i 0.997730 + 0.0673420i $$0.0214519\pi$$
−0.997730 + 0.0673420i $$0.978548\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 16884.0i 1.02593i 0.858409 + 0.512966i $$0.171453\pi$$
−0.858409 + 0.512966i $$0.828547\pi$$
$$648$$ 0 0
$$649$$ 23128.0 1.39885
$$650$$ 0 0
$$651$$ −2340.00 −0.140878
$$652$$ 0 0
$$653$$ 4018.00i 0.240791i 0.992726 + 0.120395i $$0.0384163\pi$$
−0.992726 + 0.120395i $$0.961584\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 29646.0i − 1.76043i
$$658$$ 0 0
$$659$$ 19071.0 1.12732 0.563658 0.826009i $$-0.309394\pi$$
0.563658 + 0.826009i $$0.309394\pi$$
$$660$$ 0 0
$$661$$ 17424.0 1.02529 0.512644 0.858601i $$-0.328666\pi$$
0.512644 + 0.858601i $$0.328666\pi$$
$$662$$ 0 0
$$663$$ 1260.00i 0.0738075i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 6208.00i − 0.360382i
$$668$$ 0 0
$$669$$ −35388.0 −2.04511
$$670$$ 0 0
$$671$$ −41890.0 −2.41005
$$672$$ 0 0
$$673$$ 5382.00i 0.308263i 0.988050 + 0.154131i $$0.0492579\pi$$
−0.988050 + 0.154131i $$0.950742\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 4496.00i − 0.255237i −0.991823 0.127618i $$-0.959267\pi$$
0.991823 0.127618i $$-0.0407333\pi$$
$$678$$ 0 0
$$679$$ −39052.0 −2.20718
$$680$$ 0 0
$$681$$ 54756.0 3.08114
$$682$$ 0 0
$$683$$ 3249.00i 0.182020i 0.995850 + 0.0910099i $$0.0290095\pi$$
−0.995850 + 0.0910099i $$0.970991\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 44964.0i − 2.49706i
$$688$$ 0 0
$$689$$ 504.000 0.0278677
$$690$$ 0 0
$$691$$ 13399.0 0.737658 0.368829 0.929497i $$-0.379759\pi$$
0.368829 + 0.929497i $$0.379759\pi$$
$$692$$ 0 0
$$693$$ 82836.0i 4.54066i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 585.000i 0.0317912i
$$698$$ 0 0
$$699$$ 28998.0 1.56911
$$700$$ 0 0
$$701$$ −18148.0 −0.977804 −0.488902 0.872339i $$-0.662602\pi$$
−0.488902 + 0.872339i $$0.662602\pi$$
$$702$$ 0 0
$$703$$ − 21582.0i − 1.15787i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 6084.00i 0.323638i
$$708$$ 0 0
$$709$$ −4868.00 −0.257858 −0.128929 0.991654i $$-0.541154\pi$$
−0.128929 + 0.991654i $$0.541154\pi$$
$$710$$ 0 0
$$711$$ −22356.0 −1.17921
$$712$$ 0 0
$$713$$ 1940.00i 0.101898i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 24624.0i 1.28257i
$$718$$ 0 0
$$719$$ −17366.0 −0.900755 −0.450377 0.892838i $$-0.648711\pi$$
−0.450377 + 0.892838i $$0.648711\pi$$
$$720$$ 0 0
$$721$$ −30472.0 −1.57398
$$722$$ 0 0
$$723$$ 15057.0i 0.774517i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 21824.0i − 1.11335i −0.830729 0.556676i $$-0.812076\pi$$
0.830729 0.556676i $$-0.187924\pi$$
$$728$$ 0 0
$$729$$ 19683.0 1.00000
$$730$$ 0 0
$$731$$ −1940.00 −0.0981580
$$732$$ 0 0
$$733$$ 31428.0i 1.58366i 0.610744 + 0.791828i $$0.290870\pi$$
−0.610744 + 0.791828i $$0.709130\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 14927.0i 0.746056i
$$738$$ 0 0
$$739$$ −14292.0 −0.711420 −0.355710 0.934596i $$-0.615761\pi$$
−0.355710 + 0.934596i $$0.615761\pi$$
$$740$$ 0 0
$$741$$ 27468.0 1.36176
$$742$$ 0 0
$$743$$ 13950.0i 0.688797i 0.938824 + 0.344398i $$0.111917\pi$$
−0.938824 + 0.344398i $$0.888083\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 6534.00i 0.320036i
$$748$$ 0 0
$$749$$ −29250.0 −1.42693
$$750$$ 0 0
$$751$$ 38736.0 1.88215 0.941076 0.338194i $$-0.109816\pi$$
0.941076 + 0.338194i $$0.109816\pi$$
$$752$$ 0 0
$$753$$ − 48195.0i − 2.33243i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 3664.00i − 0.175919i −0.996124 0.0879593i $$-0.971965\pi$$
0.996124 0.0879593i $$-0.0280345\pi$$
$$758$$ 0 0
$$759$$ 103014. 4.92644
$$760$$ 0 0
$$761$$ 19557.0 0.931591 0.465795 0.884892i $$-0.345768\pi$$
0.465795 + 0.884892i $$0.345768\pi$$
$$762$$ 0 0
$$763$$ − 32084.0i − 1.52231i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 10976.0i − 0.516715i
$$768$$ 0 0
$$769$$ 13283.0 0.622883 0.311442 0.950265i $$-0.399188\pi$$
0.311442 + 0.950265i $$0.399188\pi$$
$$770$$ 0 0
$$771$$ 49410.0 2.30799
$$772$$ 0 0
$$773$$ − 24840.0i − 1.15580i −0.816108 0.577900i $$-0.803873\pi$$
0.816108 0.577900i $$-0.196127\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 46332.0i − 2.13919i
$$778$$ 0 0
$$779$$ 12753.0 0.586552
$$780$$ 0 0
$$781$$ 36108.0 1.65435
$$782$$ 0 0
$$783$$ − 7776.00i − 0.354906i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 18044.0i − 0.817280i −0.912696 0.408640i $$-0.866003\pi$$
0.912696 0.408640i $$-0.133997\pi$$
$$788$$ 0 0
$$789$$ −28350.0 −1.27920
$$790$$ 0 0
$$791$$ −14742.0 −0.662661
$$792$$ 0 0
$$793$$ 19880.0i 0.890239i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 6174.00i − 0.274397i −0.990544 0.137198i $$-0.956190\pi$$
0.990544 0.137198i $$-0.0438098\pi$$
$$798$$ 0 0
$$799$$ −340.000 −0.0150542
$$800$$ 0 0
$$801$$ −4374.00 −0.192943
$$802$$ 0 0
$$803$$ 32391.0i 1.42348i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 1584.00i − 0.0690947i
$$808$$ 0 0
$$809$$ 1998.00 0.0868306 0.0434153 0.999057i $$-0.486176\pi$$
0.0434153 + 0.999057i $$0.486176\pi$$
$$810$$ 0 0
$$811$$ 7156.00 0.309841 0.154921 0.987927i $$-0.450488\pi$$
0.154921 + 0.987927i $$0.450488\pi$$
$$812$$ 0 0
$$813$$ − 21546.0i − 0.929460i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 42292.0i 1.81103i
$$818$$ 0 0
$$819$$ 39312.0 1.67726
$$820$$ 0 0
$$821$$ 27922.0 1.18695 0.593474 0.804853i $$-0.297756\pi$$
0.593474 + 0.804853i $$0.297756\pi$$
$$822$$ 0 0
$$823$$ 22636.0i 0.958738i 0.877613 + 0.479369i $$0.159134\pi$$
−0.877613 + 0.479369i $$0.840866\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 26559.0i − 1.11674i −0.829591 0.558372i $$-0.811426\pi$$
0.829591 0.558372i $$-0.188574\pi$$
$$828$$ 0 0
$$829$$ −12580.0 −0.527046 −0.263523 0.964653i $$-0.584885\pi$$
−0.263523 + 0.964653i $$0.584885\pi$$
$$830$$ 0 0
$$831$$ 56304.0 2.35038
$$832$$ 0 0
$$833$$ − 1665.00i − 0.0692543i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 2430.00i 0.100350i
$$838$$ 0 0
$$839$$ −11344.0 −0.466792 −0.233396 0.972382i $$-0.574984\pi$$
−0.233396 + 0.972382i $$0.574984\pi$$
$$840$$ 0 0
$$841$$ −23365.0 −0.958014
$$842$$ 0 0
$$843$$ − 43218.0i − 1.76573i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 55900.0i − 2.26771i
$$848$$ 0 0
$$849$$ −19107.0 −0.772380
$$850$$ 0 0
$$851$$ −38412.0 −1.54729
$$852$$ 0 0
$$853$$ − 14786.0i − 0.593509i −0.954954 0.296754i $$-0.904096\pi$$
0.954954 0.296754i $$-0.0959043\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 29259.0i − 1.16624i −0.812386 0.583120i $$-0.801832\pi$$
0.812386 0.583120i $$-0.198168\pi$$
$$858$$ 0 0
$$859$$ −13651.0 −0.542219 −0.271109 0.962549i $$-0.587391\pi$$
−0.271109 + 0.962549i $$0.587391\pi$$
$$860$$ 0 0
$$861$$ 27378.0 1.08367
$$862$$ 0 0
$$863$$ 29016.0i 1.14451i 0.820074 + 0.572257i $$0.193932\pi$$
−0.820074 + 0.572257i $$0.806068\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 43992.0i 1.72324i
$$868$$ 0 0
$$869$$ 24426.0 0.953504
$$870$$ 0 0
$$871$$ 7084.00 0.275582
$$872$$ 0 0
$$873$$ 81108.0i 3.14443i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 21412.0i 0.824438i 0.911085 + 0.412219i $$0.135246\pi$$
−0.911085 + 0.412219i $$0.864754\pi$$
$$878$$ 0 0
$$879$$ −79506.0 −3.05082
$$880$$ 0 0
$$881$$ −1170.00 −0.0447427 −0.0223713 0.999750i $$-0.507122\pi$$
−0.0223713 + 0.999750i $$0.507122\pi$$
$$882$$ 0 0
$$883$$ − 12655.0i − 0.482304i −0.970487 0.241152i $$-0.922475\pi$$
0.970487 0.241152i $$-0.0775253\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 32764.0i − 1.24026i −0.784500 0.620128i $$-0.787081\pi$$
0.784500 0.620128i $$-0.212919\pi$$
$$888$$ 0 0
$$889$$ 61308.0 2.31294
$$890$$ 0 0
$$891$$ 43011.0 1.61720
$$892$$ 0 0
$$893$$ 7412.00i 0.277753i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 48888.0i − 1.81976i
$$898$$ 0 0
$$899$$ −320.000 −0.0118716
$$900$$ 0 0
$$901$$ −90.0000 −0.00332779
$$902$$ 0 0
$$903$$ 90792.0i 3.34592i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 29844.0i 1.09256i 0.837602 + 0.546281i $$0.183957\pi$$
−0.837602 + 0.546281i $$0.816043\pi$$
$$908$$ 0 0
$$909$$ 12636.0 0.461067
$$910$$ 0 0
$$911$$ −15628.0 −0.568363 −0.284182 0.958770i $$-0.591722\pi$$
−0.284182 + 0.958770i $$0.591722\pi$$
$$912$$ 0 0
$$913$$ − 7139.00i − 0.258780i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 43992.0i − 1.58424i
$$918$$ 0 0
$$919$$ 42974.0 1.54253 0.771263 0.636517i $$-0.219625\pi$$
0.771263 + 0.636517i $$0.219625\pi$$
$$920$$ 0 0
$$921$$ 12321.0 0.440815
$$922$$ 0 0
$$923$$ − 17136.0i − 0.611092i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 63288.0i 2.24234i
$$928$$ 0 0
$$929$$ −13342.0 −0.471191 −0.235596 0.971851i $$-0.575704\pi$$
−0.235596 + 0.971851i $$0.575704\pi$$
$$930$$ 0 0
$$931$$ −36297.0 −1.27775
$$932$$ 0 0
$$933$$ 93834.0i 3.29259i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 3005.00i − 0.104770i −0.998627 0.0523848i $$-0.983318\pi$$
0.998627 0.0523848i $$-0.0166822\pi$$
$$938$$ 0 0
$$939$$ −32166.0 −1.11789
$$940$$ 0 0
$$941$$ 16204.0 0.561355 0.280678 0.959802i $$-0.409441\pi$$
0.280678 + 0.959802i $$0.409441\pi$$
$$942$$ 0 0
$$943$$ − 22698.0i − 0.783827i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 30200.0i 1.03629i 0.855292 + 0.518146i $$0.173378\pi$$
−0.855292 + 0.518146i $$0.826622\pi$$
$$948$$ 0 0
$$949$$ 15372.0 0.525813
$$950$$ 0 0
$$951$$ −81324.0 −2.77299
$$952$$ 0 0
$$953$$ 29583.0i 1.00555i 0.864418 + 0.502774i $$0.167687\pi$$
−0.864418 + 0.502774i $$0.832313\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 16992.0i 0.573953i
$$958$$ 0 0
$$959$$ 5954.00 0.200485
$$960$$ 0 0
$$961$$ −29691.0 −0.996643
$$962$$ 0 0
$$963$$ 60750.0i 2.03286i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 6480.00i 0.215494i 0.994178 + 0.107747i $$0.0343637\pi$$
−0.994178 + 0.107747i $$0.965636\pi$$
$$968$$ 0 0
$$969$$ −4905.00 −0.162612
$$970$$ 0 0
$$971$$ 40171.0 1.32765 0.663825 0.747888i $$-0.268932\pi$$
0.663825 + 0.747888i $$0.268932\pi$$
$$972$$ 0 0
$$973$$ − 72306.0i − 2.38235i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 50801.0i 1.66353i 0.555129 + 0.831765i $$0.312669\pi$$
−0.555129 + 0.831765i $$0.687331\pi$$
$$978$$ 0 0
$$979$$ 4779.00 0.156014
$$980$$ 0 0
$$981$$ −66636.0 −2.16873
$$982$$ 0 0
$$983$$ − 58338.0i − 1.89287i −0.322891 0.946436i $$-0.604655\pi$$
0.322891 0.946436i $$-0.395345\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 15912.0i 0.513156i
$$988$$ 0 0
$$989$$ 75272.0 2.42013
$$990$$ 0 0
$$991$$ 51202.0 1.64126 0.820628 0.571462i $$-0.193624\pi$$
0.820628 + 0.571462i $$0.193624\pi$$
$$992$$ 0 0
$$993$$ − 92097.0i − 2.94321i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1764.00i 0.0560345i 0.999607 + 0.0280173i $$0.00891934\pi$$
−0.999607 + 0.0280173i $$0.991081\pi$$
$$998$$ 0 0
$$999$$ −48114.0 −1.52378
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 400.4.c.b.49.2 2
4.3 odd 2 200.4.c.b.49.1 2
5.2 odd 4 400.4.a.t.1.1 1
5.3 odd 4 400.4.a.a.1.1 1
5.4 even 2 inner 400.4.c.b.49.1 2
12.11 even 2 1800.4.f.w.649.2 2
20.3 even 4 200.4.a.j.1.1 yes 1
20.7 even 4 200.4.a.b.1.1 1
20.19 odd 2 200.4.c.b.49.2 2
40.3 even 4 1600.4.a.c.1.1 1
40.13 odd 4 1600.4.a.by.1.1 1
40.27 even 4 1600.4.a.bz.1.1 1
40.37 odd 4 1600.4.a.b.1.1 1
60.23 odd 4 1800.4.a.bh.1.1 1
60.47 odd 4 1800.4.a.c.1.1 1
60.59 even 2 1800.4.f.w.649.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.b.1.1 1 20.7 even 4
200.4.a.j.1.1 yes 1 20.3 even 4
200.4.c.b.49.1 2 4.3 odd 2
200.4.c.b.49.2 2 20.19 odd 2
400.4.a.a.1.1 1 5.3 odd 4
400.4.a.t.1.1 1 5.2 odd 4
400.4.c.b.49.1 2 5.4 even 2 inner
400.4.c.b.49.2 2 1.1 even 1 trivial
1600.4.a.b.1.1 1 40.37 odd 4
1600.4.a.c.1.1 1 40.3 even 4
1600.4.a.by.1.1 1 40.13 odd 4
1600.4.a.bz.1.1 1 40.27 even 4
1800.4.a.c.1.1 1 60.47 odd 4
1800.4.a.bh.1.1 1 60.23 odd 4
1800.4.f.w.649.1 2 60.59 even 2
1800.4.f.w.649.2 2 12.11 even 2